x
/
y
is the quotient of x
x
\/
y
evaluates to the roundedx % y
evaluates to the modular(x,y)
(x,y,{v})
(x,y)
(x,y)
(x,y)
(x,n)
(x,n)
(x)
(x,{&v})
(x,{&v})
(x, {n})
(x, {n})
({x = []})
({x})
({x = []})
(a,b)
(t,{v = 'x})
(t,{v = 'x})
(a,b,c,{D = 0.})
({x = []})
({x}*)
(x)
({x}*)
({x}*)
(x, {n})
(x, {n})
(x, {n})
(x)
(x,y)
(x,{n = -1})
(x,y)
(x,y)
(x,{n})
(x,n)
(x,y)
(x)
(x,{v})
(x)
(x,n)
(x)
(z)
(x)
(x)
(x)
(x)
(x)
(x)
(x,{v})
(x)
(x)
(x)
(x)
(x)
(n,k)
(x,p)
(x)
(x,{n})
(x)
(x,{&e})
(x,v)
(x)
(x)
(x)
(x,{&e})
(x,p)
(
name,{v})
({x})
({x})
(
name,{v})
(x)
(x)
(x)
(x,y)
(x)
(x)
(x)
(x)
(x)
(x)
(n, {v = 'x})
(x)
(x)
(
nu,x)
(
nu,x)
(
nu,x)
(
nu,x)
(n,x)
(
nu,x)
(
nu,x)
(x)
(x)
(x)
(x)
(x)
(x,{n})
(x)
(z,{
flag = 0})
(x)
(x)
(s)
(x)
(G,t,{m = 0})
(A,n,{m = 0})
(A,{m = 0})
(a,b,x)
(s,x,{g})
(s,x)
(y)
(x)
(x)
(m,x,{
flag = 0})
(x)
(x)
(x)
(x)
(x)
(x)
(x,n,{&z})
(x)
(x)
(q,z)
(q,k)
(x,{
flag = 0})
(s)
(s)
({x = []})
(x, {B})
(x, {B})
(x,y)
(x)
(x,y)
(
cyc,
chi)
(
cyc, a,b)
(G,
chi, x, {z}))
(
cyc,
chi)
(
cyc, a,b)
(
cyc,
chi)
(x,{y})
(x)
(x, {n = -1})
(n,{
flag = 0})
(n,{
flag = 0})
(x,y)
(p = a,b,
expr,{c})
(x,y)
(x)
(x)
(f,{e})
(x,p)
(x,{p},{a})
(x)
(x,{
flag = 0})
(x,p,{
flag = 0})
(q,{v})
(p,n,{v = 'x})
(x,g,{o})
(x,{o})
(x, {&o})
(x)
(x,{y})
(x,y)
(x,y,{p})
(x)
(x,s,{&N})
(x,{k},{&n})
(x)
(x,{
flag = 0})
(x,{&n})
(x,{
flag})
(x,{&n})
(x,{&n})
(x)
(x,{&N})
(x,y)
(x,{y})
(x,b,{&z})
(x)
(x)
(n)
(x)
(x)
(k,{a = k},{n = k}))
(x,{p},{a})
(x)
(n)
(x)
(n)
(D,{
flag = 0})
(x,y)
(x)
(x,y,L)
(x,n,{L})
(x,n)
(x,p)
(Q,p)
(x)
(D)
(D)
(D,{v = 'x})
(D,f)
(x)
(D)
(n)
({x = []})
(x,{k = 1})
(x)
(x,n)
(n,k,{
flag = 1})
(h,k)
(G,
chi, N)
(G,
chi)
(G,
chi, {
flag = 0})
(
bid,m)
(
bid,
chi)
(
bid,m)
(P, N, X, {B = N})
(x,g,{o})
(x,{o})
(n)
(n,{
flag = 0})
(e, {r = 0})
(E,
z1,
z2)
(E,n)
(E,n)
(E,{p})
(E,
z1,
z2)
(E,{p})
(E,v)
(x,v)
(x,v)
(
name)
(E,n,{v = 'x})
(w,k,{
flag = 0})
(w)
(P)
(j)
(E)
(E)
(E,{p},{
flag})
(E)
(E,P,{Q})
(E,x)
(E)
(x,{D = 1})
(E,{&v})
(E,P,n,{&Q}))
(f, g)
(E,z)
(E,{p})
(x)
(E,p)
(E,P,G,{o})
(E,s,{A = 1})
(E,{&v})
(E, {
flag = 0})
(e)
(N,{x},{y})
(E,z,n)
(E,z)
(E,P)
(E,z,{o})
(E,x)
(E, p, n, {s = 0}, {r = 0}, {D = 1})
(E,p,n)
(E, p,n, P,{Q})
(E,p,n,v)
(E,p,n,P)
(E,p,n)
(w, {
flag = 0})
(E,P)
(E,z,n)
(E,{p})
(N)
(L,{z = 'x},{
flag = 0})
(E,
z1,
z2)
(E, P, Q, m)
(E)
(E,{P})
(E, P, Q, m)
(w,{z = 'x},{
flag = 0})
(E,n,{v = 'x})
(w,{z = 'x})
(E,z)
(
PQ,{p})
(X)
(Q,p,n)
L
-functionsL
and theta functionsL
-functionsL
-functionsL
functionsL
-functions of algebraic varieties(L,s,{D = 0})
(L,n)
(
nf,
gal,M,n)
(L,{t})
(
obj)
(
L1,
L2)
(M)
(F)
(L,t)
(L,s,{D = 0})
(L)
(
L1,
L2)
(L, {m = -1})
(Q)
(
data)
(
data,t,{m = 0})
(M,Q,{H})
(M, {
flag = 0})
(M)
(M,s,{p})
(M, c)
(M, v, {H})
(M)
(M)
(M)
(M,p,{H})
(M,s)
(M)
(
Mp,
PHI,
path)
(
mu, {s = 0}, {r = 0})
(M, p, n, {
flag})
(
Mp,
PHI, {D = 1})
(
mu, {i = 0})
(M)
(M,p)
(M,{H})
(
Mp,
phi)
(
bnf,{
flag = 0})
(
bnf)
(
nf,m)
(
bnf,x)
(
bnf,x,{
flag = 1})
(
bnf,x,{
flag = 1})
(
bnf,
sfu,x)
(
bnf,x)
(
bnf, l)
(
nf, A, l)
(
nf,
pr)
(
bnf)
(
bnf)
(
bnf,S)
(
bnr, {H}, {
flag = 0})
(
bnr,g,{v})
(A,{B},{C})
(
bnf,
list)
(A,{B},{C},{
flag = 0})
(
bnr,
chi)
(A,{B},{C},{
flag = 0})
(
bnr,
mat, H)
(
bnr,
aut)
(
bnf,f,{
flag = 0})
(A,{B},{C})
(
bnr,
gal, H)
(
bnr,x,{
flag = 1})
(
bnr,
chi,{
flag = 0})
(
nf,b)
(x,t)
(
gal,{
flag})
(
gal,
perm,{
flag},{v = y})
(a,{b},{s})
(
gal)
(
gal,{
flag = 0})
(
gal,
subgrp)
(
gal,
perm)
(G,{
flag = 0},{v})
(G)
(
nf,x,y)
(
nf,x,{y})
(
nf,x,{
flag})
(
nf,x,{y})
(
nf,x,y)
(
nf,x,y,{
flag = 0})
(
nf,x)
(
nf,f,{e},{
flag = 0})
(
nf,
gal,
pr)
(
nf,u,{v})
(
nf,A,B)
(
nf,x)
(
nf,
bound,{
flag = 4})
(
nf,
list,
arch)
(
nf,x,y,{
flag = 0})
(
nf,x)
(
nf,x)
(
nf,x,k,{
flag = 0})
(
nf,p,{f = 0})
(
nf,
pr,k)
(
nf,
gal,
pr)
(
nf,I,{v = 0})
(
nf,x,{a})
(
nf,x,
pr)
(
nf,x)
(
nf,x)
(z)
(x,p)
(
nf,x)
(T)
(
nf,x)
(
nf)
(
nf,P,Q,{
flag = 0})
(
nf,x)
(T)
(
nf,x,y)
(
nf,x,y)
(
nf,x,y)
(
nf,x,y,
pr)
(
nf,x,y)
(
nf,x,y)
(
nf,x,y)
(
nf,x,y,
pr)
(
nf,x)
(
nf,x,k)
(
nf,x,k,
pr)
(
nf,a,
id)
(
nf,x,
pr)
(
nf,x)
(
nf,x,
pr,{&y})
(
nf,T)
(
nf,f,{e})
(
nf,Q,
pr)
(
nf,
aut,x)
(
nf,{
flag = 0},{d})
(
nf,
Lpr,
Ld,
pl,{v = 'x})
(
nf,x,{
flag = 0})
(
nf,x,
detx)
(
pol,{
flag = 0})
(
nf,x)
(x,y)
(x,y)
(
nf,
pr,a,n)
(
nf,x,
pr)
(
nf,x,
pr)
(
nf,
pr)
(
nf,x,
pr)
(
nf)
(
nf)
(
nf,x,{
flag = 0})
(
nf,a,b,P)
(
nf,{d})
(
pol,{d = 0})
(P,Q,{
flag = 0})
(T)
(T,{
flag = 0})
(T,{
flag = 0})
(T,{
flag = 0})
(x)
(x)
(
rnf,x)
(
bnf,M)
(
rnf,x)
(
bnf,
pol)
(
nf,M)
(
nf,
pol)
(
rnf,x)
(
rnf,x,{
flag = 0})
(
rnf,x)
(
rnf,x)
(
rnf,x)
(
rnf,x,{
flag = 0})
(
nf,
pol,{
flag = 0})
(
bnf,x)
(
rnf,x)
(
rnf,x)
(
rnf,x)
(
rnf,x)
(
rnf,x,y)
(
rnf,x)
(
rnf,x)
(
rnf,
pr)
(
rnf,x,{
flag = 0})
(
rnf,x)
(
rnf,x,{
flag = 0})
(
nf,
pol,{
flag = 0})
(
nf,T)
(
bnf,x)
(
rnf)
(T,a,{
flag = 0})
(
pol,
polrel,{
flag = 2})
(
nf,
pol,
order)
(
bnr,
pol)
(
nf,
pol)
(
nf,
pol,{
flag = 0})
(
nf,
pol,{
flag = 0})
(
nf,
pol)
(
nf,x)
(
al)
(
al,x,y)
(
al,x)
(
al)
(
al)
(
al)
(
al,x)
(
al)
(
al)
(
al,b,{v = 'x})
(
al)
(
al)
(
al)
(
al)
(
al,x,y)
(
al,x,y)
(
gal, {p = 0})
(
al,
pl)
(
al)
(
al)
(B, C, {v}, {
flag = 1})
(
al,x)
(
al)
(
mt,p = 0)
(
al)
(
al,x,y,{&z})
(
al)
(
al,m)
(
al,x)
(
al,x,y)
(
al)
(
al,x)
(
al,x)
(
al,T,b)
(
al,x,n)
(
al)
(
al,I,{
flag = 0})
(
al)
(
al)
(
al,b)
(
al)
(
al,{
flag = 0})
(
al)
(
al)
(
al,x)
(
al,x)
(
al,x,y)
(
al,B)
(
mt, {p = 0})
(
al,x)
(
al)
(A,B,{v})
(x,{v})
(x,v,d,{n = 1})
(x)
(
pol,p,r)
(x,{v})
(
pol,a)
(p, N, {
flag = 0})
(n,{
flag = 1},{a = 'x})
(x,n,{v})
(n,{a = 'x})
(f)
(x,{v})
(
pol,{v})
(f)
(f)
(A, B, p, e)
(n,{a = 'x})
(X,{Y},{t = 'x},{&e})
(f)
(f)
(
pol)
(x,{v})
(n,{a = 'x})
(
pol)
(x,y,{v},{
flag = 0})
(A,B,{v})
(x)
(
pol,p,{
flag = 0})
(x,p,r)
(n,d,{v = 'x})
(x,y)
(x,n)
(n,{v = 'x})
(n,m)
(x,y)
(x)
(s)
(x,y,z)
(x,y,z)
(x,v,w)
(f,{v})
(P,{
flag = 0})
(z,k,{
flag = 0})
(A,{v = 'x},{
flag = 5})
(x,{y})
(v,q,b,
expr)
(v,{
flag = 0})
(M,{
flag = 0})
(x)
(v)
(x,{
flag = 0})
(B)
(x)
(x,{
flag = 0})
(M,{
flag},{v = 'x})
(x)
(n)
(M,{
flag = 0})
(x,d)
(x,d)
(Q,v)
(n)
(x,{
flag = 0})
(x)
(x)
(x,y)
(x,y)
(x)
(x,{
flag = 0})
(x,{
flag = 0})
(x,d)
(x,y)
(n,{q})
(M,{
flag = 0})
(x)
(A,{p = 0})
(x)
(X,{
flag = 0})
(M,B)
(M,D,B,{
flag = 0})
(x)
(x)
(A,{v = 'x})
(x)
(
qfa,{
flag})
(x,y,{q})
({q},x,{y})
(q)
(A)
(x,{
flag = 0})
(G,{
flag = 0})
(x,{b},{m},{
flag = 0})
(x,{q})
(G,V)
(G,
sol, {
flag = 0})
(G)
(q,B,{
flag = 0})
(x)
(G)
(s,p,r)
(f,X,{Y})
(x,y)
(x)
(x,y)
(S,x,{
flag = 0})
(x,y)
(x)
(x,y,{z})
(v)
(X = a,
expr)
(t = a,b,f,{m = 0})
({n})
(a,b,{m = 0})
(X = a,b,
expr,{
flag = 0})
(X = a,b,
expr,{x = 1})
(X = a,b,
expr)
(X = a,
expr,{
flag = 0})
(X = a,b,
expr)
(X = a,b,
step,
expr,{
flag = 0})
(X = a,b,
expr,{x = 0})
(X = a,
expr,{
flag = 0})
(n,X,
expr)
(n,d,
expr)
(X = a,
expr)
(X = a,
expr,{
flag = 0})
(w,
x2,
y2)
(w)
(w,c)
(
sourcew,
destw,
dx,
dy,{
flag = 0})
(w)
(
list, {
flag = 0})
(
listx,
listy,{
flag = 0})
({
flag = 0})
(w,{x},{y},{
flag = 0})
(w)
(w,X,Y,{
flag = 0})
(w,
type)
(w,x,y)
(w,X,Y)
(w,
size)
(w,
type)
(w,
dx,
dy)
(w,X = a,b,
expr,{
flag = 0},{n = 0})
(w,
dx,
dy)
(w,
dx,
dy)
(w,
dx,
dy)
(w,
x1,
x2,
y1,
y2)
(
list, {
flag = 0})
(
listx,
listy,{
flag = 0})
({n = 1})
()
({n = 1})
()
({n = 1})
(A{,n})
(X = a,b,
seq)
(n = a,{b},
seq)
(n,X,
seq)
(E,a,b,
seq,{
flag = 0})
(X = k,
seq,{a = k},{n = k})
(p = a,{b},
seq)
(X = a,b,s,
seq)
(X = v,
seq,{
flag = 0})
({n = 1})
({x = 0})
(a,
seq)
(a,
seq)
(
fmt,{x}*)
(
sym,
str)
(
newsym,
sym)
({s = 0})
(f, A)
(f, A)
(E)
(
str)
(
str)
(f, A)
()
(s)
()
()
()
()
()
(
list of variables)
(x,...,z)
()
(
sym)
({n})
(L,x,n)
(L)
(
list,{n})
(
list,x,{n})
(L,{
flag = 0})
(p)
(p)
(M,x)
(M,x)
(M,x,{&z})
(M,x,y)
(
fmt,{x}*)
(f, A, {
flag = 0})
()
(n)
(
str)
(x)
()
()
(
key)
(
filename,{x})
libPARI - Functions and Operations Available in PARI and GP
The functions and operators available in PARI and in the GP/PARI calculator are numerous and ever-expanding. Here is a description of the ones available in version 2.9.1. It should be noted that many of these functions accept quite different types as arguments, but others are more restricted. The list of acceptable types will be given for each function or class of functions. Except when stated otherwise, it is understood that a function or operation which should make natural sense is legal.
On the other hand, many routines list explicit preconditions for some of their
argument, e.g. p
is a prime number, or q
is a positive definite quadratic
form. For reasons of efficiency, all trust the user input and only perform
minimal sanity checks. When a precondition is not satisfied, any of the
following may occur: a regular exception is raised, the PARI stack overflows, a
SIGSEGV
or SIGBUS
signal is generated, or we enter an infinite
loop. The function can also quietly return a mathematically meaningless
result: junk in, junk out.
In this chapter, we will describe the functions according to a rough classification. The general entry looks something like:
foo(x,{
flag = 0})
: short description.
The library syntax is GEN foo(GEN x, long fl = 0)
.
This means that the GP function foo
has one mandatory argument x
, and
an optional one, flag, whose default value is 0. (The {}
should not be
typed, it is just a convenient notation we will use throughout to denote
optional arguments.) That is, you can type foo(x,2)
, or foo(x)
,
which is then understood to mean foo(x,0)
. As well, a comma or closing
parenthesis, where an optional argument should have been, signals to GP it
should use the default. Thus, the syntax foo(x,)
is also accepted as a
synonym for our last expression. When a function has more than one optional
argument, the argument list is filled with user supplied values, in order.
When none are left, the defaults are used instead. Thus, assuming that
foo
's prototype had been
foo({x = 1},{y = 2},{z = 3}),
typing in foo(6,4)
would give
you foo(6,4,3)
. In the rare case when you want to set some far away
argument, and leave the defaults in between as they stand, you can use the
``empty arg'' trick alluded to above: foo(6,,1)
would yield
foo(6,2,1)
. By the way, foo()
by itself yields
foo(1,2,3)
as was to be expected.
In this rather special case of a function having no mandatory argument, you
can even omit the ()
: a standalone foo
would be enough (though we
do not recommend it for your scripts, for the sake of clarity). In defining
GP syntax, we strove to put optional arguments at the end of the argument
list (of course, since they would not make sense otherwise), and in order of
decreasing usefulness so that, most of the time, you will be able to ignore
them.
Finally, an optional argument (between braces) followed by a star, like
{
x}*
, means that any number of such arguments (possibly none) can
be given. This is in particular used by the various print
routines.
@3Flags. A flag is an argument which, rather than conveying actual information to the routine, instructs it to change its default behavior, e.g. return more or less information. All such flags are optional, and will be called flag in the function descriptions to follow. There are two different kind of flags
@3* generic: all valid values for the flag are individually
described (``If flag is equal to 1
, then...'').
@3* binary: use customary binary notation as a
compact way to represent many toggles with just one integer. Let
(p_0,...,p_n)
be a list of switches (i.e. of properties which take either
the value 0
or 1
), the number 2^3 + 2^5 = 40
means that p_3
and p_5
are set (that is, set to 1
), and none of the others are (that is, they
are set to 0
). This is announced as ``The binary digits of flag mean 1:
p_0
, 2: p_1
, 4: p_2
'', and so on, using the available consecutive
powers of 2
.
@3Mnemonics for flags. Numeric flags as mentioned above are
obscure, error-prone, and quite rigid: should the authors
want to adopt a new flag numbering scheme (for instance when noticing
flags with the same meaning but different numeric values across a set of
routines), it would break backward compatibility. The only advantage of
explicit numeric values is that they are fast to type, so their use is only
advised when using the calculator gp
.
As an alternative, one can replace a numeric flag by a character string containing symbolic identifiers. For a generic flag, the mnemonic corresponding to the numeric identifier is given after it as in
fun(x, {flag = 0} ):
If flag is equal to 1 = AGM, use an agm formula ...
@3
which means that one can use indifferently fun(x, 1)
or
fun(x, "AGM")
.
For a binary flag, mnemonics corresponding to the various toggles are given
after each of them. They can be negated by prepending no_
to the
mnemonic, or by removing such a prefix. These toggles are grouped together
using any punctuation character (such as ','
or ';'
). For instance (taken
from description of ploth(X = a,b,
expr,{
flag = 0},{n = 0})
)
Binary digits of flags mean: C<1 = Parametric>, C<2 = Recursive>,...
@3so that, instead of 1
, one could use the mnemonic
"Parametric; no_Recursive"
, or simply "Parametric"
since
Recursive
is unset by default (default value of flag is 0
,
i.e. everything unset). People used to the bit-or notation in languages like
C may also use the form "Parametric | no_Recursive"
.
@3Pointers. \varsidx{pointer} If a parameter in the function prototype is prefixed with a & sign, as in
foo(x,&e)
@3it means that, besides the normal return value, the function may
assign a value to e
as a side effect. When passing the argument, the &
sign has to be typed in explicitly. As of version 2.9.1, this pointer
argument is optional for all documented functions, hence the & will always
appear between brackets as in Z_issquare
(x,{&e})
.
@3About library programming.
The library function foo
, as defined at the beginning of this
section, is seen to have two mandatory arguments, x
and flag: no function
seen in the present chapter has been implemented so as to accept a variable
number of arguments, so all arguments are mandatory when programming with the
library (usually, variants are provided corresponding to the various flag values).
We include an = default value
token in the prototype to signal how a missing
argument should be encoded. Most of the time, it will be a NULL
pointer, or
-1 for a variable number. Refer to the User's Guide to the PARI library
for general background and details.
\subseckbd{+/
-} The expressions +
x
and -
x
refer
to monadic operators (the first does nothing, the second negates x
).
The library syntax is GEN
gneg(GEN x)
for -
x
.
\subseckbd{+} The expression x
+
y
is the sum of x
and y
.
Addition between a scalar type x
and a t_COL
or t_MAT
y
returns
respectively [y[1] + x, y[2],...]
and y + x Id
. Other additions
between a scalar type and a vector or a matrix, or between vector/matrices of
incompatible sizes are forbidden.
The library syntax is GEN
gadd(GEN x, GEN y)
.
\subseckbd{-} The expression x
-
y
is the difference of x
and y
. Subtraction between a scalar type x
and a t_COL
or t_MAT
y
returns respectively [y[1] - x, y[2],...]
and y - x Id
.
Other subtractions between a scalar type and a vector or a matrix, or
between vector/matrices of incompatible sizes are forbidden.
The library syntax is GEN
gsub(GEN x, GEN y)
for x
-
y
.
\subseckbd{*} The expression x
*
y
is the product of x
and y
. Among the prominent impossibilities are multiplication between
vector/matrices of incompatible sizes, between a t_INTMOD
or t_PADIC
Restricted to scalars, *
is commutative; because of vector and matrix
operations, it is not commutative in general.
Multiplication between two t_VEC
s or two t_COL
s is not
allowed; to take the scalar product of two vectors of the same length,
transpose one of the vectors (using the operator ~
or the function
mattranspose
, see Label se:linear_algebra) and multiply a line vector
by a column vector:
? a = [1,2,3]; ? a * a *** at top-level: a*a *** ^-- *** _*_: forbidden multiplication t_VEC * t_VEC. ? a * a~ %2 = 14
If x,y
are binary quadratic forms, compose them; see also
qfbnucomp
and qfbnupow
. If x,y
are t_VECSMALL
of the same
length, understand them as permutations and compose them.
The library syntax is GEN
gmul(GEN x, GEN y)
for x
*
y
.
Also available is GEN
gsqr(GEN x)
for x
*
x
.
x
/
y
is the quotient of x
and y
. In addition to the impossibilities for multiplication, note that if
the divisor is a matrix, it must be an invertible square matrix, and in that
case the result is x*y^{-1}
. Furthermore note that the result is as exact
as possiblein particular, division of two integers always gives a rational
number (which may be an integer if the quotient is exact) and not the
Euclidean quotient (see x
\
y
for that), and similarly the
quotient of two polynomials is a rational function in general. To obtain the
approximate real value of the quotient of two integers, add 0.
to the
result; to obtain the approximate p
-adic value of the quotient of two
integers, add O(p^k)
to the result; finally, to obtain the
Taylor series expansion of the quotient of two polynomials, add
O(X^k)
to the result or use the taylor
function
(see Label se:taylor).
The library syntax is GEN
gdiv(GEN x, GEN y)
for x
/
y
.
\subseckbd{} The expression x \y
is the
Euclidean quotient of x
and y
. If y
is a real scalar, this is
defined as floor(x/y)
if y > 0
, and ceil(x/y)
if
y < 0
and the division is not exact. Hence the remainder
x - (x\y)*y
is in [0, |y|[
.
Note that when y
is an integer and x
a polynomial, y
is first promoted
to a polynomial of degree 0
. When x
is a vector or matrix, the operator
is applied componentwise.
The library syntax is GEN
gdivent(GEN x, GEN y)
for x
\
y
.
x
\/
y
evaluates to the rounded
Euclidean quotient of x
and y
. This is the same as x \y
except for scalar divisionthe quotient is such that the corresponding
remainder is smallest in absolute value and in case of a tie the quotient
closest to + oo
is chosen (hence the remainder would belong to
]{-}|y|/2, |y|/2]
).
When x
is a vector or matrix, the operator is applied componentwise.
The library syntax is GEN
gdivround(GEN x, GEN y)
for x
\/
y
.
x % y
evaluates to the modular
Euclidean remainder of x
and y
, which we now define. When x
or y
is a non-integral real number, x%y
is defined as
x - (x\y)*y
. Otherwise, if y
is an integer, this is
the smallest
non-negative integer congruent to x
modulo y
. (This actually coincides
with the previous definition if and only if x
is an integer.) If y
is a
polynomial, this is the polynomial of smallest degree congruent to
x
modulo y
. For instance? (1/2) % 3 %1 = 2 ? 0.5 % 3 %2 = 0.5000000000000000000000000000 ? (1/2) % 3.0 %3 = 1/2
Note that when y
is an integer and x
a polynomial, y
is first promoted
to a polynomial of degree 0
. When x
is a vector or matrix, the operator
is applied componentwise.
The library syntax is GEN
gmod(GEN x, GEN y)
for x
%
y
.
\subseckbd{^} The expression x^n
is powering.
@3* If the exponent n
is an integer, then exact operations are performed
using binary (left-shift) powering techniques. If x
is a p
-adic number, its
precision will increase if v_p(n) > 0
. Powering a binary quadratic form
(types t_QFI
and t_QFR
) returns a representative of the class, which is
always reduced if the input was. (In particular, x^1
returns x
itself, whether it is reduced or not.)
PARI is able to rewrite the multiplication x * x
of two identical
objects as x^2
, or sqr(x)
. Here, identical means the operands are
two different labels referencing the same chunk of memory; no equality test
is performed. This is no longer true when more than two arguments are
involved.
@3* If the exponent n
is not an integer, powering is treated as the
transcendental function exp (n
log x)
, and in particular acts
componentwise on vector or matrices, even square matrices ! (See
Label se:trans.)
@3* As an exception, if the exponent is a rational number p/q
and x
an
integer modulo a prime or a p
-adic number, return a solution y
of
y^q = x^p
if it exists. Currently, q
must not have large prime factors.
Beware that
? Mod(7,19)^(1/2) %1 = Mod(11, 19) /* is any square root */ ? sqrt(Mod(7,19)) %2 = Mod(8, 19) /* is the smallest square root */ ? Mod(7,19)^(3/5) %3 = Mod(1, 19) ? %3^(5/3) %4 = Mod(1, 19) /* Mod(7,19) is just another cubic root */
@3* If the exponent is a negative integer, an inverse must be computed.
For non-invertible t_INTMOD
x
, this will fail and implicitly exhibit a
non trivial factor of the modulus:
? Mod(4,6)^(-1) *** at top-level: Mod(4,6)^(-1) *** ^----- *** _^_: impossible inverse modulo: Mod(2, 6).
(Here, a factor 2 is obtained directly. In general, take the gcd of the
representative and the modulus.) This is most useful when performing
complicated operations modulo an integer N
whose factorization is
unknown. Either the computation succeeds and all is well, or a factor d
is discovered and the computation may be restarted modulo d
or N/d
.
For non-invertible t_POLMOD
x
, the behaviour is the same:
? Mod(x^2, x^3-x)^(-1) *** at top-level: Mod(x^2,x^3-x)^(-1) *** ^----- *** _^_: impossible inverse in RgXQ_inv: Mod(x^2, x^3 - x).
@3Note that the underlying algorihm (subresultant) assumes the base ring is a domain:
? a = Mod(3*y^3+1, 4); b = y^6+y^5+y^4+y^3+y^2+y+1; c = Mod(a,b); ? c^(-1) *** at top-level: Mod(a,b)^(-1) *** ^----- *** _^_: impossible inverse modulo: Mod(2, 4).
In fact c
is invertible, but Z/4
Z is not a domain and the algorithm
fails. It is possible for the algorithm to succeed in such situations
and any returned result will be correct, but chances are an error
will occur first. In this specific case, one should work with 2
-adics.
In general, one can also try the following approach
? inversemod(a, b) = { my(m, v = variable(b)); m = polsylvestermatrix(polrecip(a), polrecip(b)); m = matinverseimage(m, matid(#m)[,1]); Polrev(m[1..poldegree(b)], v); } ? inversemod(a,b) %2 = Mod(2,4)*y^5 + Mod(3,4)*y^3 + Mod(1,4)*y^2 + Mod(3,4)*y + Mod(2,4)
This is not guaranteed to work either since matinverseimage
must also
invert pivots. See Label se:linear_algebra.
For a t_MAT
x
, the matrix is expected to be square and invertible, except
in the special case x^(-1)
which returns a left inverse if one exists
(rectangular x
with full column rank).
? x = Mat([1;2]) %1 = [1]
[2]
? x^(-1) %2 = [1 0]
The library syntax is GEN
gpow(GEN x, GEN n, long prec)
for x^n
.
(x,y)
Gives the result of a comparison between arbitrary objects x
and y
(as -1
, 0
or 1
). The underlying order relation is transitive,
the function returns 0
if and only if x === y
, and its
restriction to integers coincides with the customary one. Besides that,
it has no useful mathematical meaning.
In case all components are equal up to the smallest length of the operands,
the more complex is considered to be larger. More precisely, the longest is
the largest; when lengths are equal, we have matrix >
vector >
scalar.
For example:
? cmp(1, 2) %1 = -1 ? cmp(2, 1) %2 = 1 ? cmp(1, 1.0) \\ note that 1 == 1.0, but (1===1.0) is false. %3 = -1 ? cmp(x + Pi, []) %4 = -1
@3This function is mostly useful to handle sorted lists or
vectors of arbitrary objects. For instance, if v
is a vector, the
construction vecsort(v, cmp)
is equivalent to Set(v)
.
The library syntax is GEN
cmp_universal(GEN x, GEN y)
.
(x,y,{v})
Creates a column vector with two components, the first being the Euclidean
quotient (x \y
), the second the Euclidean remainder
(x - (x\y)*y
), of the division of x
by y
. This avoids the
need to do two divisions if one needs both the quotient and the remainder.
If v
is present, and x
, y
are multivariate
polynomials, divide with respect to the variable v
.
Beware that divrem(x,y)[2]
is in general not the same as
x % y
; no GP operator corresponds to it:
? divrem(1/2, 3)[2] %1 = 1/2 ? (1/2) % 3 %2 = 2 ? divrem(Mod(2,9), 3)[2] *** at top-level: divrem(Mod(2,9),3)[2 *** ^-------------------- *** forbidden division t_INTMOD \ t_INT. ? Mod(2,9) % 6 %3 = Mod(2,3)
The library syntax is GEN
divrem(GEN x, GEN y, long v = -1)
where v
is a variable number.
Also available is GEN
gdiventres(GEN x, GEN y)
when v
is
not needed.
(x,y)
Gives the result of a lexicographic comparison
between x
and y
(as -1
, 0
or 1
). This is to be interpreted in quite
a wide sense: It is admissible to compare objects of different types
(scalars, vectors, matrices), provided the scalars can be compared, as well
as vectors/matrices of different lengths. The comparison is recursive.
In case all components are equal up to the smallest length of the operands,
the more complex is considered to be larger. More precisely, the longest is
the largest; when lengths are equal, we have matrix >
vector >
scalar.
For example:
? lex([1,3], [1,2,5]) %1 = 1 ? lex([1,3], [1,3,-1]) %2 = -1 ? lex([1], [[1]]) %3 = -1 ? lex([1], [1]~) %4 = 0
The library syntax is GEN
lexcmp(GEN x, GEN y)
.
(x,y)
Creates the maximum of x
and y
when they can be compared.
The library syntax is GEN
gmax(GEN x, GEN y)
.
(x,y)
Creates the minimum of x
and y
when they can be compared.
The library syntax is GEN
gmin(GEN x, GEN y)
.
powers(x,n,{
x0})
For non-negative n
, return the vector with n+1
components
[1,x,...,x^n]
if x0
is omitted, and [x_0, x_0*x, ..., x_0*x^n]
otherwise.
? powers(Mod(3,17), 4) %1 = [Mod(1, 17), Mod(3, 17), Mod(9, 17), Mod(10, 17), Mod(13, 17)] ? powers(Mat([1,2;3,4]), 3) %2 = [[1, 0; 0, 1], [1, 2; 3, 4], [7, 10; 15, 22], [37, 54; 81, 118]] ? powers(3, 5, 2) %3 = [2, 6, 18, 54, 162, 486]
@3When n < 0
, the function returns the empty vector []
.
The library syntax is GEN
gpowers0(GEN x, long n, GEN x0 = NULL)
.
Also available is
GEN
gpowers(GEN x, long n)
when x0
is NULL
.
(x,n)
Shifts x
componentwise left by n
bits if n >= 0
and right by |n|
bits if n < 0
. May be abbreviated as x
<<
n
or x
>>
(-n)
.
A left shift by n
corresponds to multiplication by 2^n
. A right shift of an
integer x
by |n|
corresponds to a Euclidean division of x
by 2^{|n|}
with a remainder of the same sign as x
, hence is not the same (in general) as
x \ 2^n
.
The library syntax is GEN
gshift(GEN x, long n)
.
(x,n)
Multiplies x
by 2^n
. The difference with
shift
is that when n < 0
, ordinary division takes place, hence for
example if x
is an integer the result may be a fraction, while for shifts
Euclidean division takes place when n < 0
hence if x
is an integer the result
is still an integer.
The library syntax is GEN
gmul2n(GEN x, long n)
.
(x)
sign (0
, 1
or -1
) of x
, which must be of
type integer, real or fraction; t_QUAD
with positive discriminants and
t_INFINITY
are also supported.
The library syntax is GEN
gsigne(GEN x)
.
(x,{&v})
If x
is a vector or a matrix, returns the largest entry of x
,
otherwise returns a copy of x
. Error if x
is empty.
If v
is given, set it to the index of a largest entry (indirect maximum),
when x
is a vector. If x
is a matrix, set v
to coordinates [i,j]
such that x[i,j]
is a largest entry. This flag is ignored if x
is not a
vector or matrix.
? vecmax([10, 20, -30, 40]) %1 = 40 ? vecmax([10, 20, -30, 40], &v); v %2 = 4 ? vecmax([10, 20; -30, 40], &v); v %3 = [2, 2]
The library syntax is GEN
vecmax0(GEN x, GEN *v = NULL)
.
When v
is not needed, the function GEN
vecmax(GEN x)
is
also available.
(x,{&v})
If x
is a vector or a matrix, returns the smallest entry of x
,
otherwise returns a copy of x
. Error if x
is empty.
If v
is given, set it to the index of a smallest entry (indirect minimum),
when x
is a vector. If x
is a matrix, set v
to coordinates [i,j]
such
that x[i,j]
is a smallest entry. This is ignored if x
is not a vector or
matrix.
? vecmin([10, 20, -30, 40]) %1 = -30 ? vecmin([10, 20, -30, 40], &v); v %2 = 3 ? vecmin([10, 20; -30, 40], &v); v %3 = [2, 1]
The library syntax is GEN
vecmin0(GEN x, GEN *v = NULL)
.
When v
is not needed, the function GEN
vecmin(GEN x)
is also
available.
The six
standard comparison operators <=
, <
, >=
, >
,
==
, !=
are available in GP. The result is 1 if the comparison is
true, 0 if it is false. The operator ==
is quite liberal : for
instance, the integer 0, a 0 polynomial, and a vector with 0 entries are all
tested equal.
The extra operator ===
tests whether two objects are identical and is
much stricter than ==
: objects of different type or length are never
identical.
For the purpose of comparison, t_STR
objects are compared using
the standard lexicographic order, and comparing them to objects
of a different type raises an exception.
GP accepts <>
as a synonym for !=
. On the other hand, =
is
definitely not a synonym for ==
: it is the assignment statement.
The standard boolean operators ||
(inclusive or), &&
(and) and !
(not) are also available.
Many of the conversion functions are rounding or truncating operations. In this case, if the argument is a rational function, the result is the Euclidean quotient of the numerator by the denominator, and if the argument is a vector or a matrix, the operation is done componentwise. This will not be restated for every function.
(x, {n})
Transforms the object x
into a column vector. The dimension of the
resulting vector can be optionally specified via the extra parameter n
.
If n
is omitted or 0
, the dimension depends on the type of x
; the
vector has a single component, except when x
is
@3* a vector or a quadratic form (in which case the resulting vector is simply the initial object considered as a row vector),
@3* a polynomial or a power series. In the case of a polynomial, the
coefficients of the vector start with the leading coefficient of the
polynomial, while for power series only the significant coefficients are
taken into account, but this time by increasing order of degree.
In this last case, Vec
is the reciprocal function of Pol
and
Ser
respectively,
@3* a matrix (the column of row vector comprising the matrix is returned),
@3* a character string (a vector of individual characters is returned).
In the last two cases (matrix and character string), n
is meaningless and
must be omitted or an error is raised. Otherwise, if n
is given, 0
entries are appended at the end of the vector if n > 0
, and prepended at
the beginning if n < 0
. The dimension of the resulting vector is |n|
.
Note that the function Colrev
does not exist, use Vecrev
.
The library syntax is GEN
gtocol0(GEN x, long n)
.
GEN
gtocol(GEN x)
is also available.
(x, {n})
As Col(x, -n)
, then reverse the result. In particular,
Colrev
is the reciprocal function of Polrev
: the
coefficients of the vector start with the constant coefficient of the
polynomial and the others follow by increasing degree.
The library syntax is GEN
gtocolrev0(GEN x, long n)
.
GEN
gtocolrev(GEN x)
is also available.
({x = []})
Transforms a (row or column) vector x
into a list, whose components are
the entries of x
. Similarly for a list, but rather useless in this case.
For other types, creates a list with the single element x
. Note that,
except when x
is omitted, this function creates a small memory leak; so,
either initialize all lists to the empty list, or use them sparingly.
The library syntax is GEN
gtolist(GEN x = NULL)
.
The variant GEN
mklist(void)
creates an empty list.
({x})
A ``Map'' is an associative array, or dictionary: a data
type composed of a collection of (key, value) pairs, such that
each key appears just once in the collection. This function
converts the matrix [a_1,b_1;a_2,b_2;...;a_n,b_n]
to the map a_i:--->b_i
.
? M = Map(factor(13!)); ? mapget(M,3) %2 = 5
@3If the argument x
is omitted, creates an empty map, which
may be filled later via mapput
.
The library syntax is GEN
gtomap(GEN x = NULL)
.
({x = []})
Transforms the object x
into a matrix.
If x
is already a matrix, a copy of x
is created.
If x
is a row (resp. column) vector, this creates a 1-row (resp.
1-column) matrix, unless all elements are column (resp. row) vectors
of the same length, in which case the vectors are concatenated sideways
and the attached big matrix is returned.
If x
is a binary quadratic form, creates the attached 2 x 2
matrix. Otherwise, this creates a 1 x 1
matrix containing x
.
? Mat(x + 1) %1 = [x + 1] ? Vec( matid(3) ) %2 = [[1, 0, 0]~, [0, 1, 0]~, [0, 0, 1]~] ? Mat(%) %3 = [1 0 0]
[0 1 0]
[0 0 1] ? Col( [1,2; 3,4] ) %4 = [[1, 2], [3, 4]]~ ? Mat(%) %5 = [1 2]
[3 4] ? Mat(Qfb(1,2,3)) %6 = [1 1]
[1 3]
The library syntax is GEN
gtomat(GEN x = NULL)
.
(a,b)
In its basic form, creates an intmod or a polmod (a mod b)
; b
must
be an integer or a polynomial. We then obtain a t_INTMOD
and a
t_POLMOD
respectively:
? t = Mod(2,17); t^8 %1 = Mod(1, 17) ? t = Mod(x,x^2+1); t^2 %2 = Mod(-1, x^2+1)
@3If a % b
makes sense and yields a result of the
appropriate type (t_INT
or scalar/t_POL
), the operation succeeds as
well:
? Mod(1/2, 5) %3 = Mod(3, 5) ? Mod(7 + O(3^6), 3) %4 = Mod(1, 3) ? Mod(Mod(1,12), 9) %5 = Mod(1, 3) ? Mod(1/x, x^2+1) %6 = Mod(-1, x^2+1) ? Mod(exp(x), x^4) %7 = Mod(1/6*x^3 + 1/2*x^2 + x + 1, x^4)
If a
is a complex object, ``base change'' it to Z/b
Z or K[x]/(b)
,
which is equivalent to, but faster than, multiplying it by Mod(1,b)
:
? Mod([1,2;3,4], 2) %8 = [Mod(1, 2) Mod(0, 2)]
[Mod(1, 2) Mod(0, 2)] ? Mod(3*x+5, 2) %9 = Mod(1, 2)*x + Mod(1, 2) ? Mod(x^2 + y*x + y^3, y^2+1) %10 = Mod(1, y^2 + 1)*x^2 + Mod(y, y^2 + 1)*x + Mod(-y, y^2 + 1)
This function is not the same as x
%
y
, the result of which
has no knowledge of the intended modulus y
. Compare
? x = 4 % 5; x + 1 %1 = 5 ? x = Mod(4,5); x + 1 %2 = Mod(0,5)
Note that such ``modular'' objects can be lifted via lift
or
centerlift
. The modulus of a t_INTMOD
or t_POLMOD
z
can
be recovered via z.mod
.
The library syntax is GEN
gmodulo(GEN a, GEN b)
.
(t,{v = 'x})
Transforms the object t
into a polynomial with main variable v
. If t
is a scalar, this gives a constant polynomial. If t
is a power series with
non-negative valuation or a rational function, the effect is similar to
truncate
, i.e. we chop off the O(X^k)
or compute the Euclidean
quotient of the numerator by the denominator, then change the main variable
of the result to v
.
The main use of this function is when t
is a vector: it creates the
polynomial whose coefficients are given by t
, with t[1]
being the leading
coefficient (which can be zero). It is much faster to evaluate
Pol
on a vector of coefficients in this way, than the corresponding
formal expression a_n X^n +...+ a_0
, which is evaluated naively exactly
as written (linear versus quadratic time in n
). Polrev
can be used if
one wants x[1]
to be the constant coefficient:
? Pol([1,2,3]) %1 = x^2 + 2*x + 3 ? Polrev([1,2,3]) %2 = 3*x^2 + 2*x + 1
The reciprocal function of Pol
(resp. Polrev
) is Vec
(resp.
Vecrev
).
? Vec(Pol([1,2,3])) %1 = [1, 2, 3] ? Vecrev( Polrev([1,2,3]) ) %2 = [1, 2, 3]
@3Warning. This is not a substitution function. It will not
transform an object containing variables of higher priority than v
.
? Pol(x + y, y) *** at top-level: Pol(x+y,y) *** ^---------- *** Pol: variable must have higher priority in gtopoly.
The library syntax is GEN
gtopoly(GEN t, long v = -1)
where v
is a variable number.
(t,{v = 'x})
Transform the object t
into a polynomial
with main variable v
. If t
is a scalar, this gives a constant polynomial.
If t
is a power series, the effect is identical to truncate
, i.e. it
chops off the O(X^k)
.
The main use of this function is when t
is a vector: it creates the
polynomial whose coefficients are given by t
, with t[1]
being the
constant term. Pol
can be used if one wants t[1]
to be the leading
coefficient:
? Polrev([1,2,3]) %1 = 3*x^2 + 2*x + 1 ? Pol([1,2,3]) %2 = x^2 + 2*x + 3
The reciprocal function of Pol
(resp. Polrev
) is Vec
(resp.
Vecrev
).
The library syntax is GEN
gtopolyrev(GEN t, long v = -1)
where v
is a variable number.
(a,b,c,{D = 0.})
Creates the binary quadratic form
ax^2+bxy+cy^2
. If b^2-4ac > 0
, initialize Shanks' distance
function to D
. Negative definite forms are not implemented,
use their positive definite counterpart instead.
The library syntax is GEN
Qfb0(GEN a, GEN b, GEN c, GEN D = NULL, long prec)
.
Also available are
GEN
qfi(GEN a, GEN b, GEN c)
(assumes b^2-4ac < 0
) and
GEN
qfr(GEN a, GEN b, GEN c, GEN D)
(assumes b^2-4ac > 0
).
Ser(s,{v = 'x},{d =
seriesprecision})
Transforms the object s
into a power series with main variable v
(x
by default) and precision (number of significant terms) equal to
d >= 0
(d = seriesprecision
by default). If s
is a
scalar, this gives a constant power series in v
with precision d
.
If s
is a polynomial, the polynomial is truncated to d
terms if needed
? Ser(1, 'y, 5) %1 = 1 + O(y^5) ? Ser(x^2,, 5) %2 = x^2 + O(x^7) ? T = polcyclo(100) %3 = x^40 - x^30 + x^20 - x^10 + 1 ? Ser(T, 'x, 11) %4 = 1 - x^10 + O(x^11)
@3The function is more or less equivalent with multiplication by
1 + O(v^d)
in theses cases, only faster.
If s
is a vector, on the other hand, the coefficients of the vector are
understood to be the coefficients of the power series starting from the
constant term (as in Polrev
(x)
), and the precision d
is ignored:
in other words, in this case, we convert t_VEC
/ t_COL
to the power
series whose significant terms are exactly given by the vector entries.
Finally, if s
is already a power series in v
, we return it verbatim,
ignoring d
again. If d
significant terms are desired in the last two
cases, convert/truncate to t_POL
first.
? v = [1,2,3]; Ser(v, t, 7) %5 = 1 + 2*t + 3*t^2 + O(t^3) \\ 3 terms: 7 is ignored! ? Ser(Polrev(v,t), t, 7) %6 = 1 + 2*t + 3*t^2 + O(t^7) ? s = 1+x+O(x^2); Ser(s, x, 7) %7 = 1 + x + O(x^2) \\ 2 terms: 7 ignored ? Ser(truncate(s), x, 7) %8 = 1 + x + O(x^7)
The warning given for Pol
also applies here: this is not a substitution
function.
The library syntax is GEN
gtoser(GEN s, long v = -1, long precdl)
where v
is a variable number.
({x = []})
Converts x
into a set, i.e. into a row vector, with strictly increasing
entries with respect to the (somewhat arbitrary) universal comparison function
cmp
. Standard container types t_VEC
, t_COL
, t_LIST
and
t_VECSMALL
are converted to the set with corresponding elements. All
others are converted to a set with one element.
? Set([1,2,4,2,1,3]) %1 = [1, 2, 3, 4] ? Set(x) %2 = [x] ? Set(Vecsmall([1,3,2,1,3])) %3 = [1, 2, 3]
The library syntax is GEN
gtoset(GEN x = NULL)
.
({x}*)
Converts its argument list into a
single character string (type t_STR
, the empty string if x
is omitted).
To recover an ordinary GEN
from a string, apply eval
to it. The
arguments of Str
are evaluated in string context, see Label se:strings.
? x2 = 0; i = 2; Str(x, i) %1 = "x2" ? eval(%) %2 = 0
This function is mostly useless in library mode. Use the pair
strtoGEN
/GENtostr
to convert between GEN
and char*
.
The latter returns a malloced string, which should be freed after usage.
(x)
Converts x
to a string, translating each integer
into a character.
? Strchr(97) %1 = "a" ? Vecsmall("hello world") %2 = Vecsmall([104, 101, 108, 108, 111, 32, 119, 111, 114, 108, 100]) ? Strchr(%) %3 = "hello world"
The library syntax is GEN
Strchr(GEN x)
.
({x}*)
Converts its argument list into a
single character string (type t_STR
, the empty string if x
is omitted).
Then perform environment expansion, see Label se:envir.
This feature can be used to read environment variable values.
? Strexpand("$HOME/doc") %1 = "/home/pari/doc"
The individual arguments are read in string context, see Label se:strings.
({x}*)
Translates its arguments to TeX
format, and concatenates the results into a single character string (type
t_STR
, the empty string if x
is omitted).
The individual arguments are read in string context, see Label se:strings.
(x, {n})
Transforms the object x
into a row vector. The dimension of the
resulting vector can be optionally specified via the extra parameter n
.
If n
is omitted or 0
, the dimension depends on the type of x
; the
vector has a single component, except when x
is
@3* a vector or a quadratic form: returns the initial object considered as a row vector,
@3* a polynomial or a power series: returns a vector consisting of the coefficients.
In the case of a polynomial, the coefficients of the vector start with the leading
coefficient of the polynomial, while for power series only the significant coefficients
are taken into account, but this time by increasing order of degree.
Vec
is the reciprocal function of Pol
for a polynomial and of
Ser
for a power series,
@3* a matrix: returns the vector of columns comprising the matrix,
@3* a character string: returns the vector of individual characters,
@3* a map: returns the vector of the domain of the map,
@3* an error context (t_ERROR
): returns the error components, see
iferr
.
In the last four cases (matrix, character string, map, error), n
is
meaningless and must be omitted or an error is raised. Otherwise, if n
is
given, 0
entries are appended at the end of the vector if n > 0
, and
prepended at the beginning if n < 0
. The dimension of the resulting vector
is |n|
. Variant: GEN
gtovec(GEN x)
is also available.
The library syntax is GEN
gtovec0(GEN x, long n)
.
(x, {n})
As Vec(x, -n)
, then reverse the result. In particular,
Vecrev
is the reciprocal function of Polrev
: the
coefficients of the vector start with the constant coefficient of the
polynomial and the others follow by increasing degree.
The library syntax is GEN
gtovecrev0(GEN x, long n)
.
GEN
gtovecrev(GEN x)
is also available.
(x, {n})
Transforms the object x
into a row vector of type t_VECSMALL
. The
dimension of the resulting vector can be optionally specified via the extra
parameter n
.
This acts as Vec
(x,n)
, but only on a limited set of objects:
the result must be representable as a vector of small integers.
If x
is a character string, a vector of individual characters in ASCII
encoding is returned (Strchr
yields back the character string).
The library syntax is GEN
gtovecsmall0(GEN x, long n)
.
GEN
gtovecsmall(GEN x)
is also available.
(x)
Outputs the vector of the binary digits of |x|
. Here x
can be an
integer, a real number (in which case the result has two components, one for
the integer part, one for the fractional part) or a vector/matrix.
? binary(10) %1 = [1, 0, 1, 0]
? binary(3.14) %2 = [[1, 1], [0, 0, 1, 0, 0, 0, [...]]
? binary([1,2]) %3 = [[1], [1, 0]]
@3By convention, 0
has no digits:
? binary(0) %4 = []
The library syntax is GEN
binaire(GEN x)
.
(x,y)
Bitwise and
of two integers x
and y
, that is the integer
sum_i (x_i and y_i) 2^i
Negative numbers behave 2
-adically, i.e. the result is the 2
-adic limit
of bitand
(x_n,y_n)
, where x_n
and y_n
are non-negative integers
tending to x
and y
respectively. (The result is an ordinary integer,
possibly negative.)
? bitand(5, 3) %1 = 1 ? bitand(-5, 3) %2 = 3 ? bitand(-5, -3) %3 = -7
The library syntax is GEN
gbitand(GEN x, GEN y)
.
Also available is
GEN
ibitand(GEN x, GEN y)
, which returns the bitwise and
of |x|
and |y|
, two integers.
(x,{n = -1})
bitwise negation of an integer x
,
truncated to n
bits, n >= 0
, that is the integer
sum_{i = 0}^{n-1} not(x_i) 2^i.
The special case n = -1
means no truncation: an infinite sequence of
leading 1
is then represented as a negative number.
See Label se:bitand for the behavior for negative arguments.
The library syntax is GEN
gbitneg(GEN x, long n)
.
(x,y)
Bitwise negated imply of two integers x
and
y
(or not
(x ==> y)
), that is the integer
sum
(x_i and not(y_i)) 2^i
See Label se:bitand for the behavior for negative arguments.
The library syntax is GEN
gbitnegimply(GEN x, GEN y)
.
Also available is
GEN
ibitnegimply(GEN x, GEN y)
, which returns the bitwise negated
imply of |x|
and |y|
, two integers.
(x,y)
bitwise (inclusive)
or
of two integers x
and y
, that is the integer
sum
(x_i or y_i) 2^i
See Label se:bitand for the behavior for negative arguments.
The library syntax is GEN
gbitor(GEN x, GEN y)
.
Also available is
GEN
ibitor(GEN x, GEN y)
, which returns the bitwise ir
of |x|
and |y|
, two integers.
(x,{n})
The function behaves differently according to whether n
is
present and positive or not. If n
is missing, the function returns the
(floating point) precision in bits of the PARI object x
. If x
is an
exact object, the function returns +oo
.
? bitprecision(exp(1e-100)) %1 = 512 \\ 512 bits ? bitprecision( [ exp(1e-100), 0.5 ] ) %2 = 128 \\ minimal accuracy among components ? bitprecision(2 + x) %3 = +oo \\ exact object
If n
is present and positive, the function creates a new object equal to x
with the new bit-precision roughly n
. In fact, the smallest multiple of 64
(resp. 32 on a 32-bit machine) larger than or equal to n
.
For x
a vector or a matrix, the operation is
done componentwise; for series and polynomials, the operation is done
coefficientwise. For real x
, n
is the number of desired significant
bits. If n
is smaller than the precision of x
, x
is truncated,
otherwise x
is extended with zeros. For exact or non-floating point types,
no change.
? bitprecision(Pi, 10) \\ actually 64 bits ~ 19 decimal digits %1 = 3.141592653589793239 ? bitprecision(1, 10) %2 = 1 ? bitprecision(1 + O(x), 10) %3 = 1 + O(x) ? bitprecision(2 + O(3^5), 10) %4 = 2 + O(3^5)
The library syntax is GEN
bitprecision0(GEN x, long n)
.
(x,n)
Outputs the n-th
bit of x
starting
from the right (i.e. the coefficient of 2^n
in the binary expansion of x
).
The result is 0 or 1.
? bittest(7, 0) %1 = 1 \\ the bit 0 is 1 ? bittest(7, 2) %2 = 1 \\ the bit 2 is 1 ? bittest(7, 3) %3 = 0 \\ the bit 3 is 0
See Label se:bitand for the behavior at negative arguments.
The library syntax is GEN
gbittest(GEN x, long n)
.
For a t_INT
x
, the variant long
bittest(GEN x, long n)
is
generally easier to use, and if furthermore n >= 0
the low-level function
ulong
int_bit(GEN x, long n)
returns bittest(abs(x),n)
.
(x,y)
Bitwise (exclusive) or
of two integers x
and y
, that is the integer
sum (x_i xor y_i) 2^i
See Label se:bitand for the behavior for negative arguments.
The library syntax is GEN
gbitxor(GEN x, GEN y)
.
Also available is
GEN
ibitxor(GEN x, GEN y)
, which returns the bitwise xor
of |x|
and |y|
, two integers.
(x)
Ceiling of x
. When x
is in R, the result is the
smallest integer greater than or equal to x
. Applied to a rational
function, ceil(x)
returns the Euclidean quotient of the numerator by
the denominator.
The library syntax is GEN
gceil(GEN x)
.
(x,{v})
Same as lift
, except that t_INTMOD
and t_PADIC
components
are lifted using centered residues:
@3* for a t_INTMOD
x\in
Z/n
Z, the lift y
is such that
-n/2 < y <= n/2
.
@3* a t_PADIC
x
is lifted in the same way as above (modulo
p^padicprec(x)
) if its valuation v
is non-negative; if not, returns
the fraction p^v
centerlift
(x p^{-v})
; in particular, rational
reconstruction is not attempted. Use bestappr
for this.
For backward compatibility, centerlift(x,'v)
is allowed as an alias
for lift(x,'v)
.
The library syntax is centerlift(GEN x)
.
(x)
Returns the characteristic of the base ring over which x
is defined (as
defined by t_INTMOD
and t_FFELT
components). The function raises an
exception if incompatible primes arise from t_FFELT
and t_PADIC
components.
? characteristic(Mod(1,24)*x + Mod(1,18)*y) %1 = 6
The library syntax is GEN
characteristic(GEN x)
.
(x,n)
Extracts the n-th
-component of x
. This is to be understood
as follows: every PARI type has one or two initial code words. The
components are counted, starting at 1, after these code words. In particular
if x
is a vector, this is indeed the n-th
-component of x
, if
x
is a matrix, the n-th
column, if x
is a polynomial, the
n-th
coefficient (i.e. of degree n-1
), and for power series,
the n-th
significant coefficient.
For polynomials and power series, one should rather use polcoeff
, and
for vectors and matrices, the [$] operator. Namely, if
x is a
vector, then x[n] represents the
n-th component of
x. If
x is a matrix, x[m,n] represents the coefficient of row m and
column n of the matrix, x[m,] represents the
m-th
row of
x, and x[,n] represents the
n-th
column of
x$.
Using of this function requires detailed knowledge of the structure of the different PARI types, and thus it should almost never be used directly. Some useful exceptions:
? x = 3 + O(3^5); ? component(x, 2) %2 = 81 \\ p^(p-adic accuracy) ? component(x, 1) %3 = 3 \\ p ? q = Qfb(1,2,3); ? component(q, 1) %5 = 1
The library syntax is GEN
compo(GEN x, long n)
.
(x)
Conjugate of x
. The meaning of this
is clear, except that for real quadratic numbers, it means conjugation in the
real quadratic field. This function has no effect on integers, reals,
intmods, fractions or p
-adics. The only forbidden type is polmod
(see conjvec
for this).
The library syntax is GEN
gconj(GEN x)
.
(z)
Conjugate vector representation of z
. If z
is a
polmod, equal to Mod
(a,T)
, this gives a vector of length
degree(T)
containing:
@3* the complex embeddings of z
if T
has rational coefficients,
i.e. the a(r[i])
where r = polroots(T)
;
@3* the conjugates of z
if T
has some intmod coefficients;
@3if z
is a finite field element, the result is the vector of
conjugates [z,z^p,z^{p^2},...,z^{p^{n-1}}]
where n = degree(T)
.
@3If z
is an integer or a rational number, the result is z
. If
z
is a (row or column) vector, the result is a matrix whose columns are
the conjugate vectors of the individual elements of z
.
The library syntax is GEN
conjvec(GEN z, long prec)
.
(x)
Denominator of x
. The meaning of this
is clear when x
is a rational number or function. If x
is an integer
or a polynomial, it is treated as a rational number or function,
respectively, and the result is equal to 1
. For polynomials, you
probably want to use
denominator( content(x) )
instead. As for modular objects, t_INTMOD
and t_PADIC
have
denominator 1
, and the denominator of a t_POLMOD
is the denominator
of its (minimal degree) polynomial representative.
If x
is a recursive structure, for instance a vector or matrix, the lcm
of the denominators of its components (a common denominator) is computed.
This also applies for t_COMPLEX
s and t_QUAD
s.
@3Warning. Multivariate objects are created according to variable
priorities, with possibly surprising side effects (x/y
is a polynomial, but
y/x
is a rational function). See Label se:priority.
The library syntax is GEN
denom(GEN x)
.
digits(x,{b = 10})
Outputs the vector of the digits of |x|
in base b
, where x
and b
are
integers (b = 10
by default). See fromdigits
for the reverse
operation.
? digits(123) %1 = [1, 2, 3, 0]
? digits(10, 2) \\ base 2 %2 = [1, 0, 1, 0]
@3By convention, 0
has no digits:
? digits(0) %3 = []
The library syntax is GEN
digits(GEN x, GEN b = NULL)
.
(x)
Floor of x
. When x
is in R, the result is the
largest integer smaller than or equal to x
. Applied to a rational function,
floor(x)
returns the Euclidean quotient of the numerator by the
denominator.
The library syntax is GEN
gfloor(GEN x)
.
(x)
Fractional part of x
. Identical to
x-floor(x)
. If x
is real, the result is in [0,1[
.
The library syntax is GEN
gfrac(GEN x)
.
fromdigits(x,{b = 10})
Gives the integer formed by the elements of x
seen as the digits of a
number in base b
(b = 10
by default). This is the reverse of digits
:
? digits(1234,5) %1 = [1,4,4,1,4] ? fromdigits([1,4,4,1,4],5) %2 = 1234
@3By convention, 0
has no digits:
? fromdigits([]) %3 = 0
The library syntax is GEN
fromdigits(GEN x, GEN b = NULL)
.
(x)
If x
is a t_INT
, return the binary Hamming weight of |x|
. Otherwise
x
must be of type t_POL
, t_VEC
, t_COL
, t_VECSMALL
, or
t_MAT
and the function returns the number of non-zero coefficients of
x
.
? hammingweight(15) %1 = 4 ? hammingweight(x^100 + 2*x + 1) %2 = 3 ? hammingweight([Mod(1,2), 2, Mod(0,3)]) %3 = 2 ? hammingweight(matid(100)) %4 = 100
The library syntax is long
hammingweight(GEN x)
.
(x)
Imaginary part of x
. When x
is a quadratic number, this is the
coefficient of omega in the ``canonical'' integral basis (1,
omega)
.
The library syntax is GEN
gimag(GEN x)
.
(x)
Length of x
; #
x
is a shortcut for length
(x)
.
This is mostly useful for
@3* vectors: dimension (0 for empty vectors),
@3* lists: number of entries (0 for empty lists),
@3* matrices: number of columns,
@3* character strings: number of actual characters (without
trailing \0
, should you expect it from C
char*
).
? #"a string" %1 = 8 ? #[3,2,1] %2 = 3 ? #[] %3 = 0 ? #matrix(2,5) %4 = 5 ? L = List([1,2,3,4]); #L %5 = 4
The routine is in fact defined for arbitrary GP types, but is awkward and
useless in other cases: it returns the number of non-code words in x
, e.g.
the effective length minus 2 for integers since the t_INT
type has two code
words.
The library syntax is long
glength(GEN x)
.
(x,{v})
If v
is omitted, lifts intmods from Z/n
Z in Z,
p
-adics from Q_p
to Q (as truncate
), and polmods to
polynomials. Otherwise, lifts only polmods whose modulus has main
variable v
. t_FFELT
are not lifted, nor are List elements: you may
convert the latter to vectors first, or use apply(lift,L)
. More
generally, components for which such lifts are meaningless (e.g. character
strings) are copied verbatim.
? lift(Mod(5,3)) %1 = 2 ? lift(3 + O(3^9)) %2 = 3 ? lift(Mod(x,x^2+1)) %3 = x ? lift(Mod(x,x^2+1)) %4 = x
Lifts are performed recursively on an object components, but only
by one level: once a t_POLMOD
is lifted, the components of
the result are not lifted further.
? lift(x * Mod(1,3) + Mod(2,3)) %4 = x + 2 ? lift(x * Mod(y,y^2+1) + Mod(2,3)) %5 = y*x + Mod(2, 3) \\ do you understand this one? ? lift(x * Mod(y,y^2+1) + Mod(2,3), 'x) %6 = Mod(y, y^2 + 1)*x + Mod(Mod(2, 3), y^2 + 1) ? lift(%, y) %7 = y*x + Mod(2, 3)
@3To recursively lift all components not only by one level,
but as long as possible, use liftall
. To lift only t_INTMOD
s and
t_PADIC
s components, use liftint
. To lift only t_POLMOD
s
components, use liftpol
. Finally, centerlift
allows to lift
t_INTMOD
s and t_PADIC
s using centered residues (lift of smallest
absolute value).
The library syntax is GEN
lift0(GEN x, long v = -1)
where v
is a variable number.
Also available is GEN
lift(GEN x)
corresponding to
lift0(x,-1)
.
(x)
Recursively lift all components of x
from Z/n
Z to Z,
from Q_p
to Q (as truncate
), and polmods to
polynomials. t_FFELT
are not lifted, nor are List elements: you may
convert the latter to vectors first, or use apply(liftall,L)
. More
generally, components for which such lifts are meaningless (e.g. character
strings) are copied verbatim.
? liftall(x * (1 + O(3)) + Mod(2,3)) %1 = x + 2 ? liftall(x * Mod(y,y^2+1) + Mod(2,3)*Mod(z,z^2)) %2 = y*x + 2*z
The library syntax is GEN
liftall(GEN x)
.
(x)
Recursively lift all components of x
from Z/n
Z to Z and
from Q_p
to Q (as truncate
).
t_FFELT
are not lifted, nor are List elements: you may
convert the latter to vectors first, or use apply(liftint,L)
. More
generally, components for which such lifts are meaningless (e.g. character
strings) are copied verbatim.
? liftint(x * (1 + O(3)) + Mod(2,3)) %1 = x + 2 ? liftint(x * Mod(y,y^2+1) + Mod(2,3)*Mod(z,z^2)) %2 = Mod(y, y^2 + 1)*x + Mod(Mod(2*z, z^2), y^2 + 1)
The library syntax is GEN
liftint(GEN x)
.
(x)
Recursively lift all components of x
which are polmods to
polynomials. t_FFELT
are not lifted, nor are List elements: you may
convert the latter to vectors first, or use apply(liftpol,L)
. More
generally, components for which such lifts are meaningless (e.g. character
strings) are copied verbatim.
? liftpol(x * (1 + O(3)) + Mod(2,3)) %1 = (1 + O(3))*x + Mod(2, 3) ? liftpol(x * Mod(y,y^2+1) + Mod(2,3)*Mod(z,z^2)) %2 = y*x + Mod(2, 3)*z
The library syntax is GEN
liftpol(GEN x)
.
(x)
Algebraic norm of x
, i.e. the product of x
with
its conjugate (no square roots are taken), or conjugates for polmods. For
vectors and matrices, the norm is taken componentwise and hence is not the
L^2
-norm (see norml2
). Note that the norm of an element of
R is its square, so as to be compatible with the complex norm.
The library syntax is GEN
gnorm(GEN x)
.
(x)
Numerator of x
. The meaning of this
is clear when x
is a rational number or function. If x
is an integer
or a polynomial, it is treated as a rational number or function,
respectively, and the result is x
itself. For polynomials, you
probably want to use
numerator( content(x) )
instead.
In other cases, numerator(x)
is defined to be
denominator(x)*x
. This is the case when x
is a vector or a
matrix, but also for t_COMPLEX
or t_QUAD
. In particular since a
t_PADIC
or t_INTMOD
has denominator 1
, its numerator is
itself.
@3Warning. Multivariate objects are created according to variable
priorities, with possibly surprising side effects (x/y
is a polynomial, but
y/x
is a rational function). See Label se:priority.
The library syntax is GEN
numer(GEN x)
.
(n,k)
Generates the k
-th permutation (as a row vector of length n
) of the
numbers 1
to n
. The number k
is taken modulo n!
, i.e. inverse
function of permtonum
. The numbering used is the standard lexicographic
ordering, starting at 0
.
The library syntax is GEN
numtoperm(long n, GEN k)
.
Returns an object meaning + oo
, for use in functions such as
intnum
. It can be negated (-oo
represents - oo
), and
compared to real numbers (t_INT
, t_FRAC
, t_REAL
), with the
expected meaning: + oo
is greater than any real number and - oo
is
smaller.
The library syntax is GEN
mkoo()
.
(x,p)
Returns the absolute p
-adic precision of the object x
; this is the
minimum precision of the components of x
. The result is +oo
if x
is an exact object (as a p
-adic):
? padicprec((1 + O(2^5)) * x + (2 + O(2^4)), 2) %1 = 4 ? padicprec(x + 2, 2) %2 = +oo ? padicprec(2 + x + O(x^2), 2) %3 = +oo
@3The function raises an exception if it encounters
an object incompatible with p
-adic computations:
? padicprec(O(3), 2) *** at top-level: padicprec(O(3),2) *** ^----------------- *** padicprec: inconsistent moduli in padicprec: 3 != 2
? padicprec(1.0, 2) *** at top-level: padicprec(1.0,2) *** ^---------------- *** padicprec: incorrect type in padicprec (t_REAL).
The library syntax is GEN
gppadicprec(GEN x, GEN p)
.
Also available is the function long
padicprec(GEN x, GEN p)
,
which returns LONG_MAX
if x = 0
and the p
-adic precision as a
long
integer.
(x)
Given a permutation x
on n
elements, gives the number k
such that
x = numtoperm(n,k)
, i.e. inverse function of numtoperm
.
The numbering used is the standard lexicographic ordering, starting at 0
.
The library syntax is GEN
permtonum(GEN x)
.
(x,{n})
The function behaves differently according to whether n
is
present and positive or not. If n
is missing, the function returns the
precision in decimal digits of the PARI object x
. If x
is an exact
object, the function returns +oo
.
? precision(exp(1e-100)) %1 = 154 \\ 154 significant decimal digits ? precision(2 + x) %2 = +oo \\ exact object ? precision(0.5 + O(x)) %3 = 38 \\ floating point accuracy, NOT series precision ? precision( [ exp(1e-100), 0.5 ] ) %4 = 38 \\ minimal accuracy among components
If n
is present, the function creates a new object equal to x
with a new
floating point precision n
: n
is the number of desired significant
decimal digits. If n
is smaller than the precision of a t_REAL
component of x
, it is truncated, otherwise it is extended with zeros.
For exact or non-floating point types, no change.
The library syntax is GEN
precision0(GEN x, long n)
.
Also available are GEN
gprec(GEN x, long n)
and
long
precision(GEN x)
. In both, the accuracy is expressed in
words (32-bit or 64-bit depending on the architecture).
\subsec{random({N = 2^{{31}}})
}
Returns a random element in various natural sets depending on the
argument N
.
@3* t_INT
: returns an integer
uniformly distributed between 0
and N-1
. Omitting the argument
is equivalent to random(2^31)
.
@3* t_REAL
: returns a real number in [0,1[
with the same accuracy as
N
(whose mantissa has the same number of significant words).
@3* t_INTMOD
: returns a random intmod for the same modulus.
@3* t_FFELT
: returns a random element in the same finite field.
@3* t_VEC
of length 2
, N = [a,b]
: returns an integer uniformly
distributed between a
and b
.
@3* t_VEC
generated by ellinit
over a finite field k
(coefficients are t_INTMOD
s modulo a prime or t_FFELT
s): returns a
``random'' k
-rational affine point on the curve. More precisely
if the curve has a single point (at infinity!) we return it; otherwise
we return an affine point by drawing an abscissa uniformly at
random until ellordinate
succeeds. Note that this is definitely not a
uniform distribution over E(k)
, but it should be good enough for
applications.
@3* t_POL
return a random polynomial of degree at most the degree of N
.
The coefficients are drawn by applying random
to the leading
coefficient of N
.
? random(10) %1 = 9 ? random(Mod(0,7)) %2 = Mod(1, 7) ? a = ffgen(ffinit(3,7), 'a); random(a) %3 = a^6 + 2*a^5 + a^4 + a^3 + a^2 + 2*a ? E = ellinit([3,7]*Mod(1,109)); random(E) %4 = [Mod(103, 109), Mod(10, 109)] ? E = ellinit([1,7]*a^0); random(E) %5 = [a^6 + a^5 + 2*a^4 + 2*a^2, 2*a^6 + 2*a^4 + 2*a^3 + a^2 + 2*a] ? random(Mod(1,7)*x^4) %6 = Mod(5, 7)*x^4 + Mod(6, 7)*x^3 + Mod(2, 7)*x^2 + Mod(2, 7)*x + Mod(5, 7)
These variants all depend on a single internal generator, and are
independent from your operating system's random number generators.
A random seed may be obtained via getrand
, and reset
using setrand
: from a given seed, and given sequence of random
s,
the exact same values will be generated. The same seed is used at each
startup, reseed the generator yourself if this is a problem. Note that
internal functions also call the random number generator; adding such a
function call in the middle of your code will change the numbers produced.
@3Technical note.
Up to
version 2.4 included, the internal generator produced pseudo-random numbers
by means of linear congruences, which were not well distributed in arithmetic
progressions. We now
use Brent's XORGEN algorithm, based on Feedback Shift Registers, see
brent/random.html>http://wwwmaths.anu.edu.au/~brent/random.html. The generator has period
2^{4096}-1
, passes the Crush battery of statistical tests of L'Ecuyer and
Simard, but is not suitable for cryptographic purposes: one can reconstruct
the state vector from a small sample of consecutive values, thus predicting
the entire sequence.
The library syntax is GEN
genrand(GEN N = NULL)
.
Also available: GEN
ellrandom(GEN E)
and GEN
ffrandom(GEN a)
.
(x)
Real part of x
. In the case where x
is a quadratic number, this is the
coefficient of 1
in the ``canonical'' integral basis (1,
omega)
.
The library syntax is GEN
greal(GEN x)
.
(x,{&e})
If x
is in R, rounds x
to the nearest integer (rounding to
+ oo
in case of ties), then and sets e
to the number of error bits,
that is the binary exponent of the difference between the original and the
rounded value (the ``fractional part''). If the exponent of x
is too large
compared to its precision (i.e. e > 0
), the result is undefined and an error
occurs if e
was not given.
@3Important remark. Contrary to the other truncation functions, this function operates on every coefficient at every level of a PARI object. For example
truncate((2.4*X^2-1.7)/(X)) = 2.4*X,
whereas
round((2.4*X^2-1.7)/(X)) = (2*X^2-2)/(X).
An important use of round
is to get exact results after an approximate
computation, when theory tells you that the coefficients must be integers.
The library syntax is GEN
round0(GEN x, GEN *e = NULL)
.
Also available are GEN
grndtoi(GEN x, long *e)
and
GEN
ground(GEN x)
.
(x,v)
Returns the absolute precision of x
with respec to power series
in the variable v
; this is the
minimum precision of the components of x
. The result is +oo
if x
is an exact object (as a series in v
):
? serprec(x + O(y^2), y) %1 = 2 ? serprec(x + 2, x) %2 = +oo ? serprec(2 + x + O(x^2), y) %3 = +oo
The library syntax is GEN
gpserprec(GEN x, long v)
where v
is a variable number.
Also available is long
serprec(GEN x, GEN p)
, which returns
LONG_MAX
if x = 0
and the series precision as a long
integer.
(x)
This function simplifies x
as much as it can. Specifically, a complex or
quadratic number whose imaginary part is the integer 0 (i.e. not Mod(0,2)
or 0.E-28
) is converted to its real part, and a polynomial of degree 0
is converted to its constant term. Simplifications occur recursively.
This function is especially useful before using arithmetic functions, which expect integer arguments:
? x = 2 + y - y %1 = 2 ? isprime(x) *** at top-level: isprime(x) *** ^---------- *** isprime: not an integer argument in an arithmetic function ? type(x) %2 = "t_POL" ? type(simplify(x)) %3 = "t_INT"
Note that GP results are simplified as above before they are stored in the
history. (Unless you disable automatic simplification with \y
, that is.)
In particular
? type(%1) %4 = "t_INT"
The library syntax is GEN
simplify(GEN x)
.
(x)
Outputs the total number of bytes occupied by the tree representing the
PARI object x
.
The library syntax is long
gsizebyte(GEN x)
.
Also available is long
gsizeword(GEN x)
returning a
number of words.
(x)
This function is DEPRECATED, essentially meaningless, and provided for backwards compatibility only. Don't use it!
outputs a quick upper bound for the number of decimal digits of (the
components of) x
, off by at most 1
. More precisely, for a positive
integer x
, it computes (approximately) the ceiling of
floor(1 +
log _2 x)
log _{10}2,
To count the number of decimal digits of a positive integer x
, use
#digits(x)
. To estimate (recursively) the size of x
, use
normlp(x)
.
The library syntax is long
sizedigit(GEN x)
.
(x,{&e})
Truncates x
and sets e
to the number of
error bits. When x
is in R, this means that the part after the decimal
point is chopped away, e
is the binary exponent of the difference between
the original and the truncated value (the ``fractional part''). If the
exponent of x
is too large compared to its precision (i.e. e > 0
), the
result is undefined and an error occurs if e
was not given. The function
applies componentwise on vector / matrices; e
is then the maximal number of
error bits. If x
is a rational function, the result is the ``integer part''
(Euclidean quotient of numerator by denominator) and e
is not set.
Note a very special use of truncate
: when applied to a power series, it
transforms it into a polynomial or a rational function with denominator
a power of X
, by chopping away the O(X^k)
. Similarly, when applied to
a p
-adic number, it transforms it into an integer or a rational number
by chopping away the O(p^k)
.
The library syntax is GEN
trunc0(GEN x, GEN *e = NULL)
.
The following functions are also available: GEN
gtrunc(GEN x)
and GEN
gcvtoi(GEN x, long *e)
.
(x,p)
Computes the highest
exponent of p
dividing x
. If p
is of type integer, x
must be an
integer, an intmod whose modulus is divisible by p
, a fraction, a
q
-adic number with q = p
, or a polynomial or power series in which case the
valuation is the minimum of the valuation of the coefficients.
If p
is of type polynomial, x
must be of type polynomial or rational
function, and also a power series if x
is a monomial. Finally, the
valuation of a vector, complex or quadratic number is the minimum of the
component valuations.
If x = 0
, the result is +oo
if x
is an exact object. If x
is a
p
-adic numbers or power series, the result is the exponent of the zero.
Any other type combinations gives an error.
The library syntax is GEN
gpvaluation(GEN x, GEN p)
.
Also available is
long
gvaluation(GEN x, GEN p)
, which returns LONG_MAX
if x = 0
and the valuation as a long
integer.
(
name,{v})
Return a variable name whose priority is higher
than the priority of v
(of all existing variables if v
is omitted).
This is a counterpart to varlower
.
? Pol([x,x], t) *** at top-level: Pol([x,x],t) *** ^------------ *** Pol: incorrect priority in gtopoly: variable x <= t ? t = varhigher("t", x); ? Pol([x,x], t) %3 = x*t + x
@3This routine is useful since new GP variables directly created by the interpreter always have lower priority than existing GP variables. When some basic objects already exist in a variable that is incompatible with some function requirement, you can now create a new variable with a suitable priority instead of changing variables in existing objects:
? K = nfinit(x^2+1); ? rnfequation(K,y^2-2) *** at top-level: rnfequation(K,y^2-2) *** ^-------------------- *** rnfequation: incorrect priority in rnfequation: variable y >= x ? y = varhigher("y", x); ? rnfequation(K, y^2-2) %3 = y^4 - 2*y^2 + 9
@3Caution 1. The name is an arbitrary character string, only used for display purposes and need not be related to the GP variable holding the result, nor to be a valid variable name. In particular the name can not be used to retrieve the variable, it is not even present in the parser's hash tables.
? x = varhigher("#"); ? x^2 %2 = #^2
@3Caution 2. There are a limited number of variables and if no
existing variable with the given display name has the requested
priority, the call to varhigher
uses up one such slot. Do not create
new variables in this way unless it's absolutely necessary,
reuse existing names instead and choose sensible priority requirements:
if you only need a variable with higher priority than x
, state so
rather than creating a new variable with highest priority.
\\ quickly use up all variables ? n = 0; while(1,varhigher("tmp"); n++) *** at top-level: n=0;while(1,varhigher("tmp");n++) *** ^------------------- *** varhigher: no more variables available. *** Break loop: type 'break' to go back to GP prompt break> n 65510 \\ infinite loop: here we reuse the same 'tmp' ? n = 0; while(1,varhigher("tmp", x); n++)
The library syntax is GEN
varhigher(const char *name, long v = -1)
where v
is a variable number.
({x})
Gives the main variable of the object x
(the variable with the highest
priority used in x
), and p
if x
is a p
-adic number. Return 0
if
x
has no variable attached to it.
? variable(x^2 + y) %1 = x ? variable(1 + O(5^2)) %2 = 5 ? variable([x,y,z,t]) %3 = x ? variable(1) %4 = 0
@3The construction
if (!variable(x),...)
@3can be used to test whether a variable is attached to x
.
If x
is omitted, returns the list of user variables known to the
interpreter, by order of decreasing priority. (Highest priority is initially
x
, which come first until varhigher
is used.) If varhigher
or varlower
are used, it is quite possible to end up with different
variables (with different priorities) printed in the same way: they
will then appear multiple times in the output:
? varhigher("y"); ? varlower("y"); ? variable() %4 = [y, x, y]
@3Using v = variable()
then v[1]
, v[2]
,
etc. allows to recover and use existing variables.
The library syntax is GEN
gpolvar(GEN x = NULL)
.
However, in library mode, this function should not be used for x
non-NULL
, since gvar
is more appropriate. Instead, for
x
a p
-adic (type t_PADIC
), p
is gel(x,2)
; otherwise, use
long
gvar(GEN x)
which returns the variable number of x
if
it exists, NO_VARIABLE
otherwise, which satisfies the property
varncmp(NO_VARIABLE, v) > 0
for all valid variable number
v
, i.e. it has lower priority than any variable.
({x})
Returns the list of all variables occuring in object x
(all user
variables known to the interpreter if x
is omitted), sorted by
decreasing priority.
? variables([x^2 + y*z + O(t), a+x]) %1 = [x, y, z, t, a]
@3The construction
if (!variables(x),...)
@3can be used to test whether a variable is attached to x
.
If varhigher
or varlower
are used, it is quite possible to end up
with different variables (with different priorities) printed in the same
way: they will then appear multiple times in the output:
? y1 = varhigher("y"); ? y2 = varlower("y"); ? variables(y*y1*y2) %4 = [y, y, y]
The library syntax is GEN
variables_vec(GEN x = NULL)
.
Also available is GEN
variables_vecsmall(GEN x)
which returns
the (sorted) variable numbers instead of the attached monomials of degree 1.
(
name,{v})
Return a variable name whose priority is lower
than the priority of v
(of all existing variables if v
is omitted).
This is a counterpart to varhigher
.
New GP variables directly created by the interpreter always have lower priority than existing GP variables, but it is not easy to check whether an identifier is currently unused, so that the corresponding variable has the expected priority when it's created! Thus, depending on the session history, the same command may fail or succeed:
? t; z; \\ now t > z ? rnfequation(t^2+1,z^2-t) *** at top-level: rnfequation(t^2+1,z^ *** ^-------------------- *** rnfequation: incorrect priority in rnfequation: variable t >= t
@3Restart and retry:
? z; t; \\ now z > t ? rnfequation(t^2+1,z^2-t) %2 = z^4 + 1
@3It is quite annoying for package authors, when trying to define a base ring, to notice that the package may fail for some users depending on their session history. The safe way to do this is as follows:
? z; t; \\ In new session: now z > t ... ? t = varlower("t", 'z); ? rnfequation(t^2+1,z^2-2) %2 = z^4 - 2*z^2 + 9 ? variable() %3 = [x, y, z, t]
? t; z; \\ In new session: now t > z ... ? t = varlower("t", 'z); \\ create a new variable, still printed "t" ? rnfequation(t^2+1,z^2-2) %2 = z^4 - 2*z^2 + 9 ? variable() %3 = [x, y, t, z, t]
@3Now both constructions succeed. Note that in the
first case, varlower
is essentially a no-op, the existing variable t
has correct priority. While in the second case, two different variables are
displayed as t
, one with higher priority than z
(created in the first
line) and another one with lower priority (created by varlower
).
@3Caution 1. The name is an arbitrary character string, only used for display purposes and need not be related to the GP variable holding the result, nor to be a valid variable name. In particular the name can not be used to retrieve the variable, it is not even present in the parser's hash tables.
? x = varlower("#"); ? x^2 %2 = #^2
@3Caution 2. There are a limited number of variables and if no
existing variable with the given display name has the requested
priority, the call to varlower
uses up one such slot. Do not create
new variables in this way unless it's absolutely necessary,
reuse existing names instead and choose sensible priority requirements:
if you only need a variable with higher priority than x
, state so
rather than creating a new variable with highest priority.
\\ quickly use up all variables ? n = 0; while(1,varlower("x"); n++) *** at top-level: n=0;while(1,varlower("x");n++) *** ^------------------- *** varlower: no more variables available. *** Break loop: type 'break' to go back to GP prompt break> n 65510 \\ infinite loop: here we reuse the same 'tmp' ? n = 0; while(1,varlower("tmp", x); n++)
The library syntax is GEN
varlower(const char *name, long v = -1)
where v
is a variable number.
Since the values of transcendental functions cannot be exactly represented,
these functions will always return an inexact object: a real number,
a complex number, a p
-adic number or a power series. All these objects
have a certain finite precision.
As a general rule, which of course in some cases may have exceptions, transcendental functions operate in the following way:
@3* If the argument is either a real number or an inexact complex number
(like 1.0 + I
or Pi*I
but not 2 - 3*I
), then the
computation is done with the precision of the argument.
In the example below, we see that changing the precision to 50
digits does
not matter, because x
only had a precision of 19
digits.
? \p 15 realprecision = 19 significant digits (15 digits displayed) ? x = Pi/4 %1 = 0.785398163397448 ? \p 50 realprecision = 57 significant digits (50 digits displayed) ? sin(x) %2 = 0.7071067811865475244
Note that even if the argument is real, the result may be complex
(e.g. acos(2.0)
or acosh(0.0)
). See each individual
function help for the definition of the branch cuts and choice of principal
value.
@3* If the argument is either an integer, a rational, an exact complex
number or a quadratic number, it is first converted to a real
or complex number using the current precision, which can be
view and manipulated using the defaults realprecision
(in decimal
digits) or realbitprecision
(in bits). This precision can be changed
indifferently
@3* in decimal digits: use \p
or default(realprecision,...)
.
@3* in bits: use \pb
or default(realbitprecision,...)
.
After this conversion, the computation proceeds as above for real or complex arguments.
In library mode, the realprecision
does not matter; instead the
precision is taken from the prec
parameter which every transcendental
function has. As in gp
, this prec
is not used when the argument
to a function is already inexact. Note that the argument prec stands
for the length in words of a real number, including codewords. Hence we must
have prec >= 3
. (Some functions allow a bitprec
argument
instead which allow finer granularity.)
Some accuracies attainable on 32-bit machines cannot be attained
on 64-bit machines for parity reasons. For example the default gp
accuracy
is 28 decimal digits on 32-bit machines, corresponding to prec having
the value 5, but this cannot be attained on 64-bit machines.
@3* If the argument is a polmod (representing an algebraic number),
then the function is evaluated for every possible complex embedding of that
algebraic number. A column vector of results is returned, with one component
for each complex embedding. Therefore, the number of components equals
the degree of the t_POLMOD
modulus.
@3* If the argument is an intmod or a p
-adic, at present only a
few functions like sqrt
(square root), sqr
(square), log
,
exp
, powering, teichmuller
(Teichmüller character) and
agm
(arithmetic-geometric mean) are implemented.
Note that in the case of a 2
-adic number, sqr(x)
may not be
identical to x*x
: for example if x = 1+O(2^5)
and y = 1+O(2^5)
then
x*y = 1+O(2^5)
while sqr(x) = 1+O(2^6)
. Here, x * x
yields the
same result as sqr(x)
since the two operands are known to be
identical. The same statement holds true for p
-adics raised to the
power n
, where v_p(n) > 0
.
@3Remark. If we wanted to be strictly consistent with
the PARI philosophy, we should have x*y = (4 mod 8)
and sqr(x) =
(4 mod 32)
when both x
and y
are congruent to 2
modulo 4
.
However, since intmod is an exact object, PARI assumes that the modulus
must not change, and the result is hence (0 mod 4)
in both cases. On
the other hand, p
-adics are not exact objects, hence are treated
differently.
@3* If the argument is a polynomial, a power series or a rational function,
it is, if necessary, first converted to a power series using the current
series precision, held in the default seriesprecision
. This precision
(the number of significant terms) can be changed using \ps
or
default(seriesprecision,...)
. Then the Taylor series expansion of the
function around X = 0
(where X
is the main variable) is computed to a
number of terms depending on the number of terms of the argument and the
function being computed.
Under gp
this again is transparent to the user. When programming in
library mode, however, it is strongly advised to perform an explicit
conversion to a power series first, as in x = gtoser(x, seriesprec)
,
where the number of significant terms seriesprec
can be specified
explicitly. If you do not do this, a global variable precdl
is used
instead, to convert polynomials and rational functions to a power series with
a reasonable number of terms; tampering with the value of this global
variable is deprecated and strongly discouraged.
@3* If the argument is a vector or a matrix, the result is the componentwise evaluation of the function. In particular, transcendental functions on square matrices, which are not implemented in the present version 2.9.1, will have a different name if they are implemented some day.
\subseckbd{^} If y
is not of type integer, x^y
has the same
effect as exp(y*log(x))
. It can be applied to p
-adic numbers as well
as to the more usual types.
The library syntax is GEN
gpow(GEN x, GEN n, long prec)
for x^n
.
Catalan's constant G =
sum_{n >= 0}((-1)^n)/((2n+1)^2) = 0.91596...
.
Note that Catalan
is one of the few reserved names which cannot be
used for user variables.
The library syntax is GEN
mpcatalan(long prec)
.
Euler's constant gamma = 0.57721...
. Note that
Euler
is one of the few reserved names which cannot be used for
user variables.
The library syntax is GEN
mpeuler(long prec)
.
The complex number sqrt {-1}
.
The library syntax is GEN
gen_I()
.
The constant pi (3.14159...
). Note that Pi
is one of the few
reserved names which cannot be used for user variables.
The library syntax is GEN
mppi(long prec)
.
(x)
Absolute value of x
(modulus if x
is complex).
Rational functions are not allowed. Contrary to most transcendental
functions, an exact argument is not converted to a real number before
applying abs
and an exact result is returned if possible.
? abs(-1) %1 = 1 ? abs(3/7 + 4/7*I) %2 = 5/7 ? abs(1 + I) %3 = 1.414213562373095048801688724
If x
is a polynomial, returns -x
if the leading coefficient is
real and negative else returns x
. For a power series, the constant
coefficient is considered instead.
The library syntax is GEN
gabs(GEN x, long prec)
.
(x)
Principal branch of cos ^{-1}(x) = -i
log (x + i
sqrt {1-x^2})
.
In particular, Re (acos(x))\in [0,
pi]
and if x\in
R and |x| > 1
,
then acos(x)
is complex. The branch cut is in two pieces:
]- oo ,-1]
, continuous with quadrant II, and [1,+ oo [
, continuous
with quadrant IV. We have acos(x) =
pi/2 - asin(x)
for all
x
.
The library syntax is GEN
gacos(GEN x, long prec)
.
(x)
Principal branch of cosh ^{-1}(x) = 2
log (
sqrt {(x+1)/2} +
sqrt {(x-1)/2})
. In particular,
Re (acosh(x)) >= 0
and
Im (acosh(x))\in ]-
pi,
pi]
; if x\in
R and x < 1
, then
acosh(x)
is complex.
The library syntax is GEN
gacosh(GEN x, long prec)
.
(x,y)
Arithmetic-geometric mean of x
and y
. In the
case of complex or negative numbers, the optimal AGM is returned
(the largest in absolute value over all choices of the signs of the square
roots). p
-adic or power series arguments are also allowed. Note that
a p
-adic agm exists only if x/y
is congruent to 1 modulo p
(modulo
16 for p = 2
). x
and y
cannot both be vectors or matrices.
The library syntax is GEN
agm(GEN x, GEN y, long prec)
.
(x)
Argument of the complex number x
, such that -
pi <
arg (x) <=
pi.
The library syntax is GEN
garg(GEN x, long prec)
.
(x)
Principal branch of sin ^{-1}(x) = -i
log (ix +
sqrt {1 - x^2})
.
In particular, Re (asin(x))\in [-
pi/2,
pi/2]
and if x\in
R and
|x| > 1
then asin(x)
is complex. The branch cut is in two pieces:
]- oo ,-1]
, continuous with quadrant II, and [1,+ oo [
continuous
with quadrant IV. The function satisfies i asin(x) =
asinh(ix)
.
The library syntax is GEN
gasin(GEN x, long prec)
.
(x)
Principal branch of sinh ^{-1}(x) =
log (x +
sqrt {1+x^2})
. In
particular Im (asinh(x))\in [-
pi/2,
pi/2]
.
The branch cut is in two pieces: ]-i oo ,-i]
, continuous with quadrant
III and [+i,+i oo [
, continuous with quadrant I.
The library syntax is GEN
gasinh(GEN x, long prec)
.
(x)
Principal branch of tan^{-1}(x) =
log ((1+ix)/(1-ix)) /
2i
. In particular the real part of atan(x)
belongs to
]-
pi/2,
pi/2[
.
The branch cut is in two pieces:
]-i oo ,-i[
, continuous with quadrant IV, and ]i,+i oo [
continuous
with quadrant II. The function satisfies atan(x) =
-iatanh(ix)
for all x != +- i
.
The library syntax is GEN
gatan(GEN x, long prec)
.
(x)
Principal branch of tanh^{-1}(x) =
log ((1+x)/(1-x)) / 2
. In
particular the imaginary part of atanh(x)
belongs to
[-
pi/2,
pi/2]
; if x\in
R and |x| > 1
then atanh(x)
is complex.
The library syntax is GEN
gatanh(GEN x, long prec)
.
(x)
Bernoulli number B_x
,
where B_0 = 1
, B_1 = -1/2
, B_2 = 1/6
,..., expressed as a rational number.
The argument x
should be of type integer.
The library syntax is GEN
bernfrac(long x)
.
(n, {v = 'x})
Bernoulli polynomial B_n
in variable v
.
? bernpol(1) %1 = x - 1/2 ? bernpol(3) %2 = x^3 - 3/2*x^2 + 1/2*x
The library syntax is GEN
bernpol(long n, long v = -1)
where v
is a variable number.
(x)
Bernoulli number
B_x
, as bernfrac
, but B_x
is returned as a real number
(with the current precision).
The library syntax is GEN
bernreal(long x, long prec)
.
(x)
This routine is obsolete, kept for backward compatibility only.
The library syntax is GEN
bernvec(long x)
.
(
nu,x)
H^1
-Bessel function of index nu and argument x
.
The library syntax is GEN
hbessel1(GEN nu, GEN x, long prec)
.
(
nu,x)
H^2
-Bessel function of index nu and argument x
.
The library syntax is GEN
hbessel2(GEN nu, GEN x, long prec)
.
(
nu,x)
I
-Bessel function of index nu and
argument x
. If x
converts to a power series, the initial factor
(x/2)^
nu/
Gamma(
nu+1)
is omitted (since it cannot be represented in PARI
when nu is not integral).
The library syntax is GEN
ibessel(GEN nu, GEN x, long prec)
.
(
nu,x)
J
-Bessel function of index nu and
argument x
. If x
converts to a power series, the initial factor
(x/2)^
nu/
Gamma(
nu+1)
is omitted (since it cannot be represented in PARI
when nu is not integral).
The library syntax is GEN
jbessel(GEN nu, GEN x, long prec)
.
(n,x)
J
-Bessel function of half integral index.
More precisely, besseljh(n,x)
computes J_{n+1/2}(x)
where n
must be of type integer, and x
is any element of C. In the
present version 2.9.1, this function is not very accurate when x
is small.
The library syntax is GEN
jbesselh(GEN n, GEN x, long prec)
.
(
nu,x)
K
-Bessel function of index nu and argument x
.
The library syntax is GEN
kbessel(GEN nu, GEN x, long prec)
.
(
nu,x)
N
-Bessel function of index nu and argument x
.
The library syntax is GEN
nbessel(GEN nu, GEN x, long prec)
.
(x)
Cosine of x
.
The library syntax is GEN
gcos(GEN x, long prec)
.
(x)
Hyperbolic cosine of x
.
The library syntax is GEN
gcosh(GEN x, long prec)
.
(x)
Cotangent of x
.
The library syntax is GEN
gcotan(GEN x, long prec)
.
(x)
Hyperbolic cotangent of x
.
The library syntax is GEN
gcotanh(GEN x, long prec)
.
(x)
Principal branch of the dilogarithm of x
,
i.e. analytic continuation of the power series log _2(x) =
sum_{n >= 1}x^n/n^2
.
The library syntax is GEN
dilog(GEN x, long prec)
.
(x,{n})
Exponential integral int_x^ oo (e^{-t})/(t)dt =
incgam(0, x)
, where the latter expression extends the function
definition from real x > 0
to all complex x != 0
.
If n
is present, we must have x > 0
; the function returns the
n
-dimensional vector [eint1(x),...,eint1(nx)]
. Contrary to
other transcendental functions, and to the default case (n
omitted), the
values are correct up to a bounded absolute, rather than relative,
error 10^{-n}
, where n
is precision
(x)
if x
is a t_REAL
and defaults to realprecision
otherwise. (In the most important
application, to the computation of L
-functions via approximate functional
equations, those values appear as weights in long sums and small individual
relative errors are less useful than controlling the absolute error.) This is
faster than repeatedly calling eint1(i * x)
, but less precise.
The library syntax is GEN
veceint1(GEN x, GEN n = NULL, long prec)
.
Also available is GEN
eint1(GEN x, long prec)
.
(x)
Complementary error function, analytic continuation of
(2/
sqrt pi)
int_x^ oo e^{-t^2}dt = incgam(1/2,x^2)/
sqrt pi,
where the latter expression extends the function definition from real x
to
all complex x != 0
.
The library syntax is GEN
gerfc(GEN x, long prec)
.
(z,{
flag = 0})
Variants of Dedekind's eta function.
If flag = 0
, return prod_{n = 1}^ oo (1-q^n)
, where q
depends on x
in the following way:
@3* q = e^{2i
pi x}
if x
is a complex number (which must then
have positive imaginary part); notice that the factor q^{1/24}
is
missing!
@3* q = x
if x
is a t_PADIC
, or can be converted to a
power series (which must then have positive valuation).
If flag is non-zero, x
is converted to a complex number and we return the
true eta function, q^{1/24}
prod_{n = 1}^ oo (1-q^n)
,
where q = e^{2i
pi x}
.
The library syntax is GEN
eta0(GEN z, long flag, long prec)
.
Also available is GEN
trueeta(GEN x, long prec)
(flag = 1
).
(x)
Exponential of x
.
p
-adic arguments with positive valuation are accepted.
The library syntax is GEN
gexp(GEN x, long prec)
.
For a t_PADIC
x
, the function
GEN
Qp_exp(GEN x)
is also available.
(x)
Return exp (x)-1
, computed in a way that is also accurate
when the real part of x
is near 0
.
A naive direct computation would suffer from catastrophic cancellation;
PARI's direct computation of exp (x)
alleviates this well known problem at
the expense of computing exp (x)
to a higher accuracy when x
is small.
Using expm1
is recommended instead:
? default(realprecision, 10000); x = 1e-100; ? a = expm1(x); time = 4 ms. ? b = exp(x)-1; time = 28 ms. ? default(realprecision, 10040); x = 1e-100; ? c = expm1(x); \\ reference point ? abs(a-c)/c \\ relative error in expm1(x) %7 = 0.E-10017 ? abs(b-c)/c \\ relative error in exp(x)-1 %8 = 1.7907031188259675794 E-9919
@3As the example above shows, when x
is near 0
,
expm1
is both faster and more accurate than exp(x)-1
.
The library syntax is GEN
gexpm1(GEN x, long prec)
.
(s)
For s
a complex number, evaluates Euler's gamma
function
Gamma(s) =
int_0^ oo t^{s-1}
exp (-t)dt.
Error if s
is a non-positive integer, where Gamma has a pole.
For s
a t_PADIC
, evaluates the Morita gamma function at s
, that
is the unique continuous p
-adic function on the p
-adic integers
extending Gamma_p(k) = (-1)^k
prod_{j < k}'j
, where the prime means that p
does not divide j
.
? gamma(1/4 + O(5^10)) %1= 1 + 4*5 + 3*5^4 + 5^6 + 5^7 + 4*5^9 + O(5^10) ? algdep(%,4) %2 = x^4 + 4*x^2 + 5
The library syntax is GEN
ggamma(GEN s, long prec)
.
For a t_PADIC
x
, the function GEN
Qp_gamma(GEN x)
is
also available.
(x)
Gamma function evaluated at the argument x+1/2
.
The library syntax is GEN
ggammah(GEN x, long prec)
.
(G,t,{m = 0})
Returns the value at t
of the inverse Mellin transform
G
initialized by gammamellininvinit
.
? G = gammamellininvinit([0]); ? gammamellininv(G, 2) - 2*exp(-Pi*2^2) %2 = -4.484155085839414627 E-44
The alternative shortcut
gammamellininv(A,t,m)
@3for
gammamellininv(gammamellininvinit(A,m), t)
@3is available.
The library syntax is GEN
gammamellininv(GEN G, GEN t, long m, long bitprec)
.
(A,n,{m = 0})
Return the first n
terms of the asymptotic expansion at infinity
of the m
-th derivative K^{(m)}(t)
of the inverse Mellin transform of the
function
f(s) =
Gamma_
R(s+a_1)...
Gamma_
R(s+a_d) ,
where A
is the vector [a_1,...,a_d]
and
Gamma_
R(s) =
pi^{-s/2}
Gamma(s/2)
(Euler's gamma
).
The result is a vector
[M[1]...M[n]]
with M[1] = 1, such that
K^{(m)}(t) =
sqrt {2^{d+1}/d}t^{a+m(2/d-1)}e^{-d
pi t^{2/d}}
sum_{n >= 0} M[n+1] (
pi t^{2/d})^{-n}
with a = (1-d+
sum_{1 <= j <= d}a_j)/d
.
The library syntax is GEN
gammamellininvasymp(GEN A, long precdl, long n)
.
(A,{m = 0})
Initialize data for the computation by gammamellininv
of
the m
-th derivative of the inverse Mellin transform of the function
f(s) =
Gamma_
R(s+a_1)...
Gamma_
R(s+a_d)
where A
is the vector [a_1,...,a_d]
and
Gamma_
R(s) =
pi^{-s/2}
Gamma(s/2)
(Euler's gamma
). This is the
special case of Meijer's G
functions used to compute L
-values via the
approximate functional equation.
@3Caveat. Contrary to the PARI convention, this function guarantees an absolute (rather than relative) error bound.
For instance, the inverse Mellin transform of Gamma_
R(s)
is
2
exp (-
pi z^2)
:
? G = gammamellininvinit([0]); ? gammamellininv(G, 2) - 2*exp(-Pi*2^2) %2 = -4.484155085839414627 E-44
The inverse Mellin transform of Gamma_
R(s+1)
is
2 z
exp (-
pi z^2)
, and its second derivative is
4
pi z
exp (-
pi z^2)(2
pi z^2 - 3)
:
? G = gammamellininvinit([1], 2); ? a(z) = 4*Pi*z*exp(-Pi*z^2)*(2*Pi*z^2-3); ? b(z) = gammamellininv(G,z); ? t(z) = b(z) - a(z); ? t(3/2) %3 = -1.4693679385278593850 E-39
The library syntax is GEN
gammamellininvinit(GEN A, long m, long bitprec)
.
(a,b,x)
U
-confluent hypergeometric function with
parameters a
and b
. The parameters a
and b
can be complex but
the present implementation requires x
to be positive.
The library syntax is GEN
hyperu(GEN a, GEN b, GEN x, long prec)
.
(s,x,{g})
Incomplete gamma function int_x^ oo e^{-t}t^{s-1}dt
, extended by
analytic continuation to all complex x, s
not both 0
. The relative error
is bounded in terms of the precision of s
(the accuracy of x
is ignored
when determining the output precision). When g
is given, assume that
g =
Gamma(s)
. For small |x|
, this will speed up the computation.
The library syntax is GEN
incgam0(GEN s, GEN x, GEN g = NULL, long prec)
.
Also available is GEN
incgam(GEN s, GEN x, long prec)
.
(s,x)
Complementary incomplete gamma function.
The arguments x
and s
are complex numbers such that s
is not a pole of
Gamma and |x|/(|s|+1)
is not much larger than 1 (otherwise the
convergence is very slow). The result returned is int_0^x
e^{-t}t^{s-1}dt
.
The library syntax is GEN
incgamc(GEN s, GEN x, long prec)
.
(y)
Lambert W
function, solution of the implicit equation xe^x = y
,
for y > 0
.
The library syntax is GEN
glambertW(GEN y, long prec)
.
(x)
Principal branch of the logarithm of the gamma function of x
. This
function is analytic on the complex plane with non-positive integers
removed, and can have much larger arguments than gamma
itself.
For x
a power series such that x(0)
is not a pole of gamma
,
compute the Taylor expansion. (PARI only knows about regular power series
and can't include logarithmic terms.)
? lngamma(1+x+O(x^2)) %1 = -0.57721566490153286060651209008240243104*x + O(x^2) ? lngamma(x+O(x^2)) *** at top-level: lngamma(x+O(x^2)) *** ^----------------- *** lngamma: domain error in lngamma: valuation != 0 ? lngamma(-1+x+O(x^2)) *** lngamma: Warning: normalizing a series with 0 leading term. *** at top-level: lngamma(-1+x+O(x^2)) *** ^-------------------- *** lngamma: domain error in intformal: residue(series, pole) != 0
The library syntax is GEN
glngamma(GEN x, long prec)
.
(x)
Principal branch of the natural logarithm of
x \in
C^*
, i.e. such that Im (
log (x))\in ]-
pi,
pi]
.
The branch cut lies
along the negative real axis, continuous with quadrant 2, i.e. such that
lim _{b\to 0^+}
log (a+bi) =
log a
for a \in
R^*
. The result is complex
(with imaginary part equal to pi) if x\in
R and x < 0
. In general,
the algorithm uses the formula
log (x) ~ (
pi)/(2agm(1, 4/s)) - m
log 2,
if s = x 2^m
is large enough. (The result is exact to B
bits provided
s > 2^{B/2}
.) At low accuracies, the series expansion near 1
is used.
p
-adic arguments are also accepted for x
, with the convention that
log (p) = 0
. Hence in particular exp (
log (x))/x
is not in general equal to
1 but to a (p-1)
-th root of unity (or +-1
if p = 2
) times a power of p
.
The library syntax is GEN
glog(GEN x, long prec)
.
For a t_PADIC
x
, the function
GEN
Qp_log(GEN x)
is also available.
(m,x,{
flag = 0})
One of the different polylogarithms, depending on flag:
If flag = 0
or is omitted: m-th
polylogarithm of x
, i.e. analytic
continuation of the power series Li_m(x) =
sum_{n >= 1}x^n/n^m
(x < 1
). Uses the functional equation linking the values at x
and 1/x
to restrict to the case |x| <= 1
, then the power series when
|x|^2 <= 1/2
, and the power series expansion in log (x)
otherwise.
Using flag, computes a modified m-th
polylogarithm of x
.
We use Zagier's notations; let Re _m
denote Re or Im depending
on whether m
is odd or even:
If flag = 1
: compute ~ D_m(x)
, defined for |x| <= 1
by
Re _m(
sum_{k = 0}^{m-1} ((-
log |x|)^k)/(k!)Li_{m-k}(x)
+((-
log |x|)^{m-1})/(m!)
log |1-x|).
If flag = 2
: compute D_m(x)
, defined for |x| <= 1
by
Re _m(
sum_{k = 0}^{m-1}((-
log |x|)^k)/(k!)Li_{m-k}(x)
-(1)/(2)((-
log |x|)^m)/(m!)).
If flag = 3
: compute P_m(x)
, defined for |x| <= 1
by
Re _m(
sum_{k = 0}^{m-1}(2^kB_k)/(k!)(
log |x|)^kLi_{m-k}(x)
-(2^{m-1}B_m)/(m!)(
log |x|)^m).
These three functions satisfy the functional equation
f_m(1/x) = (-1)^{m-1}f_m(x)
.
The library syntax is GEN
polylog0(long m, GEN x, long flag, long prec)
.
Also available is
GEN
gpolylog(long m, GEN x, long prec)
(flag = 0).
(x)
The psi-function of x
, i.e. the logarithmic derivative
Gamma'(x)/
Gamma(x)
.
The library syntax is GEN
gpsi(GEN x, long prec)
.
(x)
Sine of x
.
The library syntax is GEN
gsin(GEN x, long prec)
.
(x)
Cardinal sine of x
, i.e. sin (x)/x
if x != 0
, 1
otherwise.
Note that this function also allows to compute
(1-
cos (x)) / x^2 = sinc(x/2)^2 / 2
accurately near x = 0
.
The library syntax is GEN
gsinc(GEN x, long prec)
.
(x)
Hyperbolic sine of x
.
The library syntax is GEN
gsinh(GEN x, long prec)
.
(x)
Square of x
. This operation is not completely
straightforward, i.e. identical to x * x
, since it can usually be
computed more efficiently (roughly one-half of the elementary
multiplications can be saved). Also, squaring a 2
-adic number increases
its precision. For example,
? (1 + O(2^4))^2 %1 = 1 + O(2^5) ? (1 + O(2^4)) * (1 + O(2^4)) %2 = 1 + O(2^4)
Note that this function is also called whenever one multiplies two objects which are known to be identical, e.g. they are the value of the same variable, or we are computing a power.
? x = (1 + O(2^4)); x * x %3 = 1 + O(2^5) ? (1 + O(2^4))^4 %4 = 1 + O(2^6)
(note the difference between %2
and %3
above).
The library syntax is GEN
gsqr(GEN x)
.
(x)
Principal branch of the square root of x
, defined as sqrt {x} =
exp (
log x / 2)
. In particular, we have
Arg(sqrt(x))\in ]-
pi/2,
pi/2]
, and if x\in
R and x < 0
,
then the result is complex with positive imaginary part.
Intmod a prime p
, t_PADIC
and t_FFELT
are allowed as arguments. In
the first 2 cases (t_INTMOD
, t_PADIC
), the square root (if it
exists) which is returned is the one whose first p
-adic digit is in the
interval [0,p/2]
. For other arguments, the result is undefined.
The library syntax is GEN
gsqrt(GEN x, long prec)
.
For a t_PADIC
x
, the function
GEN
Qp_sqrt(GEN x)
is also available.
(x,n,{&z})
Principal branch of the n
th root of x
,
i.e. such that Arg(sqrtn(x))\in ]-
pi/n,
pi/n]
. Intmod
a prime and p
-adics are allowed as arguments.
If z
is present, it is set to a suitable root of unity allowing to
recover all the other roots. If it was not possible, z is
set to zero. In the case this argument is present and no n
th root exist,
0
is returned instead of raising an error.
? sqrtn(Mod(2,7), 2) %1 = Mod(3, 7) ? sqrtn(Mod(2,7), 2, &z); z %2 = Mod(6, 7) ? sqrtn(Mod(2,7), 3) *** at top-level: sqrtn(Mod(2,7),3) *** ^----------------- *** sqrtn: nth-root does not exist in gsqrtn. ? sqrtn(Mod(2,7), 3, &z) %2 = 0 ? z %3 = 0
The following script computes all roots in all possible cases:
sqrtnall(x,n)= { my(V,r,z,r2); r = sqrtn(x,n, &z); if (!z, error("Impossible case in sqrtn")); if (type(x) == "t_INTMOD" || type(x)=="t_PADIC", r2 = r*z; n = 1; while (r2!=r, r2*=z;n++)); V = vector(n); V[1] = r; for(i=2, n, V[i] = V[i-1]*z); V } addhelp(sqrtnall,"sqrtnall(x,n):compute the vector of nth-roots of x");
The library syntax is GEN
gsqrtn(GEN x, GEN n, GEN *z = NULL, long prec)
.
If x
is a t_PADIC
, the function
GEN
Qp_sqrtn(GEN x, GEN n, GEN *z)
is also available.
(x)
Tangent of x
.
The library syntax is GEN
gtan(GEN x, long prec)
.
(x)
Hyperbolic tangent of x
.
The library syntax is GEN
gtanh(GEN x, long prec)
.
teichmuller(x,{
tab})
Teichmüller character of the p
-adic number x
, i.e. the unique
(p-1)
-th root of unity congruent to x / p^{v_p(x)}
modulo p
.
If x
is of the form [p,n]
, for a prime p
and integer n
,
return the lifts to Z of the images of i + O(p^n)
for
i = 1,..., p-1
, i.e. all roots of 1
ordered by residue class modulo
p
. Such a vector can be fed back to teichmuller
, as the
optional argument tab
, to speed up later computations.
? z = teichmuller(2 + O(101^5)) %1 = 2 + 83*101 + 18*101^2 + 69*101^3 + 62*101^4 + O(101^5) ? z^100 %2 = 1 + O(101^5) ? T = teichmuller([101, 5]); ? teichmuller(2 + O(101^5), T) %4 = 2 + 83*101 + 18*101^2 + 69*101^3 + 62*101^4 + O(101^5)
@3As a rule of thumb, if more than
p / 2(
log _2(p) + hammingweight(p))
values of teichmuller
are to be computed, then it is worthwile to
initialize:
? p = 101; n = 100; T = teichmuller([p,n]); \\ instantaneous ? for(i=1,10^3, vector(p-1, i, teichmuller(i+O(p^n), T))) time = 60 ms. ? for(i=1,10^3, vector(p-1, i, teichmuller(i+O(p^n)))) time = 1,293 ms. ? 1 + 2*(log(p)/log(2) + hammingweight(p)) %8 = 22.316[...]
@3Here the precompuation induces a speedup by a factor
1293/ 60 ~ 21.5
.
@3Caveat.
If the accuracy of tab
(the argument n
above) is lower than the
precision of x
, the former is used, i.e. the cached value is not
refined to higher accuracy. It the accuracy of tab
is larger, then
the precision of x
is used:
? Tlow = teichmuller([101, 2]); \\ lower accuracy ! ? teichmuller(2 + O(101^5), Tlow) %10 = 2 + 83*101 + O(101^5) \\ no longer a root of 1
? Thigh = teichmuller([101, 10]); \\ higher accuracy ? teichmuller(2 + O(101^5), Thigh) %12 = 2 + 83*101 + 18*101^2 + 69*101^3 + 62*101^4 + O(101^5)
The library syntax is GEN
teichmuller(GEN x, GEN tab = NULL)
.
Also available are the functions GEN
teich(GEN x)
(tab
is
NULL
) as well as
GEN
teichmullerinit(long p, long n)
.
(q,z)
Jacobi sine theta-function
theta_1(z, q) = 2q^{1/4}
sum_{n >= 0} (-1)^n q^{n(n+1)}
sin ((2n+1)z).
The library syntax is GEN
theta(GEN q, GEN z, long prec)
.
(q,k)
k
-th derivative at z = 0
of theta(q,z)
.
The library syntax is GEN
thetanullk(GEN q, long k, long prec)
.
GEN
vecthetanullk(GEN q, long k, long prec)
returns the vector
of all (d^i
theta)/(dz^i)(q,0)
for all odd i = 1, 3,..., 2k-1
.
GEN
vecthetanullk_tau(GEN tau, long k, long prec)
returns
vecthetanullk_tau
at q =
exp (2i
pi tau)
.
(x,{
flag = 0})
One of Weber's three f
functions.
If flag = 0
, returns
f(x) =
exp (-i
pi/24).
eta((x+1)/2)/
eta(x) such that j = (f^{24}-16)^3/f^{24},
where j
is the elliptic j
-invariant (see the function ellj
).
If flag = 1
, returns
f_1(x) =
eta(x/2)/
eta(x) such that j = (f_1^{24}+16)^3/f_1^{24}.
Finally, if flag = 2
, returns
f_2(x) =
sqrt {2}
eta(2x)/
eta(x) such that j = (f_2^{24}+16)^3/f_2^{24}.
Note the identities f^8 = f_1^8+f_2^8
and ff_1f_2 =
sqrt 2
.
The library syntax is GEN
weber0(GEN x, long flag, long prec)
.
Also available are GEN
weberf(GEN x, long prec)
,
GEN
weberf1(GEN x, long prec)
and GEN
weberf2(GEN x, long prec)
.
(s)
For s
a complex number, Riemann's zeta
function zeta(s) =
sum_{n >= 1}n^{-s}
,
computed using the Euler-Maclaurin summation formula, except
when s
is of type integer, in which case it is computed using
Bernoulli numbers for s <= 0
or s > 0
and
even, and using modular forms for s > 0
and odd.
For s
a p
-adic number, Kubota-Leopoldt zeta function at s
, that
is the unique continuous p
-adic function on the p
-adic integers
that interpolates the values of (1 - p^{-k})
zeta(k)
at negative
integers k
such that k = 1 (mod p-1)
(resp. k
is odd) if
p
is odd (resp. p = 2
).
The library syntax is GEN
gzeta(GEN s, long prec)
.
(s)
For s
a vector of positive integers such that s[1] >= 2
,
returns the multiple zeta value (MZV)
zeta(s_1,..., s_k) =
sum_{n_1 > ... > n_k > 0} n_1^{-s_1}...n_k^{-s_k}.
? zetamult([2,1]) - zeta(3)
\\ Euler's identity
%1 = 0.E-38
The library syntax is GEN
zetamult(GEN s, long prec)
.
These functions are by definition functions whose natural domain of
definition is either Z (or Z_{ > 0}
). The way these functions are used is
completely different from transcendental functions in that there are no
automatic type conversions: in general only integers are accepted as
arguments. An integer argument N
can be given in the following alternate
formats:
@3* t_MAT
: its factorization fa = factor(N)
,
@3* t_VEC
: a pair [N, fa]
giving both the integer and
its factorization.
This allows to compute different arithmetic functions at a given N
while factoring the latter only once.
? N = 10!; faN = factor(N); ? eulerphi(N) %2 = 829440 ? eulerphi(faN) %3 = 829440 ? eulerphi(S = [N, faN]) %4 = 829440 ? sigma(S) %5 = 15334088
All arithmetic functions in the narrow sense of the word --- Euler's
totient function, the Moebius function,
the sums over divisors or powers of divisors etc.--- call, after trial
division by small primes, the same versatile factoring machinery described
under factorint
. It includes Shanks SQUFOF, Pollard Rho,
ECM and MPQS stages, and has an early exit option for the
functions moebius and (the integer function underlying)
issquarefree. This machinery relies on a fairly strong
probabilistic primality test, see ispseudoprime
, but you may also set
default(factor_proven, 1)
@3to ensure that all tentative factorizations are fully proven. This should not slow down PARI too much, unless prime numbers with hundreds of decimal digits occur frequently in your application.
The following functions compute the order of an element in a finite group:
ellorder
(the rational points on an elliptic curve defined over a
finite field), fforder
(the multiplicative group of a finite field),
znorder
(the invertible elements in Z/n
Z). The following functions
compute discrete logarithms in the same groups (whenever this is meaningful)
elllog
, fflog
, znlog
.
All such functions allow an optional argument specifying an integer
N
, representing the order of the group. (The order functions also
allows any non-zero multiple of the order, with a minor loss of efficiency.)
That optional argument follows the same format as given above:
@3* t_INT
: the integer N
,
@3* t_MAT
: the factorization fa = factor(N)
,
@3* t_VEC
: this is the preferred format and provides both the
integer N
and its factorization in a two-component vector
[N, fa]
.
When the group is fixed and many orders or discrete logarithms will be computed, it is much more efficient to initialize this data once and for all and pass it to the relevant functions, as in
? p = nextprime(10^40); ? v = [p-1, factor(p-1)]; \\ data for discrete log & order computations ? znorder(Mod(2,p), v) %3 = 500000000000000000000000000028 ? g = znprimroot(p); ? znlog(2, g, v) %5 = 543038070904014908801878611374
The finite abelian group G = (
Z/N
Z)^*
can be written G = \oplus_{i <=
n} (
Z/d_i
Z) g_i
, with d_n | ... | d_2 | d_1
(SNF condition),
all d_i > 0
, and prod_i d_i =
phi(N)
.
The SNF condition makes the d_i
unique, but the generators g_i
, of
respective order d_i
, are definitely not unique. The \oplus
notation
means that all elements of G
can be written uniquely as prod_i g_i^{n_i}
where n_i \in
Z/d_i
Z. The g_i
are the so-called SNF generators
of G
.
@3* a character on the abelian group
\oplus (
Z/d_j
Z) g_j
is given by a row vector chi = [a_1,...,a_n]
of integers 0 <= a_i <
d_i
such that chi(g_j) = e(a_j / d_j)
for all j
, with the standard
notation e(x) :=
exp (2i
pi x)
.
In other words,
chi(
prod g_j^{n_j}) = e(
sum a_j n_j / d_j)
.
This will be generalized to more general abelian groups in later sections
(Hecke characters), but in the present case of (
Z/N
Z)^*
, there is a useful
alternate convention : namely, it is not necessary to impose the SNF
condition and we can use Chinese reminders instead. If N =
prod p^{e_p}
is
the factorization of N
into primes, the so-called Conrey generators
of G
are the generators of the (
Z/p^{e_p}
Z)^*
lifted to (
Z/N
Z)^*
by
requesting that they be congruent to 1
modulo N/p^{e_p}
(for p
odd we
take the smallest positive primitive root, and for p = 2
we take -1
if
e_2 > 1
and additionally 5
if e_2 > 2
). We can again write G =
\oplus_{i <= n} (
Z/D_i
Z) G_i
, where again prod_i D_i =
phi(N)
. These
generators don't satisfy the SNF condition in general since their orders are
now (p-1)p^{e_p-1}
for p
odd; for p = 2
, the generator -1
has order
2
and 5
has order 2^{e_2-2}
(e_2 > 2)
. Nevertheless, any m\in
(
Z/N
Z)^*
can be uniquely decomposed as prod G_i^{m_i}
for some m_i
modulo D_i
and we can define a character by chi(G_j) = e(m_j / D_j)
for
all j
.
@3* The column vector of the m_j
, 0 <= m_j < D_j
is called the
Conrey logarithm of m
(discrete logarithm in terms of the Conrey
generators). Note that discrete logarithms in PARI/GP are always expressed as
t_COL
s.
@3* The attached character is called the Conrey character
attached to m
.
To sum up a Dirichlet character can be defined by a t_INT
(the Conrey
label m
), a t_COL
(the Conrey logarithm of m
, in terms of the Conrey
generators) or a t_VEC
(in terms of the SNF generators). The t_COL
format, i.e. Conrey logarithms, is the preferred (fastest) representation.
Concretely, this works as follows:
G = idealstar(,N)
initializes (
Z/N
Z)^*
, which must be given as
first arguments to all functions handling Dirichlet characters.
znconreychar
transforms t_INT
and t_COL
to a SNF character.
znconreylog
transforms t_INT
and t_VEC
to a Conrey logarithm.
znconreyexp
transforms t_VEC
and t_COL
to a Conrey label.
Also available are charconj
, chardiv
, charmul
,
charker
, chareval
, charorder
, zncharinduce
,
znconreyconductor
(also computes the primitive character attached to
the input character). The prefix char
indicates that the function
applies to all characters, the prefix znchar
that it is specific to
Dirichlet characters (on (
Z/N
Z)^*
) and the prefix znconrey
that it
is specific to Conrey representation.
({x = []})
Adds the integers contained in the
vector x
(or the single integer x
) to a special table of
``user-defined primes'', and returns that table. Whenever factor
is
subsequently called, it will trial divide by the elements in this table.
If x
is empty or omitted, just returns the current list of extra
primes.
The entries in x
must be primes: there is no internal check, even if
the factor_proven
default is set. To remove primes from the list use
removeprimes
.
The library syntax is GEN
addprimes(GEN x = NULL)
.
(x, {B})
Using variants of the extended Euclidean algorithm, returns a rational
approximation a/b
to x
, whose denominator is limited
by B
, if present. If B
is omitted, return the best approximation
affordable given the input accuracy; if you are looking for true rational
numbers, presumably approximated to sufficient accuracy, you should first
try that option. Otherwise, B
must be a positive real scalar (impose
0 < b <= B
).
@3* If x
is a t_REAL
or a t_FRAC
, this function uses continued
fractions.
? bestappr(Pi, 100) %1 = 22/7 ? bestappr(0.1428571428571428571428571429) %2 = 1/7 ? bestappr([Pi, sqrt(2) + 'x], 10^3) %3 = [355/113, x + 1393/985]
By definition, a/b
is the best rational approximation to x
if
|b x - a| < |v x - u|
for all integers (u,v)
with 0 < v <= B
.
(Which implies that n/d
is a convergent of the continued fraction of x
.)
@3* If x
is a t_INTMOD
modulo N
or a t_PADIC
of precision N =
p^k
, this function performs rational modular reconstruction modulo N
. The
routine then returns the unique rational number a/b
in coprime integers
|a| < N/2B
and b <= B
which is congruent to x
modulo N
. Omitting
B
amounts to choosing it of the order of sqrt {N/2}
. If rational
reconstruction is not possible (no suitable a/b
exists), returns []
.
? bestappr(Mod(18526731858, 11^10)) %1 = 1/7 ? bestappr(Mod(18526731858, 11^20)) %2 = [] ? bestappr(3 + 5 + 3*5^2 + 5^3 + 3*5^4 + 5^5 + 3*5^6 + O(5^7)) %2 = -1/3
@3In most concrete uses, B
is a prime power and we performed
Hensel lifting to obtain x
.
The function applies recursively to components of complex objects
(polynomials, vectors,...). If rational reconstruction fails for even a
single entry, return []
.
The library syntax is GEN
bestappr(GEN x, GEN B = NULL)
.
(x, {B})
Using variants of the extended Euclidean algorithm, returns a rational
function approximation a/b
to x
, whose denominator is limited
by B
, if present. If B
is omitted, return the best approximation
affordable given the input accuracy; if you are looking for true rational
functions, presumably approximated to sufficient accuracy, you should first
try that option. Otherwise, B
must be a non-negative real (impose
0 <= degree(b) <= B
).
@3* If x
is a t_RFRAC
or t_SER
, this function uses continued
fractions.
? bestapprPade((1-x^11)/(1-x)+O(x^11)) %1 = 1/(-x + 1) ? bestapprPade([1/(1+x+O(x^10)), (x^3-2)/(x^3+1)], 1) %2 = [1/(x + 1), -2]
@3* If x
is a t_POLMOD
modulo N
or a t_SER
of precision N =
t^k
, this function performs rational modular reconstruction modulo N
. The
routine then returns the unique rational function a/b
in coprime
polynomials, with degree(b) <= B
which is congruent to x
modulo
N
. Omitting B
amounts to choosing it of the order of N/2
. If rational
reconstruction is not possible (no suitable a/b
exists), returns []
.
? bestapprPade(Mod(1+x+x^2+x^3+x^4, x^4-2)) %1 = (2*x - 1)/(x - 1) ? % * Mod(1,x^4-2) %2 = Mod(x^3 + x^2 + x + 3, x^4 - 2) ? bestapprPade(Mod(1+x+x^2+x^3+x^5, x^9)) %2 = [] ? bestapprPade(Mod(1+x+x^2+x^3+x^5, x^10)) %3 = (2*x^4 + x^3 - x - 1)/(-x^5 + x^3 + x^2 - 1)
The function applies recursively to components of complex objects
(polynomials, vectors,...). If rational reconstruction fails for even a
single entry, return []
.
The library syntax is GEN
bestapprPade(GEN x, long B)
.
(x,y)
Deprecated alias for gcdext
The library syntax is GEN
gcdext0(GEN x, GEN y)
.
(x)
Number of prime divisors of the integer |x|
counted with
multiplicity:
? factor(392) %1 = [2 3]
[7 2]
? bigomega(392) %2 = 5; \\ = 3+2 ? omega(392) %3 = 2; \\ without multiplicity
The library syntax is long
bigomega(GEN x)
.
(x,y)
binomial coefficient binom{x}{y}
.
Here y
must be an integer, but x
can be any PARI object.
The library syntax is GEN
binomial(GEN x, long y)
.
The function
GEN
binomialuu(ulong n, ulong k)
is also available, and so is
GEN
vecbinome(long n)
, which returns a vector v
with n+1
components such that v[k+1] = binomial(n,k)
for k
from
0
up to n
.
(
cyc,
chi)
Let cyc represent a finite abelian group by its elementary
divisors, i.e. (d_j)
represents sum_{j <= k}
Z/d_j
Z with d_k
| ... | d_1
; any object which has a .cyc
method is also
allowed, e.g. the output of znstar
or bnrinit
. A character
on this group is given by a row vector chi = [a_1,...,a_n]
such that
chi(
prod g_j^{n_j}) =
exp (2
pi i
sum a_j n_j / d_j)
, where g_j
denotes
the generator (of order d_j
) of the j
-th cyclic component.
This function returns the conjugate character.
? cyc = [15,5]; chi = [1,1]; ? charconj(cyc, chi) %2 = [14, 4] ? bnf = bnfinit(x^2+23); ? bnf.cyc %4 = [3] ? charconj(bnf, [1]) %5 = [2]
@3For Dirichlet characters (when cyc
is
idealstar(,q)
), characters in Conrey representation are available,
see Label se:dirichletchar or ??character
:
? G = idealstar(,8); \\ (Z/8Z)^* ? charorder(G, 3) \\ Conrey label %2 = 2 ? chi = znconreylog(G, 3); ? charorder(G, chi) \\ Conrey logarithm %4 = 2
The library syntax is GEN
charconj0(GEN cyc, GEN chi)
.
Also available is
GEN
charconj(GEN cyc, GEN chi)
, when cyc
is known to
be a vector of elementary divisors and chi
a compatible character
(no checks).
(
cyc, a,b)
Let cyc represent a finite abelian group by its elementary
divisors, i.e. (d_j)
represents sum_{j <= k}
Z/d_j
Z with d_k
| ... | d_1
; any object which has a .cyc
method is also
allowed, e.g. the output of znstar
or bnrinit
. A character
on this group is given by a row vector a = [a_1,...,a_n]
such that
chi(
prod g_j^{n_j}) =
exp (2
pi i
sum a_j n_j / d_j)
, where g_j
denotes
the generator (of order d_j
) of the j
-th cyclic component.
Given two characters a
and b
, return the character
a / b = a \overline{b}
.
? cyc = [15,5]; a = [1,1]; b = [2,4]; ? chardiv(cyc, a,b) %2 = [14, 2] ? bnf = bnfinit(x^2+23); ? bnf.cyc %4 = [3] ? chardiv(bnf, [1], [2]) %5 = [2]
@3For Dirichlet characters on (
Z/N
Z)^*
, additional
representations are available (Conrey labels, Conrey logarithm),
see Label se:dirichletchar or ??character
.
If the two characters are in the same format, the
result is given in the same format, otherwise a Conrey logarithm is used.
? G = idealstar(,100); ? G.cyc %2 = [20, 2] ? a = [10, 1]; \\ usual representation for characters ? b = 7; \\ Conrey label; ? c = znconreylog(G, 11); \\ Conrey log ? chardiv(G, b,b) %6 = 1 \\ Conrey label ? chardiv(G, a,b) %7 = [0, 5]~ \\ Conrey log ? chardiv(G, a,c) %7 = [0, 14]~ \\ Conrey log
The library syntax is GEN
chardiv0(GEN cyc, GEN a, GEN b)
.
Also available is
GEN
chardiv(GEN cyc, GEN a, GEN b)
, when cyc
is known to
be a vector of elementary divisors and a, b
are compatible characters
(no checks).
(G,
chi, x, {z}))
Let G
be an abelian group structure affording a discrete logarithm
method, e.g G = idealstar(,N)
for (
Z/N
Z)^*
or a bnr
structure, let x
be an element of G
and let chi be a character of
G
(see the note below for details). This function returns the value of
chi at x
.
@3Note on characters.
Let K
be some field. If G
is an abelian group,
let chi: G \to K^*
be a character of finite order and let o
be a
multiple of the character order such that chi(n) =
zeta^{c(n)}
for some
fixed zeta\in K^*
of multiplicative order o
and a unique morphism c: G
\to (
Z/o
Z,+)
. Our usual convention is to write
G = (
Z/o_1
Z) g_1 \oplus...\oplus (
Z/o_d
Z) g_d
for some generators (g_i)
of respective order d_i
, where the group has
exponent o := lcm_i o_i
. Since zeta^o = 1
, the vector (c_i)
in
prod (
Z/o_i
Z)
defines a character chi on G
via chi(g_i) =
zeta^{c_i (o/o_i)}
for all i
. Classical Dirichlet characters have values
in K =
C and we can take zeta =
exp (2i
pi/o)
.
@3Note on Dirichlet characters.
In the special case where bid is attached to G = (
Z/q
Z)^*
(as per bid = idealstar(,q)
), the Dirichlet
character chi can be written in one of the usual 3 formats: a t_VEC
in terms of bid.gen
as above, a t_COL
in terms of the Conrey
generators, or a t_INT
(Conrey label);
see Label se:dirichletchar or ??character
.
The character value is encoded as follows, depending on the optional
argument z
:
@3* If z
is omitted: return the rational number c(x)/o
for x
coprime
to q
, where we normalize 0 <= c(x) < o
. If x
can not be mapped to the
group (e.g. x
is not coprime to the conductor of a Dirichlet or Hecke
character) we return the sentinel value -1
.
@3* If z
is an integer o
, then we assume that o
is a multiple of the
character order and we return the integer c(x)
when x
belongs
to the group, and the sentinel value -1
otherwise.
@3* z
can be of the form [
zeta, o]
, where zeta
is an o
-th root of 1
and o
is a multiple of the character order.
We return zeta^{c(x)}
if x
belongs to the group, and the sentinel
value 0
otherwise. (Note that this coincides with the usual extension
of Dirichlet characters to Z, or of Hecke characters to general ideals.)
@3* Finally, z
can be of the form [
vzeta, o]
, where
vzeta is a vector of powers zeta^0,...,
zeta^{o-1}
of some o
-th root of 1
and o
is a multiple of the character order.
As above, we return zeta^{c(x)}
after a table lookup. Or the sentinel
value 0
.
The library syntax is GEN
chareval(GEN G, GEN chi, GEN x, GEN z) = NULL)
.
(
cyc,
chi)
Let cyc represent a finite abelian group by its elementary
divisors, i.e. (d_j)
represents sum_{j <= k}
Z/d_j
Z with d_k
| ... | d_1
; any object which has a .cyc
method is also
allowed, e.g. the output of znstar
or bnrinit
. A character
on this group is given by a row vector chi = [a_1,...,a_n]
such that
chi(
prod g_j^{n_j}) =
exp (2
pi i
sum a_j n_j / d_j)
, where g_j
denotes
the generator (of order d_j
) of the j
-th cyclic component.
This function returns the kernel of chi, as a matrix K
in HNF which is a
left-divisor of matdiagonal(d)
. Its columns express in terms of
the g_j
the generators of the subgroup. The determinant of K
is the
kernel index.
? cyc = [15,5]; chi = [1,1]; ? charker(cyc, chi) %2 = [15 12]
[ 0 1]
? bnf = bnfinit(x^2+23); ? bnf.cyc %4 = [3] ? charker(bnf, [1]) %5 = [3]
@3Note that for Dirichlet characters (when cyc
is
idealstar(,q)
), characters in Conrey representation are available,
see Label se:dirichletchar or ??character
.
? G = idealstar(,8); \\ (Z/8Z)^* ? charker(G, 1) \\ Conrey label for trivial character %2 = [1 0]
[0 1]
The library syntax is GEN
charker0(GEN cyc, GEN chi)
.
Also available is
GEN
charker(GEN cyc, GEN chi)
, when cyc
is known to
be a vector of elementary divisors and chi
a compatible character
(no checks).
(
cyc, a,b)
Let cyc represent a finite abelian group by its elementary
divisors, i.e. (d_j)
represents sum_{j <= k}
Z/d_j
Z with d_k
| ... | d_1
; any object which has a .cyc
method is also
allowed, e.g. the output of znstar
or bnrinit
. A character
on this group is given by a row vector a = [a_1,...,a_n]
such that
chi(
prod g_j^{n_j}) =
exp (2
pi i
sum a_j n_j / d_j)
, where g_j
denotes
the generator (of order d_j
) of the j
-th cyclic component.
Given two characters a
and b
, return the product character ab
.
? cyc = [15,5]; a = [1,1]; b = [2,4]; ? charmul(cyc, a,b) %2 = [3, 0] ? bnf = bnfinit(x^2+23); ? bnf.cyc %4 = [3] ? charmul(bnf, [1], [2]) %5 = [0]
@3For Dirichlet characters on (
Z/N
Z)^*
, additional
representations are available (Conrey labels, Conrey logarithm), see
Label se:dirichletchar or ??character
. If the two characters are in
the same format, their
product is given in the same format, otherwise a Conrey logarithm is used.
? G = idealstar(,100); ? G.cyc %2 = [20, 2] ? a = [10, 1]; \\ usual representation for characters ? b = 7; \\ Conrey label; ? c = znconreylog(G, 11); \\ Conrey log ? charmul(G, b,b) %6 = 49 \\ Conrey label ? charmul(G, a,b) %7 = [0, 15]~ \\ Conrey log ? charmul(G, a,c) %7 = [0, 6]~ \\ Conrey log
The library syntax is GEN
charmul0(GEN cyc, GEN a, GEN b)
.
Also available is
GEN
charmul(GEN cyc, GEN a, GEN b)
, when cyc
is known to
be a vector of elementary divisors and a, b
are compatible characters
(no checks).
(
cyc,
chi)
Let cyc represent a finite abelian group by its elementary
divisors, i.e. (d_j)
represents sum_{j <= k}
Z/d_j
Z with d_k
| ... | d_1
; any object which has a .cyc
method is also
allowed, e.g. the output of znstar
or bnrinit
. A character
on this group is given by a row vector chi = [a_1,...,a_n]
such that
chi(
prod g_j^{n_j}) =
exp (2
pi i
sum a_j n_j / d_j)
, where g_j
denotes
the generator (of order d_j
) of the j
-th cyclic component.
This function returns the order of the character chi
.
? cyc = [15,5]; chi = [1,1]; ? charorder(cyc, chi) %2 = 15 ? bnf = bnfinit(x^2+23); ? bnf.cyc %4 = [3] ? charorder(bnf, [1]) %5 = 3
@3For Dirichlet characters (when cyc
is
idealstar(,q)
), characters in Conrey representation are available,
see Label se:dirichletchar or ??character
:
? G = idealstar(,100); \\ (Z/100Z)^* ? charorder(G, 7) \\ Conrey label %2 = 4
The library syntax is GEN
charorder0(GEN cyc, GEN chi)
.
Also available is
GEN
charorder(GEN cyc, GEN chi)
, when cyc
is known to
be a vector of elementary divisors and chi
a compatible character
(no checks).
(x,{y})
If x
and y
are both intmods or both polmods, creates (with the same
type) a z
in the same residue class as x
and in the same residue class as
y
, if it is possible.
? chinese(Mod(1,2), Mod(2,3)) %1 = Mod(5, 6) ? chinese(Mod(x,x^2-1), Mod(x+1,x^2+1)) %2 = Mod(-1/2*x^2 + x + 1/2, x^4 - 1)
This function also allows vector and matrix arguments, in which case the operation is recursively applied to each component of the vector or matrix.
? chinese([Mod(1,2),Mod(1,3)], [Mod(1,5),Mod(2,7)]) %3 = [Mod(1, 10), Mod(16, 21)]
For polynomial arguments in the same variable, the function is applied to each
coefficient; if the polynomials have different degrees, the high degree terms
are copied verbatim in the result, as if the missing high degree terms in the
polynomial of lowest degree had been Mod(0,1)
. Since the latter
behavior is usually not the desired one, we propose to convert the
polynomials to vectors of the same length first:
? P = x+1; Q = x^2+2*x+1; ? chinese(P*Mod(1,2), Q*Mod(1,3)) %4 = Mod(1, 3)*x^2 + Mod(5, 6)*x + Mod(3, 6) ? chinese(Vec(P,3)*Mod(1,2), Vec(Q,3)*Mod(1,3)) %5 = [Mod(1, 6), Mod(5, 6), Mod(4, 6)] ? Pol(%) %6 = Mod(1, 6)*x^2 + Mod(5, 6)*x + Mod(4, 6)
If y
is omitted, and x
is a vector, chinese
is applied recursively
to the components of x
, yielding a residue belonging to the same class as all
components of x
.
Finally chinese(x,x) = x
regardless of the type of x
; this allows
vector arguments to contain other data, so long as they are identical in both
vectors.
The library syntax is GEN
chinese(GEN x, GEN y = NULL)
.
GEN
chinese1(GEN x)
is also available.
(x)
Computes the gcd of all the coefficients of x
,
when this gcd makes sense. This is the natural definition
if x
is a polynomial (and by extension a power series) or a
vector/matrix. This is in general a weaker notion than the ideal
generated by the coefficients:
? content(2*x+y) %1 = 1 \\ = gcd(2,y) over Q[y]
If x
is a scalar, this simply returns the absolute value of x
if x
is
rational (t_INT
or t_FRAC
), and either 1
(inexact input) or x
(exact input) otherwise; the result should be identical to gcd(x, 0)
.
The content of a rational function is the ratio of the contents of the
numerator and the denominator. In recursive structures, if a
matrix or vector coefficient x
appears, the gcd is taken
not with x
, but with its content:
? content([ [2], 4*matid(3) ]) %1 = 2
@3The content of a t_VECSMALL
is computed assuming the
entries are signed integers.
The library syntax is GEN
content(GEN x)
.
contfrac(x,{b},{
nmax})
Returns the row vector whose components are the partial quotients of the
continued fraction expansion of x
. In other words, a result
[a_0,...,a_n]
means that x ~ a_0+1/(a_1+...+1/a_n)
. The
output is normalized so that a_n != 1
(unless we also have n = 0
).
The number of partial quotients n+1
is limited by nmax
. If
nmax
is omitted, the expansion stops at the last significant partial
quotient.
? \p19 realprecision = 19 significant digits ? contfrac(Pi) %1 = [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2] ? contfrac(Pi,, 3) \\ n = 2 %2 = [3, 7, 15]
x
can also be a rational function or a power series.
If a vector b
is supplied, the numerators are equal to the coefficients
of b
, instead of all equal to 1
as above; more precisely, x ~
(1/b_0)(a_0+b_1/(a_1+...+b_n/a_n))
; for a numerical continued fraction
(x
real), the a_i
are integers, as large as possible; if x
is a
rational function, they are polynomials with deg a_i =
deg b_i + 1
.
The length of the result is then equal to the length of b
, unless the next
partial quotient cannot be reliably computed, in which case the expansion
stops. This happens when a partial remainder is equal to zero (or too small
compared to the available significant digits for x
a t_REAL
).
A direct implementation of the numerical continued fraction
contfrac(x,b)
described above would be
\\ "greedy" generalized continued fraction cf(x, b) = { my( a= vector(#b), t );
x *= b[1]; for (i = 1, #b, a[i] = floor(x); t = x - a[i]; if (!t || i == #b, break); x = b[i+1] / t; ); a; }
@3There is some degree of freedom when choosing the a_i
; the
program above can easily be modified to derive variants of the standard
algorithm. In the same vein, although no builtin
function implements the related Engel expansion (a special kind of
Egyptian fraction decomposition: x = 1/a_1 + 1/(a_1a_2) +...
),
it can be obtained as follows:
\\ n terms of the Engel expansion of x engel(x, n = 10) = { my( u = x, a = vector(n) ); for (k = 1, n, a[k] = ceil(1/u); u = u*a[k] - 1; if (!u, break); ); a }
@3Obsolete hack. (don't use this): if b
is an integer, nmax
is ignored and the command is understood as contfrac(x,, b)
.
The library syntax is GEN
contfrac0(GEN x, GEN b = NULL, long nmax)
.
Also available are GEN
gboundcf(GEN x, long nmax)
,
GEN
gcf(GEN x)
and GEN
gcf2(GEN b, GEN x)
.
(x, {n = -1})
When x
is a vector or a one-row matrix, x
is considered as the list of partial quotients [a_0,a_1,...,a_n]
of a
rational number, and the result is the 2 by 2 matrix
[p_n,p_{n-1};q_n,q_{n-1}]
in the standard notation of continued fractions,
so p_n/q_n = a_0+1/(a_1+...+1/a_n)
. If x
is a matrix with two rows
[b_0,b_1,...,b_n]
and [a_0,a_1,...,a_n]
, this is then considered as a
generalized continued fraction and we have similarly
p_n/q_n = (1/b_0)(a_0+b_1/(a_1+...+b_n/a_n))
. Note that in this case one
usually has b_0 = 1
.
If n >= 0
is present, returns all convergents from p_0/q_0
up to
p_n/q_n
. (All convergents if x
is too small to compute the n+1
requested convergents.)
? a=contfrac(Pi,20) %1 = [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2] ? contfracpnqn(a,3) %2 = [3 22 333 355]
[1 7 106 113]
? contfracpnqn(a,7) %3 = [3 22 333 355 103993 104348 208341 312689]
[1 7 106 113 33102 33215 66317 99532]
The library syntax is GEN
contfracpnqn(GEN x, long n)
.
also available is GEN
pnqn(GEN x)
for n = -1
.
(n,{
flag = 0})
If n
is an integer written as
n = df^2
with d
squarefree, returns d
. If flag is non-zero,
returns the two-element row vector [d,f]
. By convention, we write 0 = 0
x 1^2
, so core(0, 1)
returns [0,1]
.
The library syntax is GEN
core0(GEN n, long flag)
.
Also available are GEN
core(GEN n)
(flag = 0
) and
GEN
core2(GEN n)
(flag = 1
)
(n,{
flag = 0})
A fundamental discriminant is an integer of the form t = 1
mod 4
or 4t = 8,12 mod 16
, with t
squarefree (i.e. 1
or the
discriminant of a quadratic number field). Given a non-zero integer
n
, this routine returns the (unique) fundamental discriminant d
such that n = df^2
, f
a positive rational number. If flag is non-zero,
returns the two-element row vector [d,f]
. If n
is congruent to
0 or 1 modulo 4, f
is an integer, and a half-integer otherwise.
By convention, coredisc(0, 1))
returns [0,1]
.
Note that quaddisc
(n)
returns the same value as coredisc
(n)
,
and also works with rational inputs n\in
Q^*
.
The library syntax is GEN
coredisc0(GEN n, long flag)
.
Also available are GEN
coredisc(GEN n)
(flag = 0
) and
GEN
coredisc2(GEN n)
(flag = 1
)
(x,y)
x
and y
being vectors of perhaps different
lengths but with y[1] != 0
considered as Dirichlet series, computes
the quotient of x
by y
, again as a vector.
The library syntax is GEN
dirdiv(GEN x, GEN y)
.
(p = a,b,
expr,{c})
Computes the Dirichlet series attached to the
Euler product of expression expr as p
ranges through the primes
from a
to b
. expr must be a polynomial or rational function in another
variable than p
(say X
) and expr(X)
is understood as the local
factor expr(p^{-s})
.
The series is output as a vector of coefficients. If c
is omitted, output
the first b
coefficients of the series; otherwise, output the first c
coefficients. The following command computes the sigma function,
attached to zeta(s)
zeta(s-1)
:
? direuler(p=2, 10, 1/((1-X)*(1-p*X))) %1 = [1, 3, 4, 7, 6, 12, 8, 15, 13, 18]
? direuler(p=2, 10, 1/((1-X)*(1-p*X)), 5) \\ fewer terms %2 = [1, 3, 4, 7, 6]
@3Setting c < b
is useless (the same effect would be
achieved by setting b = c)
. If c > b
, the computed coefficients are
``missing'' Euler factors:
? direuler(p=2, 10, 1/((1-X)*(1-p*X)), 15) \\ more terms, no longer = sigma ! %3 = [1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 0, 28, 0, 24, 24]
The library syntax is direuler(void *E, GEN (*eval)(void*,GEN), GEN a, GEN b)
(x,y)
x
and y
being vectors of perhaps different lengths representing
the Dirichlet series sum_n x_n n^{-s}
and sum_n y_n n^{-s}
,
computes the product of x
by y
, again as a vector.
? dirmul(vector(10,n,1), vector(10,n,moebius(n))) %1 = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
The product
length is the minimum of #x*v(y)
and #y*v(x)
,
where v(x)
is the index of the first non-zero coefficient.
? dirmul([0,1], [0,1]); %2 = [0, 0, 0, 1]
The library syntax is GEN
dirmul(GEN x, GEN y)
.
(x)
Creates a row vector whose components are the
divisors of x
. The factorization of x
(as output by factor
) can
be used instead.
By definition, these divisors are the products of the irreducible
factors of n
, as produced by factor(n)
, raised to appropriate
powers (no negative exponent may occur in the factorization). If n
is
an integer, they are the positive divisors, in increasing order.
The library syntax is GEN
divisors(GEN x)
.
(x)
Euler's phi (totient) function of the
integer |x|
, in other words |(
Z/x
Z)^*|
.
? eulerphi(40) %1 = 16
According to this definition we let phi(0) := 2
, since Z^ *= {-1,1}
;
this is consistent with znstar(0)
: we have
znstar(n).no = eulerphi(n)
for all n\in
Z.
The library syntax is GEN
eulerphi(GEN x)
.
factor(x,{
lim})
General factorization function, where x
is a
rational (including integers), a complex number with rational
real and imaginary parts, or a rational function (including polynomials).
The result is a two-column matrix: the first contains the irreducibles
dividing x
(rational or Gaussian primes, irreducible polynomials),
and the second the exponents. By convention, 0
is factored as 0^1
.
@3Q and Q(i)
.
See factorint
for more information about the algorithms used.
The rational or Gaussian primes are in fact pseudoprimes
(see ispseudoprime
), a priori not rigorously proven primes. In fact,
any factor which is <= 2^{64}
(whose norm is <= 2^{64}
for an
irrational Gaussian prime) is a genuine prime. Use isprime
to prove
primality of other factors, as in
? fa = factor(2^2^7 + 1) %1 = [59649589127497217 1]
[5704689200685129054721 1]
? isprime( fa[,1] ) %2 = [1, 1]~ \\ both entries are proven primes
@3
Another possibility is to set the global default factor_proven
, which
will perform a rigorous primality proof for each pseudoprime factor.
A t_INT
argument lim can be added, meaning that we look only for
prime factors p <
lim. The limit lim must be non-negative.
In this case, all but the last factor are proven primes, but the remaining
factor may actually be a proven composite! If the remaining factor is less
than lim^2
, then it is prime.
? factor(2^2^7 +1, 10^5) %3 = [340282366920938463463374607431768211457 1]
@3Deprecated feature. Setting lim = 0
is the same
as setting it to primelimit + 1
. Don't use this: it is unwise to
rely on global variables when you can specify an explicit argument.
This routine uses trial division and perfect power tests, and should not be
used for huge values of lim (at most 10^9
, say):
factorint(, 1 + 8)
will in general be faster. The latter does not
guarantee that all small
prime factors are found, but it also finds larger factors, and in a much more
efficient way.
? F = (2^2^7 + 1) * 1009 * 100003; factor(F, 10^5) \\ fast, incomplete time = 0 ms. %4 = [1009 1]
[34029257539194609161727850866999116450334371 1]
? factor(F, 10^9) \\ very slow time = 6,892 ms. %6 = [1009 1]
[100003 1]
[340282366920938463463374607431768211457 1]
? factorint(F, 1+8) \\ much faster, all small primes were found time = 12 ms. %7 = [1009 1]
[100003 1]
[340282366920938463463374607431768211457 1]
? factor(F) \\ complete factorisation time = 112 ms. %8 = [1009 1]
[100003 1]
[59649589127497217 1]
[5704689200685129054721 1]
@3Over Q, the prime factors are sorted in increasing order.
@3Rational functions.
The polynomials or rational functions to be factored must have scalar
coefficients. In particular PARI does not know how to factor
multivariate polynomials. The following domains are currently
supported: Q, R, C, Q_p
, finite fields and number fields.
See factormod
and factorff
for
the algorithms used over finite fields, factornf
for the algorithms
over number fields. Over Q, van Hoeij's method is used, which is
able to cope with hundreds of modular factors.
The routine guesses a sensible ring over which to factor: the
smallest ring containing all coefficients, taking into account quotient
structures induced by t_INTMOD
s and t_POLMOD
s (e.g. if a coefficient
in Z/n
Z is known, all rational numbers encountered are first mapped to
Z/n
Z; different moduli will produce an error). Factoring modulo a
non-prime number is not supported; to factor in Q_p
, use t_PADIC
coefficients not t_INTMOD
modulo p^n
.
? T = x^2+1; ? factor(T); \\ over Q ? factor(T*Mod(1,3)) \\ over F_3 ? factor(T*ffgen(ffinit(3,2,'t))^0) \\ over F_{3^2} ? factor(T*Mod(Mod(1,3), t^2+t+2)) \\ over F_{3^2}, again ? factor(T*(1 + O(3^6)) \\ over Q_3, precision 6 ? factor(T*1.) \\ over R, current precision ? factor(T*(1.+0.*I)) \\ over C ? factor(T*Mod(1, y^3-2)) \\ over Q(2^{1/3})
@3In most cases, it is clearer and simpler to call an
explicit variant than to rely on the generic factor
function and
the above detection mechanism:
? factormod(T, 3) \\ over F_3 ? factorff(T, 3, t^2+t+2)) \\ over F_{3^2} ? factorpadic(T, 3,6) \\ over Q_3, precision 6 ? nffactor(y^3-2, T) \\ over Q(2^{1/3}) ? polroots(T) \\ over C
Note that factorization of polynomials is done up to multiplication by a constant. In particular, the factors of rational polynomials will have integer coefficients, and the content of a polynomial or rational function is discarded and not included in the factorization. If needed, you can always ask for the content explicitly:
? factor(t^2 + 5/2*t + 1) %1 = [2*t + 1 1]
[t + 2 1]
? content(t^2 + 5/2*t + 1) %2 = 1/2
@3
The irreducible factors are sorted by increasing degree.
See also nffactor
.
The library syntax is GEN
gp_factor0(GEN x, GEN lim = NULL)
.
This function should only be used by the gp
interface. Use
directly GEN
factor(GEN x)
or GEN
boundfact(GEN x, ulong lim)
.
The obsolete function GEN
factor0(GEN x, long lim)
is kept for
backward compatibility.
(f,{e})
Gives back the factored object
corresponding to a factorization. The integer 1
corresponds to the empty
factorization.
If e
is present, e
and f
must be vectors of the same length (e
being
integral), and the corresponding factorization is the product of the
f[i]^{e[i]}
.
If not, and f
is vector, it is understood as in the preceding case with e
a vector of 1s: we return the product of the f[i]
. Finally, f
can be a
regular factorization, as produced with any factor
command. A few
examples:
? factor(12) %1 = [2 2]
[3 1]
? factorback(%) %2 = 12 ? factorback([2,3], [2,1]) \\ 2^3 * 3^1 %3 = 12 ? factorback([5,2,3]) %4 = 30
The library syntax is GEN
factorback2(GEN f, GEN e = NULL)
.
Also available is GEN
factorback(GEN f)
(case e = NULL
).
(x,p)
Factors the polynomial x
modulo the
prime p
, using distinct degree plus
Cantor-Zassenhaus. The coefficients of x
must be
operation-compatible with Z/p
Z. The result is a two-column matrix, the
first column being the irreducible polynomials dividing x
, and the second
the exponents. If you want only the degrees of the irreducible
polynomials (for example for computing an L
-function), use
factormod(x,p,1)
. Note that the factormod
algorithm is
usually faster than factorcantor
.
The library syntax is GEN
factcantor(GEN x, GEN p)
.
(x,{p},{a})
Factors the polynomial x
in the field
F_q
defined by the irreducible polynomial a
over F_p
. The
coefficients of x
must be operation-compatible with Z/p
Z. The result
is a two-column matrix: the first column contains the irreducible factors of
x
, and the second their exponents. If all the coefficients of x
are in
F_p
, a much faster algorithm is applied, using the computation of
isomorphisms between finite fields.
Either a
or p
can omitted (in which case both are ignored) if x has
t_FFELT
coefficients; the function then becomes identical to factor
:
? factorff(x^2 + 1, 5, y^2+3) \\ over F_5[y]/(y^2+3) ~ F_25 %1 = [Mod(Mod(1, 5), Mod(1, 5)*y^2 + Mod(3, 5))*x + Mod(Mod(2, 5), Mod(1, 5)*y^2 + Mod(3, 5)) 1]
[Mod(Mod(1, 5), Mod(1, 5)*y^2 + Mod(3, 5))*x + Mod(Mod(3, 5), Mod(1, 5)*y^2 + Mod(3, 5)) 1] ? t = ffgen(y^2 + Mod(3,5), 't); \\ a generator for F_25 as a t_FFELT ? factorff(x^2 + 1) \\ not enough information to determine the base field *** at top-level: factorff(x^2+1) *** ^--------------- *** factorff: incorrect type in factorff. ? factorff(x^2 + t^0) \\ make sure a coeff. is a t_FFELT %3 = [x + 2 1]
[x + 3 1] ? factorff(x^2 + t + 1) %11 = [x + (2*t + 1) 1]
[x + (3*t + 4) 1]
@3 Notice that the second syntax is easier to use and much more readable.
The library syntax is GEN
factorff(GEN x, GEN p = NULL, GEN a = NULL)
.
(x)
Factorial of x
. The expression x!
gives a result which is an integer,
while factorial(x)
gives a real number.
The library syntax is GEN
mpfactr(long x, long prec)
.
GEN
mpfact(long x)
returns x!
as a t_INT
.
(x,{
flag = 0})
Factors the integer n
into a product of
pseudoprimes (see ispseudoprime
), using a combination of the
Shanks SQUFOF and Pollard Rho method (with modifications due to
Brent), Lenstra's ECM (with modifications by Montgomery), and
MPQS (the latter adapted from the LiDIA code with the kind
permission of the LiDIA maintainers), as well as a search for pure powers.
The output is a two-column matrix as for factor
: the first column
contains the ``prime'' divisors of n
, the second one contains the
(positive) exponents.
By convention 0
is factored as 0^1
, and 1
as the empty factorization;
also the divisors are by default not proven primes is they are larger than
2^{64}
, they only failed the BPSW compositeness test (see
ispseudoprime
). Use isprime
on the result if you want to
guarantee primality or set the factor_proven
default to 1
.
Entries of the private prime tables (see addprimes
) are also included
as is.
This gives direct access to the integer factoring engine called by most arithmetical functions. flag is optional; its binary digits mean 1: avoid MPQS, 2: skip first stage ECM (we may still fall back to it later), 4: avoid Rho and SQUFOF, 8: don't run final ECM (as a result, a huge composite may be declared to be prime). Note that a (strong) probabilistic primality test is used; thus composites might not be detected, although no example is known.
You are invited to play with the flag settings and watch the internals at
work by using gp
's debug
default parameter (level 3 shows
just the outline, 4 turns on time keeping, 5 and above show an increasing
amount of internal details).
The library syntax is GEN
factorint(GEN x, long flag)
.
(x,p,{
flag = 0})
Factors the polynomial x
modulo the prime integer p
, using
Berlekamp. The coefficients of x
must be operation-compatible with
Z/p
Z. The result is a two-column matrix, the first column being the
irreducible polynomials dividing x
, and the second the exponents. If flag
is non-zero, outputs only the degrees of the irreducible polynomials
(for example, for computing an L
-function). A different algorithm for
computing the mod p
factorization is factorcantor
which is sometimes
faster.
The library syntax is GEN
factormod0(GEN x, GEN p, long flag)
.
(q,{v})
Return a t_FFELT
generator for the finite field with q
elements;
q = p^f
must be a prime power. This functions computes an irreducible
monic polynomial P\in
F_p[X]
of degree f
(via ffinit
) and
returns g = X (mod P(X))
. If v
is given, the variable name is used
to display g
, else the variable x
is used.
? g = ffgen(8, 't); ? g.mod %2 = t^3 + t^2 + 1 ? g.p %3 = 2 ? g.f %4 = 3 ? ffgen(6) *** at top-level: ffgen(6) *** ^-------- *** ffgen: not a prime number in ffgen: 6.
@3Alternative syntax: instead of a prime power q = p^f
, one may
input the pair [p,f]
:
? g = ffgen([2,4], 't); ? g.p %2 = 2 ? g.mod %3 = t^4 + t^3 + t^2 + t + 1
@3Finally, one may input
directly the polynomial P
(monic, irreducible, with t_INTMOD
coefficients), and the function returns the generator g = X (mod P(X))
,
inferring p
from the coefficients of P
. If v
is given, the
variable name is used to display g
, else the variable of the polynomial
P
is used. If P
is not irreducible, we create an invalid object and
behaviour of functions dealing with the resulting t_FFELT
is undefined; in fact, it is much more costly to test P
for
irreducibility than it would be to produce it via ffinit
.
The library syntax is GEN
ffgen(GEN q, long v = -1)
where v
is a variable number.
To create a generator for a prime finite field, the function
GEN
p_to_GEN(GEN p, long v)
returns 1+ffgen(x*Mod(1,p),v)
.
(p,n,{v = 'x})
Computes a monic polynomial of degree n
which is irreducible over
F_p
, where p
is assumed to be prime. This function uses a fast variant
of Adleman and Lenstra's algorithm.
It is useful in conjunction with ffgen
; for instance if
P = ffinit(3,2)
, you can represent elements in F_{3^2}
in term of
g = ffgen(P,'t)
. This can be abbreviated as
g = ffgen(3^2, 't)
, where the defining polynomial P
can be later
recovered as g.mod
.
The library syntax is GEN
ffinit(GEN p, long n, long v = -1)
where v
is a variable number.
(x,g,{o})
Discrete logarithm of the finite field element x
in base g
, i.e.
an e
in Z such that g^e = o
. If
present, o
represents the multiplicative order of g
, see
Label se:DLfun; the preferred format for
this parameter is [ord, factor(ord)]
, where ord
is the
order of g
. It may be set as a side effect of calling ffprimroot
.
If no o
is given, assume that g
is a primitive root. The result is
undefined if e
does not exist. This function uses
@3* a combination of generic discrete log algorithms (see znlog
)
@3* a cubic sieve index calculus algorithm for large fields of degree at
least 5
.
@3* Coppersmith's algorithm for fields of characteristic at most 5
.
? t = ffgen(ffinit(7,5)); ? o = fforder(t) %2 = 5602 \\ I<not> a primitive root. ? fflog(t^10,t) %3 = 10 ? fflog(t^10,t, o) %4 = 10 ? g = ffprimroot(t, &o); ? o \\ order is 16806, bundled with its factorization matrix %6 = [16806, [2, 1; 3, 1; 2801, 1]] ? fforder(g, o) %7 = 16806 ? fflog(g^10000, g, o) %8 = 10000
The library syntax is GEN
fflog(GEN x, GEN g, GEN o = NULL)
.
ffnbirred(q,n{,
fl = 0})
Computes the number of monic irreducible polynomials over F_q
of degree exactly n
,
(flag = 0
or omitted) or at most n
(flag = 1
).
The library syntax is GEN
ffnbirred0(GEN q, long n, long fl)
.
Also available are
GEN
ffnbirred(GEN q, long n)
(for flag = 0
)
and GEN
ffsumnbirred(GEN q, long n)
(for flag = 1
).
(x,{o})
Multiplicative order of the finite field element x
. If o
is
present, it represents a multiple of the order of the element,
see Label se:DLfun; the preferred format for
this parameter is [N, factor(N)]
, where N
is the cardinality
of the multiplicative group of the underlying finite field.
? t = ffgen(ffinit(nextprime(10^8), 5)); ? g = ffprimroot(t, &o); \\ o will be useful! ? fforder(g^1000000, o) time = 0 ms. %5 = 5000001750000245000017150000600250008403 ? fforder(g^1000000) time = 16 ms. \\ noticeably slower, same result of course %6 = 5000001750000245000017150000600250008403
The library syntax is GEN
fforder(GEN x, GEN o = NULL)
.
(x, {&o})
Return a primitive root of the multiplicative
group of the definition field of the finite field element x
(not necessarily
the same as the field generated by x
). If present, o
is set to
a vector [ord, fa]
, where ord
is the order of the group
and fa
its factorisation factor(ord)
. This last parameter is
useful in fflog
and fforder
, see Label se:DLfun.
? t = ffgen(ffinit(nextprime(10^7), 5)); ? g = ffprimroot(t, &o); ? o[1] %3 = 100000950003610006859006516052476098 ? o[2] %4 = [2 1]
[7 2]
[31 1]
[41 1]
[67 1]
[1523 1]
[10498781 1]
[15992881 1]
[46858913131 1]
? fflog(g^1000000, g, o) time = 1,312 ms. %5 = 1000000
The library syntax is GEN
ffprimroot(GEN x, GEN *o = NULL)
.
(x)
x-th
Fibonacci number.
The library syntax is GEN
fibo(long x)
.
(x,{y})
Creates the greatest common divisor of x
and y
.
If you also need the u
and v
such that x*u + y*v =
gcd (x,y)
,
use the bezout
function. x
and y
can have rather quite general
types, for instance both rational numbers. If y
is omitted and x
is a
vector, returns the gcd
of all components of x
, i.e. this is
equivalent to content(x)
.
When x
and y
are both given and one of them is a vector/matrix type,
the GCD is again taken recursively on each component, but in a different way.
If y
is a vector, resp. matrix, then the result has the same type as y
,
and components equal to gcd(x, y[i])
, resp. gcd(x, y[,i])
. Else
if x
is a vector/matrix the result has the same type as x
and an
analogous definition. Note that for these types, gcd
is not
commutative.
The algorithm used is a naive Euclid except for the following inputs:
@3* integers: use modified right-shift binary (``plus-minus'' variant).
@3* univariate polynomials with coefficients in the same number field (in particular rational): use modular gcd algorithm.
@3* general polynomials: use the subresultant algorithm if coefficient explosion is likely (non modular coefficients).
If u
and v
are polynomials in the same variable with inexact
coefficients, their gcd is defined to be scalar, so that
? a = x + 0.0; gcd(a,a) %1 = 1 ? b = y*x + O(y); gcd(b,b) %2 = y ? c = 4*x + O(2^3); gcd(c,c) %3 = 4
@3A good quantitative check to decide whether such a
gcd ``should be'' non-trivial, is to use polresultant
: a value
close to 0
means that a small deformation of the inputs has non-trivial gcd.
You may also use gcdext
, which does try to compute an approximate gcd
d
and provides u
, v
to check whether u x + v y
is close to d
.
The library syntax is GEN
ggcd0(GEN x, GEN y = NULL)
.
Also available are GEN
ggcd(GEN x, GEN y)
, if y
is not
NULL
, and GEN
content(GEN x)
, if y = NULL
.
(x,y)
Returns [u,v,d]
such that d
is the gcd of x,y
,
x*u+y*v =
gcd (x,y)
, and u
and v
minimal in a natural sense.
The arguments must be integers or polynomials.
? [u, v, d] = gcdext(32,102) %1 = [16, -5, 2] ? d %2 = 2 ? gcdext(x^2-x, x^2+x-2) %3 = [-1/2, 1/2, x - 1]
If x,y
are polynomials in the same variable and inexact
coefficients, then compute u,v,d
such that x*u+y*v = d
, where d
approximately divides both and x
and y
; in particular, we do not obtain
gcd(x,y)
which is defined to be a scalar in this case:
? a = x + 0.0; gcd(a,a) %1 = 1
? gcdext(a,a) %2 = [0, 1, x + 0.E-28]
? gcdext(x-Pi, 6*x^2-zeta(2)) %3 = [-6*x - 18.8495559, 1, 57.5726923]
@3For inexact inputs, the output is thus not well defined mathematically, but you obtain explicit polynomials to check whether the approximation is close enough for your needs.
The library syntax is GEN
gcdext0(GEN x, GEN y)
.
(x,y,{p})
Hilbert symbol of x
and y
modulo the prime p
, p = 0
meaning
the place at infinity (the result is undefined if p != 0
is not prime).
It is possible to omit p
, in which case we take p = 0
if both x
and y
are rational, or one of them is a real number. And take p = q
if one of x
, y
is a t_INTMOD
modulo q
or a q
-adic. (Incompatible
types will raise an error.)
The library syntax is long
hilbert(GEN x, GEN y, GEN p = NULL)
.
(x)
True (1) if x
is equal to 1 or to the discriminant of a quadratic
field, false (0) otherwise.
The library syntax is long
isfundamental(GEN x)
.
(x,s,{&N})
True (1) if the integer x
is an s-gonal number, false (0) if not.
The parameter s > 2
must be a t_INT
. If N
is given, set it to n
if x
is the n
-th s
-gonal number.
? ispolygonal(36, 3, &N) %1 = 1 ? N
The library syntax is long
ispolygonal(GEN x, GEN s, GEN *N = NULL)
.
(x,{k},{&n})
If k
is given, returns true (1) if x
is a k
-th power, false
(0) if not. What it means to be a k
-th power depends on the type of
x
; see issquare
for details.
If k
is omitted, only integers and fractions are allowed for x
and the
function returns the maximal k >= 2
such that x = n^k
is a perfect
power, or 0 if no such k
exist; in particular ispower(-1)
,
ispower(0)
, and ispower(1)
all return 0
.
If a third argument &n
is given and x
is indeed a k
-th power, sets
n
to a k
-th root of x
.
@3For a t_FFELT
x
, instead of omitting k
(which is
not allowed for this type), it may be natural to set
k = (x.p ^ x.f - 1) / fforder(x)
The library syntax is long
ispower(GEN x, GEN k = NULL, GEN *n = NULL)
.
Also available is
long
gisanypower(GEN x, GEN *pty)
(k
omitted).
(x)
True (1) if x
is a powerful integer, false (0) if not;
an integer is powerful if and only if its valuation at all primes dividing
x
is greater than 1.
? ispowerful(50) %1 = 0 ? ispowerful(100) %2 = 1 ? ispowerful(5^3*(10^1000+1)^2) %3 = 1
The library syntax is long
ispowerful(GEN x)
.
(x,{
flag = 0})
True (1) if x
is a prime
number, false (0) otherwise. A prime number is a positive integer having
exactly two distinct divisors among the natural numbers, namely 1 and
itself.
This routine proves or disproves rigorously that a number is prime, which can
be very slow when x
is indeed prime and has more than 1000
digits, say.
Use ispseudoprime
to quickly check for compositeness. See also
factor
. It accepts vector/matrices arguments, and is then applied
componentwise.
If flag = 0
, use a combination of Baillie-PSW pseudo primality test (see
ispseudoprime
), Selfridge ``p-1
'' test if x-1
is smooth enough, and
Adleman-Pomerance-Rumely-Cohen-Lenstra (APRCL) for general x
.
If flag = 1
, use Selfridge-Pocklington-Lehmer ``p-1
'' test and output a
primality certificate as follows: return
@3* 0 if x
is composite,
@3* 1 if x
is small enough that passing Baillie-PSW test guarantees
its primality (currently x < 2^{64}
, as checked by Jan Feitsma),
@3* 2
if x
is a large prime whose primality could only sensibly be
proven (given the algorithms implemented in PARI) using the APRCL test.
@3* Otherwise (x
is large and x-1
is smooth) output a three column
matrix as a primality certificate. The first column contains prime
divisors p
of x-1
(such that prod p^{v_p(x-1)} > x^{1/3}
), the second
the corresponding elements a_p
as in Proposition 8.3.1 in GTM 138 , and the
third the output of isprime(p,1).
The algorithm fails if one of the pseudo-prime factors is not prime, which is
exceedingly unlikely and well worth a bug report. Note that if you monitor
isprime
at a high enough debug level, you may see warnings about
untested integers being declared primes. This is normal: we ask for partial
factorisations (sufficient to prove primality if the unfactored part is not
too large), and factor
warns us that the cofactor hasn't been tested.
It may or may not be tested later, and may or may not be prime. This does
not affect the validity of the whole isprime
procedure.
If flag = 2
, use APRCL.
The library syntax is GEN
gisprime(GEN x, long flag)
.
(x,{&n})
If x = p^k
is a prime power (p
prime, k > 0
), return k
, else
return 0. If a second argument &n
is given and x
is indeed
the k
-th power of a prime p
, sets n
to p
.
The library syntax is long
isprimepower(GEN x, GEN *n = NULL)
.
(x,{
flag})
True (1) if x
is a strong pseudo
prime (see below), false (0) otherwise. If this function returns false, x
is not prime; if, on the other hand it returns true, it is only highly likely
that x
is a prime number. Use isprime
(which is of course much
slower) to prove that x
is indeed prime.
The function accepts vector/matrices arguments, and is then applied
componentwise.
If flag = 0
, checks whether x
has no small prime divisors (up to 101
included) and is a Baillie-Pomerance-Selfridge-Wagstaff pseudo prime.
Such a pseudo prime passes a Rabin-Miller test for base 2
,
followed by a Lucas test for the sequence (P,-1)
, P
smallest
positive integer such that P^2 - 4
is not a square mod x
).
There are no known composite numbers passing the above test, although it is
expected that infinitely many such numbers exist. In particular, all
composites <= 2^{64}
are correctly detected (checked using
http://www.cecm.sfu.ca/Pseudoprimes/index-2-to-64.html).
If flag > 0
, checks whether x
is a strong Miller-Rabin pseudo prime for
flag randomly chosen bases (with end-matching to catch square roots of -1
).
The library syntax is GEN
gispseudoprime(GEN x, long flag)
.
(x,{&n})
If x = p^k
is a pseudo-prime power (p
pseudo-prime as per
ispseudoprime
, k > 0
), return k
, else
return 0. If a second argument &n
is given and x
is indeed
the k
-th power of a prime p
, sets n
to p
.
More precisely, k
is always the largest integer such that x = n^k
for
some integer n
and, when n <= 2^{64}
the function returns k > 0
if and
only if n
is indeed prime. When n > 2^{64}
is larger than the threshold,
the function may return 1
even though n
is composite: it only passed
an ispseudoprime(n)
test.
The library syntax is long
ispseudoprimepower(GEN x, GEN *n = NULL)
.
(x,{&n})
True (1) if x
is a square, false (0)
if not. What ``being a square'' means depends on the type of x
: all
t_COMPLEX
are squares, as well as all non-negative t_REAL
; for
exact types such as t_INT
, t_FRAC
and t_INTMOD
, squares are
numbers of the form s^2
with s
in Z, Q and Z/N
Z respectively.
? issquare(3) \\ as an integer %1 = 0 ? issquare(3.) \\ as a real number %2 = 1 ? issquare(Mod(7, 8)) \\ in Z/8Z %3 = 0 ? issquare( 5 + O(13^4) ) \\ in Q_13 %4 = 0
If n
is given, a square root of x
is put into n
.
? issquare(4, &n) %1 = 1 ? n %2 = 2
For polynomials, either we detect that the characteristic is 2 (and check
directly odd and even-power monomials) or we assume that 2
is invertible
and check whether squaring the truncated power series for the square root
yields the original input.
For t_POLMOD
x
, we only support t_POLMOD
s of t_INTMOD
s
encoding finite fields, assuming without checking that the intmod modulus
p
is prime and that the polmod modulus is irreducible modulo p
.
? issquare(Mod(Mod(2,3), x^2+1), &n) %1 = 1 ? n %2 = Mod(Mod(2, 3)*x, Mod(1, 3)*x^2 + Mod(1, 3))
The library syntax is long
issquareall(GEN x, GEN *n = NULL)
.
Also available is long
issquare(GEN x)
. Deprecated
GP-specific functions GEN
gissquare(GEN x)
and
GEN
gissquareall(GEN x, GEN *pt)
return gen_0
and gen_1
instead of a boolean value.
(x)
True (1) if x
is squarefree, false (0) if not. Here x
can be an
integer or a polynomial.
The library syntax is long
issquarefree(GEN x)
.
(x,{&N})
True (1) if x =
phi(n)
for some integer n
, false (0)
if not.
? istotient(14) %1 = 0 ? istotient(100) %2 = 0
If N
is given, set N = n
as well.
? istotient(4, &n) %1 = 1 ? n %2 = 10
The library syntax is long
istotient(GEN x, GEN *N = NULL)
.
(x,y)
Kronecker symbol (x|y)
, where x
and y
must be of type integer. By
definition, this is the extension of Legendre symbol to Z x
Z
by total multiplicativity in both arguments with the following special rules
for y = 0, -1
or 2
:
@3* (x|0) = 1
if |x |= 1
and 0
otherwise.
@3* (x|-1) = 1
if x >= 0
and -1
otherwise.
@3* (x|2) = 0
if x
is even and 1
if x = 1,-1 mod 8
and -1
if x = 3,-3 mod 8
.
The library syntax is long
kronecker(GEN x, GEN y)
.
(x,{y})
Least common multiple of x
and y
, i.e. such
that lcm (x,y)*
gcd (x,y) = x*y
, up to units. If y
is omitted and x
is a vector, returns the lcm
of all components of x
.
For integer arguments, return the non-negative lcm.
When x
and y
are both given and one of them is a vector/matrix type,
the LCM is again taken recursively on each component, but in a different way.
If y
is a vector, resp. matrix, then the result has the same type as y
,
and components equal to lcm(x, y[i])
, resp. lcm(x, y[,i])
. Else
if x
is a vector/matrix the result has the same type as x
and an
analogous definition. Note that for these types, lcm
is not
commutative.
Note that lcm(v)
is quite different from
l = v[1]; for (i = 1, #v, l = lcm(l, v[i]))
Indeed, lcm(v)
is a scalar, but l
may not be (if one of
the v[i]
is a vector/matrix). The computation uses a divide-conquer tree
and should be much more efficient, especially when using the GMP
multiprecision kernel (and more subquadratic algorithms become available):
? v = vector(10^5, i, random); ? lcm(v); time = 546 ms. ? l = v[1]; for (i = 1, #v, l = lcm(l, v[i])) time = 4,561 ms.
The library syntax is GEN
glcm0(GEN x, GEN y = NULL)
.
(x,b,{&z})
Return the largest integer e
so that b^e <= x
, where the
parameters b > 1
and x > 0
are both integers. If the parameter z
is
present, set it to b^e
.
? logint(1000, 2) %1 = 9 ? 2^9 %2 = 512 ? logint(1000, 2, &z) %3 = 9 ? z %4 = 512
@3The number of digits used to write b
in base x
is
1 + logint(x,b)
:
? #digits(1000!, 10) %5 = 2568 ? logint(1000!, 10) %6 = 2567
@3This function may conveniently replace
floor( log(x) / log(b) )
@3which may not give the correct answer since PARI does not guarantee exact rounding.
The library syntax is long
logint0(GEN x, GEN b, GEN *z = NULL)
.
(x)
Moebius mu-function of |x|
. x
must be of type integer.
The library syntax is long
moebius(GEN x)
.
(x)
Finds the smallest pseudoprime (see
ispseudoprime
) greater than or equal to x
. x
can be of any real
type. Note that if x
is a pseudoprime, this function returns x
and not
the smallest pseudoprime strictly larger than x
. To rigorously prove that
the result is prime, use isprime
.
The library syntax is GEN
nextprime(GEN x)
.
(n)
Gives the number of unrestricted partitions of
n
, usually called p(n)
in the literature; in other words the number of
nonnegative integer solutions to a+2b+3c+.. .= n
. n
must be of type
integer and n < 10^{15}
(with trivial values p(n) = 0
for n < 0
and
p(0) = 1
). The algorithm uses the Hardy-Ramanujan-Rademacher formula.
To explicitly enumerate them, see partitions
.
The library syntax is GEN
numbpart(GEN n)
.
(x)
Number of divisors of |x|
. x
must be of type integer.
The library syntax is GEN
numdiv(GEN x)
.
(x)
Number of distinct prime divisors of |x|
. x
must be of type integer.
? factor(392) %1 = [2 3]
[7 2]
? omega(392) %2 = 2; \\ without multiplicity ? bigomega(392) %3 = 5; \\ = 3+2, with multiplicity
The library syntax is long
omega(GEN x)
.
(k,{a = k},{n = k}))
Returns the vector of partitions of the integer k
as a sum of positive
integers (parts); for k < 0
, it returns the empty set []
, and for k
= 0
the trivial partition (no parts). A partition is given by a
t_VECSMALL
, where parts are sorted in nondecreasing order:
? partitions(3) %1 = [Vecsmall([3]), Vecsmall([1, 2]), Vecsmall([1, 1, 1])]
@3correspond to 3
, 1+2
and 1+1+1
. The number
of (unrestricted) partitions of k
is given
by numbpart
:
? #partitions(50) %1 = 204226 ? numbpart(50) %2 = 204226
@3Optional parameters n
and a
are as follows:
@3* n =
nmax (resp. n = [
nmin,
nmax]
) restricts
partitions to length less than nmax (resp. length between
nmin and nmax
), where the length is the number of nonzero
entries.
@3* a =
amax (resp. a = [
amin,
amax]
) restricts the parts
to integers less than amax (resp. between amin and
amax).
? partitions(4, 2) \\ parts bounded by 2 %1 = [Vecsmall([2, 2]), Vecsmall([1, 1, 2]), Vecsmall([1, 1, 1, 1])] ? partitions(4,, 2) \\ at most 2 parts %2 = [Vecsmall([4]), Vecsmall([1, 3]), Vecsmall([2, 2])] ? partitions(4,[0,3], 2) \\ at most 2 parts %3 = [Vecsmall([4]), Vecsmall([1, 3]), Vecsmall([2, 2])]
By default, parts are positive and we remove zero entries unless
amin <= 0
, in which case nmin
is ignored and X
is of constant length
nmax:
? partitions(4, [0,3]) \\ parts between 0 and 3 %1 = [Vecsmall([0, 0, 1, 3]), Vecsmall([0, 0, 2, 2]),\ Vecsmall([0, 1, 1, 2]), Vecsmall([1, 1, 1, 1])]
The library syntax is GEN
partitions(long k, GEN a = NULL, GEN n) = NULL)
.
(x,{p},{a})
Returns the vector of distinct roots of the polynomial x
in the field
F_q
defined by the irreducible polynomial a
over F_p
. The
coefficients of x
must be operation-compatible with Z/p
Z.
Either a
or p
can omitted (in which case both are ignored) if x has
t_FFELT
coefficients:
? polrootsff(x^2 + 1, 5, y^2+3) \\ over F_5[y]/(y^2+3) ~ F_25 %1 = [Mod(Mod(3, 5), Mod(1, 5)*y^2 + Mod(3, 5)), Mod(Mod(2, 5), Mod(1, 5)*y^2 + Mod(3, 5))] ? t = ffgen(y^2 + Mod(3,5), 't); \\ a generator for F_25 as a t_FFELT ? polrootsff(x^2 + 1) \\ not enough information to determine the base field *** at top-level: polrootsff(x^2+1) *** ^----------------- *** polrootsff: incorrect type in factorff. ? polrootsff(x^2 + t^0) \\ make sure one coeff. is a t_FFELT %3 = [3, 2] ? polrootsff(x^2 + t + 1) %4 = [2*t + 1, 3*t + 4]
Notice that the second syntax is easier to use and much more readable.
The library syntax is GEN
polrootsff(GEN x, GEN p = NULL, GEN a = NULL)
.
(x)
Finds the largest pseudoprime (see
ispseudoprime
) less than or equal to x
. x
can be of any real type.
Returns 0 if x <= 1
. Note that if x
is a prime, this function returns x
and not the largest prime strictly smaller than x
. To rigorously prove that
the result is prime, use isprime
.
The library syntax is GEN
precprime(GEN x)
.
(n)
The n-th
prime number
? prime(10^9) %1 = 22801763489
@3Uses checkpointing and a naive O(n)
algorithm.
The library syntax is GEN
prime(long n)
.
(x)
The prime counting function. Returns the number of
primes p
, p <= x
.
? primepi(10) %1 = 4; ? primes(5) %2 = [2, 3, 5, 7, 11] ? primepi(10^11) %3 = 4118054813
@3Uses checkpointing and a naive O(x)
algorithm.
The library syntax is GEN
primepi(GEN x)
.
(n)
Creates a row vector whose components are the first n
prime numbers.
(Returns the empty vector for n <= 0
.) A t_VEC
n = [a,b]
is also
allowed, in which case the primes in [a,b]
are returned
? primes(10) \\ the first 10 primes %1 = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] ? primes([0,29]) \\ the primes up to 29 %2 = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] ? primes([15,30]) %3 = [17, 19, 23, 29]
The library syntax is GEN
primes0(GEN n)
.
(D,{
flag = 0})
Ordinary class number of the quadratic order of discriminant D
, for
``small'' values of D
.
@3* if D > 0
or flag = 1
, use a O(|D|^{1/2})
algorithm (compute L(1,
chi_D)
with the approximate functional equation).
This is slower than quadclassunit
as soon as |D| ~ 10^2
or
so and is not meant to be used for large D
.
@3* if D < 0
and flag = 0
(or omitted), use a O(|D|^{1/4})
algorithm (Shanks's baby-step/giant-step method). It should
be faster than quadclassunit
for small values of D
, say
|D| < 10^{18}
.
@3Important warning. In the latter case, this function only
implements part of Shanks's method (which allows to speed it up
considerably). It gives unconditionnally correct results for |D| < 2.
10^{10}
, but may give incorrect results for larger values if the class
group has many cyclic factors. We thus recommend to double-check results
using the function quadclassunit
, which is about 2 to 3 times slower in
the above range, assuming GRH. We currently have no counter-examples but
they should exist: we'd appreciate a bug report if you find one.
@3Warning. Contrary to what its name implies, this routine does not
compute the number of classes of binary primitive forms of discriminant D
,
which is equal to the narrow class number. The two notions are the same
when D < 0
or the fundamental unit varepsilon has negative norm; when D
> 0
and N
varepsilon > 0
, the number of classes of forms is twice the
ordinary class number. This is a problem which we cannot fix for backward
compatibility reasons. Use the following routine if you are only interested
in the number of classes of forms:
QFBclassno(D) = qfbclassno(D) * if (D < 0 || norm(quadunit(D)) < 0, 1, 2)
Here are a few examples:
? qfbclassno(400000028) time = 3,140 ms. %1 = 1 ? quadclassunit(400000028).no time = 20 ms. \{ much faster} %2 = 1 ? qfbclassno(-400000028) time = 0 ms. %3 = 7253 \{ correct, and fast enough} ? quadclassunit(-400000028).no time = 0 ms. %4 = 7253
See also qfbhclassno
.
The library syntax is GEN
qfbclassno0(GEN D, long flag)
.
The following functions are also available:
GEN
classno(GEN D)
(flag = 0
)
GEN
classno2(GEN D)
(flag = 1
).
@3Finally
GEN
hclassno(GEN D)
computes the class number of an imaginary
quadratic field by counting reduced forms, an O(|D|)
algorithm.
(x,y)
composition of the binary quadratic forms x
and y
, without
reduction of the result. This is useful e.g. to compute a generating
element of an ideal. The result is undefined if x
and y
do not have the
same discriminant.
The library syntax is GEN
qfbcompraw(GEN x, GEN y)
.
(x)
Hurwitz class number of x
, where
x
is non-negative and congruent to 0 or 3 modulo 4. For x > 5.
10^5
, we assume the GRH, and use quadclassunit
with default
parameters.
The library syntax is GEN
hclassno(GEN x)
.
(x,y,L)
composition of the primitive positive
definite binary quadratic forms x
and y
(type t_QFI
) using the NUCOMP
and NUDUPL algorithms of Shanks, à la Atkin. L
is any positive
constant, but for optimal speed, one should take L = |D/4|^{1/4}
, i.e.
sqrtnint(abs(D) >> 2,4)
, where D
is the common discriminant of x
and
y
. When x
and y
do not have the same discriminant, the result is
undefined.
The current implementation is slower than the generic routine for small D
,
and becomes faster when D
has about 45
bits.
The library syntax is GEN
nucomp(GEN x, GEN y, GEN L)
.
Also available is GEN
nudupl(GEN x, GEN L)
when x = y
.
(x,n,{L})
n
-th power of the primitive positive definite
binary quadratic form x
using Shanks's NUCOMP and NUDUPL algorithms;
if set, L
should be equal to sqrtnint(abs(D) >> 2,4)
, where D < 0
is
the discriminant of x
.
The current implementation is slower than the generic routine for small
discriminant D
, and becomes faster for D ~ 2^{45}
.
The library syntax is GEN
nupow(GEN x, GEN n, GEN L = NULL)
.
(x,n)
n
-th power of the binary quadratic form
x
, computed without doing any reduction (i.e. using qfbcompraw
).
Here n
must be non-negative and n < 2^{31}
.
The library syntax is GEN
qfbpowraw(GEN x, long n)
.
(x,p)
Prime binary quadratic form of discriminant
x
whose first coefficient is p
, where |p|
is a prime number.
By abuse of notation,
p = +- 1
is also valid and returns the unit form. Returns an
error if x
is not a quadratic residue mod p
, or if x < 0
and p < 0
.
(Negative definite t_QFI
are not implemented.) In the case where x > 0
,
the ``distance'' component of the form is set equal to zero according to the
current precision.
The library syntax is GEN
primeform(GEN x, GEN p, long prec)
.
qfbred(x,{
flag = 0},{d},{
isd},{
sd})
Reduces the binary quadratic form x
(updating Shanks's distance function
if x
is indefinite). The binary digits of flag are toggles meaning
1: perform a single reduction step
2: don't update Shanks's distance
The arguments d
, isd, sd, if present, supply the values of the
discriminant, floor{
sqrt {d}}
, and sqrt {d}
respectively
(no checking is done of these facts). If d < 0
these values are useless,
and all references to Shanks's distance are irrelevant.
The library syntax is GEN
qfbred0(GEN x, long flag, GEN d = NULL, GEN isd = NULL, GEN sd = NULL)
.
Also available are
GEN
redimag(GEN x)
(for definite x
),
@3and for indefinite forms:
GEN
redreal(GEN x)
GEN
rhoreal(GEN x)
( = qfbred(x,1)
),
GEN
redrealnod(GEN x, GEN isd)
( = qfbred(x,2,,isd)
),
GEN
rhorealnod(GEN x, GEN isd)
( = qfbred(x,3,,isd)
).
qfbredsl2(x,{
data})
Reduction of the (real or imaginary) binary quadratic form x
, return
[y,g]
where y
is reduced and g
in SL(2,
Z)
is such that
g.x = y
; data, if
present, must be equal to [D, sqrtint(D)]
, where D > 0
is the
discriminant of x
. In case x
is t_QFR
, the distance component is
unaffected.
The library syntax is GEN
qfbredsl2(GEN x, GEN data = NULL)
.
(Q,p)
Solve the equation Q(x,y) = p
over the integers,
where Q
is a binary quadratic form and p
a prime number.
Return [x,y]
as a two-components vector, or zero if there is no solution.
Note that this function returns only one solution and not all the solutions.
Let D =
disc Q
. The algorithm used runs in probabilistic polynomial time
in p
(through the computation of a square root of D
modulo p
); it is
polynomial time in D
if Q
is imaginary, but exponential time if Q
is
real (through the computation of a full cycle of reduced forms). In the
latter case, note that bnfisprincipal
provides a solution in heuristic
subexponential time in D
assuming the GRH.
The library syntax is GEN
qfbsolve(GEN Q, GEN p)
.
quadclassunit(D,{
flag = 0},{
tech = []})
Buchmann-McCurley's sub-exponential algorithm for computing the
class group of a quadratic order of discriminant D
.
This function should be used instead of qfbclassno
or quadregula
when D < -10^{25}
, D > 10^{10}
, or when the structure is wanted. It
is a special case of bnfinit
, which is slower, but more robust.
The result is a vector v
whose components should be accessed using member
functions:
@3* v.no
: the class number
@3* v.cyc
: a vector giving the structure of the class group as a
product of cyclic groups;
@3* v.gen
: a vector giving generators of those cyclic groups (as
binary quadratic forms).
@3* v.reg
: the regulator, computed to an accuracy which is the
maximum of an internal accuracy determined by the program and the current
default (note that once the regulator is known to a small accuracy it is
trivial to compute it to very high accuracy, see the tutorial).
The flag is obsolete and should be left alone. In older versions,
it supposedly computed the narrow class group when D > 0
, but this did not
work at all; use the general function bnfnarrow
.
Optional parameter tech is a row vector of the form [c_1, c_2]
,
where c_1 <= c_2
are non-negative real numbers which control the execution
time and the stack size, see se:GRHbnf. The parameter is used as a
threshold to balance the relation finding phase against the final linear
algebra. Increasing the default c_1
means that relations are easier
to find, but more relations are needed and the linear algebra will be
harder. The default value for c_1
is 0
and means that it is taken equal
to c_2
. The parameter c_2
is mostly obsolete and should not be changed,
but we still document it for completeness: we compute a tentative class
group by generators and relations using a factorbase of prime ideals
<= c_1 (
log |D|)^2
, then prove that ideals of norm
<= c_2 (
log |D|)^2
do
not generate a larger group. By default an optimal c_2
is chosen, so that
the result is provably correct under the GRH --- a famous result of Bach
states that c_2 = 6
is fine, but it is possible to improve on this
algorithmically. You may provide a smaller c_2
, it will be ignored
(we use the provably correct
one); you may provide a larger c_2
than the default value, which results
in longer computing times for equally correct outputs (under GRH).
The library syntax is GEN
quadclassunit0(GEN D, long flag, GEN tech = NULL, long prec)
.
If you really need to experiment with the tech parameter, it is
usually more convenient to use
GEN
Buchquad(GEN D, double c1, double c2, long prec)
(x)
Discriminant of the étale algebra Q(
sqrt {x})
, where x\in
Q^*
.
This is the same as coredisc
(d)
where d
is the integer square-free
part of x
, so x = d f^2
with f\in
Q^*
and d\in
Z.
This returns 0
for x = 0
, 1
for x
square and the discriminant of the
quadratic field Q(
sqrt {x})
otherwise.
? quaddisc(7) %1 = 28 ? quaddisc(-7) %2 = -7
The library syntax is GEN
quaddisc(GEN x)
.
(D)
Creates the quadratic
number omega = (a+
sqrt {D})/2
where a = 0
if D = 0 mod 4
,
a = 1
if D = 1 mod 4
, so that (1,
omega)
is an integral basis for the
quadratic order of discriminant D
. D
must be an integer congruent to 0 or
1 modulo 4, which is not a square.
The library syntax is GEN
quadgen(GEN D)
.
(D)
Relative equation defining the
Hilbert class field of the quadratic field of discriminant D
.
If D < 0
, uses complex multiplication (Schertz's variant).
If D > 0
Stark units are used and (in rare cases) a
vector of extensions may be returned whose compositum is the requested class
field. See bnrstark
for details.
The library syntax is GEN
quadhilbert(GEN D, long prec)
.
(D,{v = 'x})
Creates the ``canonical'' quadratic
polynomial (in the variable v
) corresponding to the discriminant D
,
i.e. the minimal polynomial of quadgen(D)
. D
must be an integer
congruent to 0 or 1 modulo 4, which is not a square.
The library syntax is GEN
quadpoly0(GEN D, long v = -1)
where v
is a variable number.
(D,f)
Relative equation for the ray
class field of conductor f
for the quadratic field of discriminant D
using analytic methods. A bnf
for x^2 - D
is also accepted in place
of D
.
For D < 0
, uses the sigma function and Schertz's method.
For D > 0
, uses Stark's conjecture, and a vector of relative equations may be
returned. See bnrstark
for more details.
The library syntax is GEN
quadray(GEN D, GEN f, long prec)
.
(x)
Regulator of the quadratic field of positive discriminant x
. Returns
an error if x
is not a discriminant (fundamental or not) or if x
is a
square. See also quadclassunit
if x
is large.
The library syntax is GEN
quadregulator(GEN x, long prec)
.
(D)
Fundamental unit of the
real quadratic field Q(
sqrt D)
where D
is the positive discriminant
of the field. If D
is not a fundamental discriminant, this probably gives
the fundamental unit of the corresponding order. D
must be an integer
congruent to 0 or 1 modulo 4, which is not a square; the result is a
quadratic number (see Label se:quadgen).
The library syntax is GEN
quadunit(GEN D)
.
(n)
Compute the value of Ramanujan's tau function at an individual n
,
assuming the truth of the GRH (to compute quickly class numbers of imaginary
quadratic fields using quadclassunit
).
Algorithm in ~{O}(n^{1/2})
using O(
log n)
space. If all values up
to N
are required, then
sum tau(n)q^n = q
prod_{n >= 1} (1-q^n)^{24}
will produce them in time ~{O}(N)
, against ~{O}(N^{3/2})
for
individual calls to ramanujantau
; of course the space complexity then
becomes ~{O}(N)
.
? tauvec(N) = Vec(q*eta(q + O(q^N))^24); ? N = 10^4; v = tauvec(N); time = 26 ms. ? ramanujantau(N) %3 = -482606811957501440000 ? w = vector(N, n, ramanujantau(n)); \\ much slower ! time = 13,190 ms. ? v == w %4 = 1
The library syntax is GEN
ramanujantau(GEN n)
.
\subsec{randomprime({N = 2^{{31}}})
}
Returns a strong pseudo prime (see ispseudoprime
) in [2,N-1]
.
A t_VEC
N = [a,b]
is also allowed, with a <= b
in which case a
pseudo prime a <= p <= b
is returned; if no prime exists in the
interval, the function will run into an infinite loop. If the upper bound
is less than 2^{64}
the pseudo prime returned is a proven prime.
The library syntax is GEN
randomprime(GEN N = NULL)
.
({x = []})
Removes the primes listed in x
from
the prime number table. In particular removeprimes(addprimes())
empties
the extra prime table. x
can also be a single integer. List the current
extra primes if x
is omitted.
The library syntax is GEN
removeprimes(GEN x = NULL)
.
(x,{k = 1})
Sum of the k-th
powers of the positive divisors of |x|
. x
and k
must be of type integer.
The library syntax is GEN
sumdivk(GEN x, long k)
.
Also available is GEN
sumdiv(GEN n)
, for k = 1
.
(x)
Returns the integer square root of x
, i.e. the largest integer y
such that y^2 <= x
, where x
a non-negative integer.
? N = 120938191237; sqrtint(N) %1 = 347761 ? sqrt(N) %2 = 347761.68741970412747602130964414095216
The library syntax is GEN
sqrtint(GEN x)
.
(x,n)
Returns the integer n
-th root of x
, i.e. the largest integer y
such
that y^n <= x
, where x
is a non-negative integer.
? N = 120938191237; sqrtnint(N, 5) %1 = 164 ? N^(1/5) %2 = 164.63140849829660842958614676939677391
@3The special case n = 2
is sqrtint
The library syntax is GEN
sqrtnint(GEN x, long n)
.
(n,k,{
flag = 1})
Stirling number of the first kind s(n,k)
(flag = 1
, default) or
of the second kind S(n,k)
(flag = 2), where n
, k
are non-negative
integers. The former is (-1)^{n-k}
times the
number of permutations of n
symbols with exactly k
cycles; the latter is
the number of ways of partitioning a set of n
elements into k
non-empty
subsets. Note that if all s(n,k)
are needed, it is much faster to compute
sum_k s(n,k) x^k = x(x-1)...(x-n+1).
Similarly, if a large number of S(n,k)
are needed for the same k
,
one should use
sum_n S(n,k) x^n = (x^k)/((1-x)...(1-kx)).
(Should be implemented using a divide and conquer product.) Here are
simple variants for n
fixed:
/* list of s(n,k), k = 1..n */ vecstirling(n) = Vec( factorback(vector(n-1,i,1-i*'x)) )
/* list of S(n,k), k = 1..n */ vecstirling2(n) = { my(Q = x^(n-1), t); vector(n, i, t = divrem(Q, x-i); Q=t[1]; simplify(t[2])); }
The library syntax is GEN
stirling(long n, long k, long flag)
.
Also available are GEN
stirling1(ulong n, ulong k)
(flag = 1
) and GEN
stirling2(ulong n, ulong k)
(flag = 2
).
(h,k)
Returns the Dedekind sum attached to the integers h
and k
,
corresponding to a fast implementation of
s(h,k) = sum(n = 1, k-1, (n/k)*(frac(h*n/k) - 1/2))
The library syntax is GEN
sumdedekind(GEN h, GEN k)
.
sumdigits(n,{B = 10})
Sum of digits in the integer n
, when written in base B > 1
.
? sumdigits(123456789) %1 = 45 ? sumdigits(123456789, 2) %1 = 16
@3Note that the sum of bits in n
is also returned by
hammingweight
. This function is much faster than
vecsum(digits(n,B))
when B
is 10
or a power of 2
, and only
slightly faster in other cases.
The library syntax is GEN
sumdigits0(GEN n, GEN B = NULL)
.
Also available is GEN
sumdigits(GEN n)
, for B = 10
.
(G,
chi, N)
Let G
be attached to (
Z/q
Z)^*
(as per G = idealstar(,q)
)
and let chi
be a Dirichlet character on (
Z/q
Z)^*
, given by
@3* a t_VEC
: a standard character on bid.gen
,
@3* a t_INT
or a t_COL
: a Conrey index in (
Z/q
Z)^*
or its
Conrey logarithm;
see Label se:dirichletchar or ??character
.
Let N
be a multiple of q
, return the character modulo N
induced by
chi
. As usual for arithmetic functions, the new modulus N
can be
given as a t_INT
, via a factorization matrix or a pair
[N, factor(N)]
, or by idealstar(,N)
.
? G = idealstar(,4); ? chi = znconreylog(G,1); \\ trivial character mod 4 ? zncharinduce(G, chi, 80) \\ now mod 80 %3 = [0, 0, 0]~ ? zncharinduce(G, 1, 80) \\ same using directly Conrey label %4 = [0, 0, 0]~ ? G2 = idealstar(,80); ? zncharinduce(G, 1, G2) \\ same %4 = [0, 0, 0]~
? chi = zncharinduce(G, 3, G2) \\ induce the non-trivial character mod 4 %5 = [1, 0, 0]~ ? znconreyconductor(G2, chi, &chi0) %6 = [4, Mat([2, 2])] ? chi0 %7 = [1]~
@3Here is a larger example:
? G = idealstar(,126000); ? label = 1009; ? chi = znconreylog(G, label) %3 = [0, 0, 0, 14, 0]~ ? N0 = znconreyconductor(G, label, &chi0) %4 = [125, Mat([5, 3])] ? chi0 \\ primitive character mod 5^3 attached to chi %5 = [14]~ ? G0 = idealstar(,N0); ? zncharinduce(G0, chi0, G) \\ induce back %7 = [0, 0, 0, 14, 0]~ ? znconreyexp(G, %) %8 = 1009
The library syntax is GEN
zncharinduce(GEN G, GEN chi, GEN N)
.
(G,
chi)
Let G
be attached to (
Z/N
Z)^*
(as per G = idealstar(,N)
)
and let chi
be a Dirichlet character on (
Z/N
Z)^*
, given by
@3* a t_VEC
: a standard character on bid.gen
,
@3* a t_INT
or a t_COL
: a Conrey index in (
Z/q
Z)^*
or its
Conrey logarithm;
see Label se:dirichletchar or ??character
.
Return 1
if and only if chi
(-1) = -1
and 0
otherwise.
? G = idealstar(,8); ? zncharisodd(G, 1) \\ trivial character %2 = 0 ? zncharisodd(G, 3) %3 = 1 ? chareval(G, 3, -1) %4 = 1/2
The library syntax is long
zncharisodd(GEN G, GEN chi)
.
(G,
chi, {
flag = 0})
Let G
be attached to (
Z/N
Z)^*
(as per G = idealstar(,N)
)
and let chi
be a Dirichlet character on (
Z/N
Z)^*
, given by
@3* a t_VEC
: a standard character on bid.gen
,
@3* a t_INT
or a t_COL
: a Conrey index in (
Z/q
Z)^*
or its
Conrey logarithm;
see Label se:dirichletchar or ??character
.
If flag = 0
, return the discriminant D
if chi
is real equal to the
Kronecker symbol (D/.)
and 0
otherwise. The discriminant D
is
fundamental if and only if chi
is primitive.
If flag = 1
, return the fundamental discriminant attached to the
corresponding primitive character.
? G = idealstar(,8); CHARS = [1,3,5,7]; \\ Conrey labels ? apply(t->znchartokronecker(G,t), CHARS) %2 = [4, -8, 8, -4] ? apply(t->znchartokronecker(G,t,1), CHARS) %3 = [1, -8, 8, -4]
The library syntax is GEN
znchartokronecker(GEN G, GEN chi, long flag)
.
(
bid,m)
Given a bid attached to (
Z/q
Z)^*
(as per
bid = idealstar(,q)
), this function returns the Dirichlet character
attached to m \in (
Z/q
Z)^*
via Conrey's logarithm, which
establishes a ``canonical'' bijection between (
Z/q
Z)^*
and its dual.
Let q =
prod_p p^{e_p}
be the factorization of q
into distinct primes.
For all odd p
with e_p > 0
, let g_p
be the element in (
Z/q
Z)^*
which is
@3* congruent to 1
mod q/p^{e_p}
,
@3* congruent mod p^{e_p}
to the smallest integer whose order
is phi(p^{e_p})
.
For p = 2
, we let g_4
(if 2^{e_2} >= 4
) and g_8
(if furthermore
(2^{e_2} >= 8
) be the elements in (
Z/q
Z)^*
which
are
@3* congruent to 1
mod q/2^{e_2}
,
@3* g_4 = -1 mod 2^{e_2}
,
@3* g_8 = 5 mod 2^{e_2}
.
Then the g_p
(and the extra g_4
and g_8
if 2^{e_2} >= 2
) are
independent
generators of (
Z/q
Z)^*
, i.e. every m
in (
Z/q
Z)^*
can be written
uniquely as prod_p g_p^{m_p}
, where m_p
is defined modulo the order
o_p
of g_p
and p \in S_q
, the set of prime divisors of q
together with 4
if 4 | q
and 8
if 8 | q
. Note that the g_p
are in general
not SNF
generators as produced by znstar
or idealstar
whenever
omega(q) >= 2
, although their number is the same. They however allow
to handle the finite abelian group (
Z/q
Z)^*
in a fast and elegant
way. (Which unfortunately does not generalize to ray class groups or Hecke
characters.)
The Conrey logarithm of m
is the vector (m_p)_{p\in S_q}
, obtained
via znconreylog
. The Conrey character chi_q(m,.)
attached to
m
mod q
maps
each g_p
, p\in S_q
to e(m_p / o_p)
, where e(x) =
exp (2i
pi x)
.
This function returns the Conrey character expressed in the standard PARI
way in terms of the SNF generators bid.gen
.
@3Note. It is useless to include the generators
in the bid, except for debugging purposes: they are well defined from
elementary matrix operations and Chinese remaindering, their explicit value
as elements in (
Z/q
Z)^*
is never used.
? G = idealstar(,8,2); /*add generators for debugging:*/ ? G.cyc %2 = [2, 2] \\ Z/2 x Z/2 ? G.gen %3 = [7, 3] ? znconreychar(G,1) \\ 1 is always the trivial character %4 = [0, 0] ? znconreychar(G,2) \\ 2 is not coprime to 8 !!! *** at top-level: znconreychar(G,2) *** ^----------------- *** znconreychar: elements not coprime in Zideallog: 2 8 *** Break loop: type 'break' to go back to GP prompt break>
? znconreychar(G,3) %5 = [0, 1] ? znconreychar(G,5) %6 = [1, 1] ? znconreychar(G,7) %7 = [1, 0]
@3We indeed get all 4 characters of (
Z/8
Z)^*
.
For convenience, we allow to input the Conrey logarithm of m
instead of m
:
? G = idealstar(,55); ? znconreychar(G,7) %2 = [7, 0] ? znconreychar(G, znconreylog(G,7)) %3 = [7, 0]
The library syntax is GEN
znconreychar(GEN bid, GEN m)
.
znconreyconductor(
bid,
chi, {&
chi0})
Let bid be attached to (
Z/q
Z)^*
(as per
bid = idealstar(,q)
) and chi
be a Dirichlet character on
(
Z/q
Z)^*
, given by
@3* a t_VEC
: a standard character on bid.gen
,
@3* a t_INT
or a t_COL
: a Conrey index in (
Z/q
Z)^*
or its
Conrey logarithm;
see Label se:dirichletchar or ??character
.
Return the conductor of chi
, as the t_INT
bid.mod
if chi
is primitive, and as a pair [N, faN]
(with faN
the
factorization of N
) otherwise.
If chi0
is present, set it to the Conrey logarithm of the attached
primitive character.
? G = idealstar(,126000); ? znconreyconductor(G,11) \\ primitive %2 = 126000 ? znconreyconductor(G,1) \\ trivial character, not primitive! %3 = [1, matrix(0,2)] ? N0 = znconreyconductor(G,1009, &chi0) \\ character mod 5^3 %4 = [125, Mat([5, 3])] ? chi0 %5 = [14]~ ? G0 = idealstar(,N0); \\ format [N,factor(N)] accepted ? znconreyexp(G0, chi0) %7 = 9 ? znconreyconductor(G0, chi0) \\ now primitive, as expected %8 = 125
@3The group G0
is not computed as part of
znconreyconductor
because it needs to be computed only once per
conductor, not once per character.
The library syntax is GEN
znconreyconductor(GEN bid, GEN chi, GEN *chi0 = NULL)
.
(
bid,
chi)
Given a bid attached to (
Z/q
Z)^*
(as per
bid = idealstar(,q)
), this function returns the Conrey exponential of
the character chi: it returns the integer
m \in (
Z/q
Z)^*
such that znconreylog(
bid, m)
is chi.
The character chi is given either as a
@3* t_VEC
: in terms of the generators bid.gen
;
@3* t_COL
: a Conrey logarithm.
? G = idealstar(,126000) ? znconreylog(G,1) %2 = [0, 0, 0, 0, 0]~ ? znconreyexp(G,%) %3 = 1 ? G.cyc \\ SNF generators %4 = [300, 12, 2, 2, 2] ? chi = [100, 1, 0, 1, 0]; \\ some random character on SNF generators ? znconreylog(G, chi) \\ in terms of Conrey generators %6 = [0, 3, 3, 0, 2]~ ? znconreyexp(G, %) \\ apply to a Conrey log %7 = 18251 ? znconreyexp(G, chi) \\ ... or a char on SNF generators %8 = 18251 ? znconreychar(G,%) %9 = [100, 1, 0, 1, 0]
The library syntax is GEN
znconreyexp(GEN bid, GEN chi)
.
(
bid,m)
Given a bid attached to (
Z/q
Z)^*
(as per
bid = idealstar(,q)
), this function returns the Conrey logarithm of
m \in (
Z/q
Z)^*
.
Let q =
prod_p p^{e_p}
be the factorization of q
into distinct primes,
where we assume e_2 = 0
or e_2 >= 2
. (If e_2 = 1
, we can ignore 2
from the factorization, as if we replaced q
by q/2
, since (
Z/q
Z)^*
~ (
Z/(q/2)
Z)^*
.)
For all odd p
with e_p > 0
, let g_p
be the element in (
Z/q
Z)^*
which is
@3* congruent to 1
mod q/p^{e_p}
,
@3* congruent mod p^{e_p}
to the smallest integer whose order
is phi(p^{e_p})
for p
odd,
For p = 2
, we let g_4
(if 2^{e_2} >= 4
) and g_8
(if furthermore
(2^{e_2} >= 8
) be the elements in (
Z/q
Z)^*
which
are
@3* congruent to 1
mod q/2^{e_2}
,
@3* g_4 = -1 mod 2^{e_2}
,
@3* g_8 = 5 mod 2^{e_2}
.
Then the g_p
(and the extra g_4
and g_8
if 2^{e_2} >= 2
) are
independent
generators of Z/q
Z^*
, i.e. every m
in (
Z/q
Z)^*
can be written
uniquely as prod_p g_p^{m_p}
, where m_p
is defined modulo the
order o_p
of g_p
and p \in S_q
, the set of prime divisors of q
together with 4
if 4 | q
and 8
if 8 | q
.
Note that the g_p
are in general not SNF
generators as produced by znstar
or idealstar
whenever
omega(q) >= 2
, although their number is the same. They however allow
to handle the finite abelian group (
Z/q
Z)^*
in a fast and elegant
way. (Which unfortunately does not generalize to ray class groups or Hecke
characters.)
The Conrey logarithm of m
is the vector (m_p)_{p\in S_q}
. The inverse
function znconreyexp
recovers the Conrey label m
from a character.
? G = idealstar(,126000); ? znconreylog(G,1) %2 = [0, 0, 0, 0, 0]~ ? znconreyexp(G, %) %3 = 1 ? znconreylog(G,2) \\ 2 is not coprime to modulus !!! *** at top-level: znconreylog(G,2) *** ^----------------- *** znconreylog: elements not coprime in Zideallog: 2 126000 *** Break loop: type 'break' to go back to GP prompt break> ? znconreylog(G,11) \\ wrt. Conrey generators %4 = [0, 3, 1, 76, 4]~ ? log11 = ideallog(,11,G) \\ wrt. SNF generators %5 = [178, 3, -75, 1, 0]~
For convenience, we allow to input the ordinary discrete log of m
,
ideallog(,m,bid)
, which allows to convert discrete logs
from bid.gen
generators to Conrey generators.
? znconreylog(G, log11) %7 = [0, 3, 1, 76, 4]~
@3We also allow a character (t_VEC
) on bid.gen
and
return its representation on the Conrey generators.
? G.cyc %8 = [300, 12, 2, 2, 2] ? chi = [10,1,0,1,1]; ? znconreylog(G, chi) %10 = [1, 3, 3, 10, 2]~ ? n = znconreyexp(G, chi) %11 = 84149 ? znconreychar(G, n) %12 = [10, 1, 0, 1, 1]
The library syntax is GEN
znconreylog(GEN bid, GEN m)
.
(P, N, X, {B = N})
N
being an integer and P\in
Z[X]
, finds all integers x
with
|x| <= X
such that
gcd (N, P(x)) >= B,
using Coppersmith's algorithm (a famous application of the LLL
algorithm). X
must be smaller than exp (
log ^2 B / (
deg (P)
log N))
:
for B = N
, this means X < N^{1/
deg (P)}
. Some x
larger than X
may
be returned if you are very lucky. The smaller B
(or the larger X
), the
slower the routine will be. The strength of Coppersmith method is the
ability to find roots modulo a general composite N
: if N
is a prime
or a prime power, polrootsmod
or polrootspadic
will be much
faster.
We shall now present two simple applications. The first one is
finding non-trivial factors of N
, given some partial information on the
factors; in that case B
must obviously be smaller than the largest
non-trivial divisor of N
.
setrand(1); \\ to make the example reproducible interval = [10^30, 10^31]; p = randomprime(interval); q = randomprime(interval); N = p*q; p0 = p % 10^20; \\ assume we know 1) p > 10^29, 2) the last 19 digits of p L = zncoppersmith(10^19*x + p0, N, 10^12, 10^29)
\\ result in 10ms. %6 = [738281386540] ? gcd(L[1] * 10^19 + p0, N) == p %7 = 1
@3and we recovered p
, faster than by trying all
possibilities < 10^{12}
.
The second application is an attack on RSA with low exponent, when the
message x
is short and the padding P
is known to the attacker. We use
the same RSA modulus N
as in the first example:
setrand(1); P = random(N); \\ known padding e = 3; \\ small public encryption exponent X = floor(N^0.3); \\ N^(1/e - epsilon) x0 = random(X); \\ unknown short message C = lift( (Mod(x0,N) + P)^e ); \\ known ciphertext, with padding P zncoppersmith((P + x)^3 - C, N, X)
\\ result in 244ms. %14 = [2679982004001230401]
? %[1] == x0 %15 = 1
@3
We guessed an integer of the order of 10^{18}
, almost instantly.
The library syntax is GEN
zncoppersmith(GEN P, GEN N, GEN X, GEN B = NULL)
.
(x,g,{o})
This functions allows two distinct modes of operation depending
on g
:
@3* if g
is the output of znstar
(with initialization),
we compute the discrete logarithm of x
with respect to the generators
contained in the structure. See ideallog
for details.
@3* else g
is an explicit element in (
Z/N
Z)^*
, we compute the
discrete logarithm of x
in (
Z/N
Z)^*
in base g
. The rest of this
entry describes the latter possibility.
The result is []
when x
is not a power of g
, though the function may
also enter an infinite loop in this case.
If present, o
represents the multiplicative order of g
, see
Label se:DLfun; the preferred format for this parameter is
[ord, factor(ord)]
, where ord
is the order of g
.
This provides a definite speedup when the discrete log problem is simple:
? p = nextprime(10^4); g = znprimroot(p); o = [p-1, factor(p-1)]; ? for(i=1,10^4, znlog(i, g, o)) time = 205 ms. ? for(i=1,10^4, znlog(i, g)) time = 244 ms. \\ a little slower
The result is undefined if g
is not invertible mod N
or if the supplied
order is incorrect.
This function uses
@3* a combination of generic discrete log algorithms (see below).
@3* in (
Z/N
Z)^*
when N
is prime: a linear sieve index calculus
method, suitable for N < 10^{50}
, say, is used for large prime divisors of
the order.
The generic discrete log algorithms are:
@3* Pohlig-Hellman algorithm, to reduce to groups of prime order q
,
where q | p-1
and p
is an odd prime divisor of N
,
@3* Shanks baby-step/giant-step (q < 2^{32}
is small),
@3* Pollard rho method (q > 2^{32}
).
The latter two algorithms require O(
sqrt {q})
operations in the group on
average, hence will not be able to treat cases where q > 10^{30}
, say.
In addition, Pollard rho is not able to handle the case where there are no
solutions: it will enter an infinite loop.
? g = znprimroot(101) %1 = Mod(2,101) ? znlog(5, g) %2 = 24 ? g^24 %3 = Mod(5, 101)
? G = znprimroot(2 * 101^10) %4 = Mod(110462212541120451003, 220924425082240902002) ? znlog(5, G) %5 = 76210072736547066624 ? G^% == 5 %6 = 1 ? N = 2^4*3^2*5^3*7^4*11; g = Mod(13, N); znlog(g^110, g) %7 = 110 ? znlog(6, Mod(2,3)) \\ no solution %8 = []
@3For convenience, g
is also allowed to be a p
-adic number:
? g = 3+O(5^10); znlog(2, g) %1 = 1015243 ? g^% %2 = 2 + O(5^10)
The library syntax is GEN
znlog0(GEN x, GEN g, GEN o = NULL)
.
The function
GEN
znlog(GEN x, GEN g, GEN o)
is also available
(x,{o})
x
must be an integer mod n
, and the
result is the order of x
in the multiplicative group (
Z/n
Z)^*
. Returns
an error if x
is not invertible.
The parameter o, if present, represents a non-zero
multiple of the order of x
, see Label se:DLfun; the preferred format for
this parameter is [ord, factor(ord)]
, where ord = eulerphi(n)
is the cardinality of the group.
The library syntax is GEN
znorder(GEN x, GEN o = NULL)
.
Also available is GEN
order(GEN x)
.
(n)
Returns a primitive root (generator) of (
Z/n
Z)^*
, whenever this
latter group is cyclic (n = 4
or n = 2p^k
or n = p^k
, where p
is an
odd prime and k >= 0
). If the group is not cyclic, the result is
undefined. If n
is a prime power, then the smallest positive primitive
root is returned. This may not be true for n = 2p^k
, p
odd.
Note that this function requires factoring p-1
for p
as above,
in order to determine the exact order of elements in
(
Z/n
Z)^*
: this is likely to be costly if p
is large.
The library syntax is GEN
znprimroot(GEN n)
.
(n,{
flag = 0})
Gives the structure of the multiplicative group (
Z/n
Z)^*
.
The output G
depends on the value of flag:
@3* flag = 0
(default), an abelian group structure [h,d,g]
,
where h =
phi(n)
is the order (G.no
), d
(G.cyc
)
is a k
-component row-vector d
of integers d_i
such that d_i > 1
,
d_i | d_{i-1}
for i >= 2
and
(
Z/n
Z)^* ~
prod_{i = 1}^k (
Z/d_i
Z),
and g
(G.gen
) is a k
-component row vector giving generators of
the image of the cyclic groups Z/d_i
Z.
@3* flag = 1
the result is a bid
structure without generators
(which are well defined but not explicitly computed, which saves time);
this allows computing discrite logarithms using znlog
(also in the
non-cyclic case!).
@3* flag = 2
same as flag = 1
with generators.
? G = znstar(40) %1 = [16, [4, 2, 2], [Mod(17, 40), Mod(21, 40), Mod(11, 40)]] ? G.no \\ eulerphi(40) %2 = 16 ? G.cyc \\ cycle structure %3 = [4, 2, 2] ? G.gen \\ generators for the cyclic components %4 = [Mod(17, 40), Mod(21, 40), Mod(11, 40)] ? apply(znorder, G.gen) %5 = [4, 2, 2]
@3According to the above definitions, znstar(0)
is
[2, [2], [-1]]
, corresponding to Z^*
.
The library syntax is GEN
znstar0(GEN n, long flag)
.
Instead the above hardcoded numerical flags, one should rather use
GEN
ZNstar(GEN N, long flag)
, where flag
is
an or-ed combination of nf_GEN
(include generators) and nf_INIT
(return a full bid
, not a group), possibly 0
. This offers
one more combination: no gen and no init.
An elliptic curve is given by a Weierstrass model
y^2+a_1xy+a_3y = x^3+a_2x^2+a_4x+a_6,
whose discriminant is non-zero. Affine points on E
are represented as
two-component vectors [x,y]
; the point at infinity, i.e. the identity
element of the group law, is represented by the one-component vector
[0]
.
Given a vector of coefficients [a_1,a_2,a_3,a_4,a_6]
, the function
ellinit
initializes and returns an ell structure. (An additional
optional argument allows to specify the base field in case it cannot be
inferred from the curve coefficients.) This structure contains data needed by
elliptic curve related functions, and is generally passed as a first argument.
Expensive data are skipped on initialization: they will be dynamically
computed when (and if) needed, and then inserted in the structure. The
precise layout of the ell structure is left undefined and should never
be used directly. The following member functions are available,
depending on the underlying domain.
@3* a1
, a2
, a3
, a4
, a6
: coefficients of the
elliptic curve.
@3* b2
, b4
, b6
, b8
: b
-invariants of the curve; in
characteristic != 2
, for Y = 2y + a_1x+a3
, the curve equation becomes
Y^2 = 4 x^3 + b_2 x^2 + 2b_4 x + b_6 = : g(x).
@3* c4
, c6
: c
-invariants of the curve; in characteristic !=
2,3
, for X = x + b_2/12
and Y = 2y + a_1x+a3
, the curve equation becomes
Y^2 = 4 X^3 - (c_4/12) X - (c_6/216).
@3* disc
: discriminant of the curve. This is only required to be
non-zero, not necessarily a unit.
@3* j
: j
-invariant of the curve.
@3These are used as follows:
? E = ellinit([0,0,0, a4,a6]); ? E.b4 %2 = 2*a4 ? E.disc %3 = -64*a4^3 - 432*a6^2
This in particular includes curves defined over Q. All member functions in
this section return data, as it is currently stored in the structure, if
present; and otherwise compute it to the default accuracy, that was fixed
at the time of ellinit (via a t_REAL
D
domain argument, or
realprecision
by default). The function ellperiods
allows to
recompute (and cache) the following data to current
realprecision
.
@3* area
: volume of the complex lattice defining E
.
@3* roots
is a vector whose three components contain the complex
roots of the right hand side g(x)
of the attached b
-model Y^2 = g(x)
.
If the roots are all real, they are ordered by decreasing value. If only one
is real, it is the first component.
@3* omega
: [
omega_1,
omega_2]
, periods forming a basis of the
complex lattice defining E
. The first component omega_1
is the
(positive) real period, in other words the integral of the Néron
differential dx/(2y+a_1x+a_3)
over the connected component of the identity component of E(
R)
.
The second component omega_2
is a complex period, such that
tau = (
omega_1)/(
omega_2)
belongs to Poincaré's
half-plane (positive imaginary part); not necessarily to the standard
fundamental domain. It is normalized so that Im (
omega_2) < 0
and either Re (
omega_2) = 0
, when E.disc > 0
(E(
R)
has two connected
components), or Re (
omega_2) =
omega_1/2
@3* eta
is a row vector containing the quasi-periods eta_1
and
eta_2
such that eta_i = 2
zeta(
omega_i/2)
, where zeta is the
Weierstrass zeta function attached to the period lattice; see
ellzeta
. In particular, the Legendre relation holds: eta_2
omega_1 -
eta_1
omega_2 = 2
pi i
.
@3Warning. As for the orientation of the basis of the period lattice,
beware that many sources use the inverse convention where omega_2/
omega_1
has positive imaginary part and our omega_2
is the negative of theirs. Our
convention tau =
omega_1/
omega_2
ensures that the action of PSL_2
is the natural
one:
[a,b;c,d].
tau = (a
tau+b)/(c
tau+d)
= (a
omega_1 + b
omega_2)/(c
omega_1 + d
omega_2),
instead of a twisted one. (Our tau
is -1/
tau in the above inverse
convention.)
_p
We advise to input a model defined over Q for such curves. In any case,
if you input an approximate model with t_PADIC
coefficients, it will be
replaced by a lift to Q (an exact model ``close'' to the one that was
input) and all quantities will then be computed in terms of this lifted
model.
For the time being only curves with multiplicative reduction (split or
non-split), i.e. v_p(j) < 0
, are supported by non-trivial functions. In
this case the curve is analytically isomorphic to \bar{
Q}_p^*/q^
Z :=
E_q(\bar{
Q}_p)
, for some p
-adic integer q
(the Tate period). In
particular, we have j(q) = j(E)
.
@3* p
is the residual characteristic
@3* roots
is a vector with a single component, equal to the p
-adic
root e_1
of the right hand side g(x)
of the attached b
-model Y^2
= g(x)
. The point (e_1,0)
corresponds to -1 \in \bar{
Q}_p^*/q^
Z
under the Tate parametrization.
@3* tate
returns [u^2,u,q,[a,b],L, Ei]
in the notation of Henniart-Mestre
(CRAS t. 308, p. 391--395, 1989): q
is as above, u\in
Q_p(
sqrt {-c_6})
is such that phi^* dx/(2y + a_1x+a3) = u dt/t
, where phi: E_q\to E
is an isomorphism (well defined up to sign) and dt/t
is the canonical
invariant differential on the Tate curve; u^2\in
Q_p
does not depend on
phi. (Technicality: if u\not\in
Q_p
, it is stored as a quadratic
t_POLMOD
.)
The parameters [a,b]
satisfy 4u^2 b.agm(
sqrt {a/b},1)^2 = 1
as in Theorem 2 (loc. cit.).
Ei
describes the sequence of 2-isogenous curves (with kernel generated
by [0,0]
) E_i: y^2 = x(x+A_i)(x+A_i-B_i)
converging quadratically towards
the singular curve E_ oo
. Finally, L
is Mazur-Tate-Teitelbaum's
L-invariant, equal to log _p q / v_p(q)
.
_q
@3* p
is the characteristic of F_q
.
@3* no
is #E(
F_q)
.
@3* cyc
gives the cycle structure of E(
F_q)
.
@3* gen
returns the generators of E(
F_q)
.
@3* group
returns [no,cyc,gen]
, i.e. E(
F_q)
as an abelian group structure.
All functions should return a correct result, whether the model is minimal or
not, but it is a good idea to stick to minimal models whenever
gcd (c_4,c_6)
is easy to factor (minor speed-up). The construction
E = ellminimalmodel(E0, &v)
@3replaces the original model E_0
by a minimal model E
,
and the variable change v
allows to go between the two models:
ellchangepoint(P0, v) ellchangepointinv(P, v)
@3respectively map the point P_0
on E_0
to its image on
E
, and the point P
on E
to its pre-image on E_0
.
A few routines --- namely ellgenerators
, ellidentify
,
ellsearch
, forell
--- require the optional package elldata
(John Cremona's database) to be installed. In that case, the function
ellinit
will allow alternative inputs, e.g. ellinit("11a1")
.
Functions using this package need to load chunks of a large database in
memory and require at least 2MB stack to avoid stack overflows.
@3* gen
returns the generators of E(
Q)
, if known (from John
Cremona's database)
@3* nf
return the nf structure attached to the number field
over which E
is defined.
@3* bnf
return the bnf structure attached to the number field
over which E
is defined or raise an error (if only an nf is available).
(e, {r = 0})
Returns the value at s = 1
of the derivative of order r
of the
L
-function of the elliptic curve e
.
? e = ellinit("11a1"); \\ order of vanishing is 0 ? ellL1(e) %2 = 0.2538418608559106843377589233 ? e = ellinit("389a1"); \\ order of vanishing is 2 ? ellL1(e) %4 = -5.384067311837218089235032414 E-29 ? ellL1(e, 1) %5 = 0 ? ellL1(e, 2) %6 = 1.518633000576853540460385214
The main use of this function, after computing at low accuracy the
order of vanishing using ellanalyticrank
, is to compute the
leading term at high accuracy to check (or use) the Birch and
Swinnerton-Dyer conjecture:
? \p18 realprecision = 18 significant digits ? e = ellinit("5077a1"); ellanalyticrank(e) time = 8 ms. %1 = [3, 10.3910994007158041] ? \p200 realprecision = 202 significant digits (200 digits displayed) ? ellL1(e, 3) time = 104 ms. %3 = 10.3910994007158041387518505103609170697263563756570092797[...]
The library syntax is GEN
ellL1_bitprec(GEN e, long r, long bitprec)
.
(E,
z1,
z2)
Sum of the points z1
and z2
on the
elliptic curve corresponding to E
.
The library syntax is GEN
elladd(GEN E, GEN z1, GEN z2)
.
(E,n)
Computes the coefficient a_n
of the L
-function of the elliptic curve
E/
Q, i.e. coefficients of a newform of weight 2 by the modularity theorem
(Taniyama-Shimura-Weil conjecture). E
must be an ell
structure
over Q as output by ellinit
. E
must be given by an integral model,
not necessarily minimal, although a minimal model will make the function
faster.
? E = ellinit([0,1]); ? ellak(E, 10) %2 = 0 ? e = ellinit([5^4,5^6]); \\ not minimal at 5 ? ellak(e, 5) \\ wasteful but works %3 = -3 ? E = ellminimalmodel(e); \\ now minimal ? ellak(E, 5) %5 = -3
@3If the model is not minimal at a number of bad primes, then
the function will be slower on those n
divisible by the bad primes.
The speed should be comparable for other n
:
? for(i=1,10^6, ellak(E,5)) time = 820 ms. ? for(i=1,10^6, ellak(e,5)) \\ 5 is bad, markedly slower time = 1,249 ms.
? for(i=1,10^5,ellak(E,5*i)) time = 977 ms. ? for(i=1,10^5,ellak(e,5*i)) \\ still slower but not so much on average time = 1,008 ms.
The library syntax is GEN
akell(GEN E, GEN n)
.
(E,n)
Computes the vector of the first n
Fourier coefficients a_k
corresponding to the elliptic curve E
defined over a number field.
If E
is defined over Q, the curve may be given by an
arbitrary model, not necessarily minimal,
although a minimal model will make the function faster. Over a more general
number field, the model must be locally minimal at all primes above 2
and 3
.
The library syntax is GEN
ellan(GEN E, long n)
.
Also available is GEN
ellanQ_zv(GEN e, long n)
, which
returns a t_VECSMALL
instead of a t_VEC
, saving on memory.
ellanalyticrank(e, {
eps})
Returns the order of vanishing at s = 1
of the L
-function of the
elliptic curve e
and the value of the first non-zero derivative. To
determine this order, it is assumed that any value less than eps
is
zero. If no value of eps
is given, a value of half the current
precision is used.
? e = ellinit("11a1"); \\ rank 0 ? ellanalyticrank(e) %2 = [0, 0.2538418608559106843377589233] ? e = ellinit("37a1"); \\ rank 1 ? ellanalyticrank(e) %4 = [1, 0.3059997738340523018204836835] ? e = ellinit("389a1"); \\ rank 2 ? ellanalyticrank(e) %6 = [2, 1.518633000576853540460385214] ? e = ellinit("5077a1"); \\ rank 3 ? ellanalyticrank(e) %8 = [3, 10.39109940071580413875185035]
The library syntax is GEN
ellanalyticrank_bitprec(GEN e, GEN eps = NULL, long bitprec)
.
(E,{p})
Let E
be an ell
structure as output by ellinit
, defined over
a number field or a finite field F_q
. The argument p
is best left
omitted if the curve is defined over a finite field, and must be a prime
number or a maximal ideal otherwise. This function computes the trace of
Frobenius t
for the elliptic curve E
, defined by the equation #E(
F_q)
= q+1 - t
(for primes of good reduction).
When the characteristic of the finite field is large, the availability of
the seadata
package will speed the computation.
If the curve is defined over Q, p
must be explicitly given and the
function computes the trace of the reduction over F_p
.
The trace of Frobenius is also the a_p
coefficient in the curve L
-series
L(E,s) =
sum_n a_n n^{-s}
, whence the function name. The equation must be
integral at p
but need not be minimal at p
; of course, a minimal model
will be more efficient.
? E = ellinit([0,1]); \\ y^2 = x^3 + 0.x + 1, defined over Q ? ellap(E, 7) \\ 7 necessary here %2 = -4 \\ #E(F_7) = 7+1-(-4) = 12 ? ellcard(E, 7) %3 = 12 \\ OK
? E = ellinit([0,1], 11); \\ defined over F_11 ? ellap(E) \\ no need to repeat 11 %4 = 0 ? ellap(E, 11) \\ ... but it also works %5 = 0 ? ellgroup(E, 13) \\ ouch, inconsistent input! *** at top-level: ellap(E,13) *** ^----------- *** ellap: inconsistent moduli in Rg_to_Fp: 11 13
? Fq = ffgen(ffinit(11,3), 'a); \\ defines F_q := F_{11^3} ? E = ellinit([a+1,a], Fq); \\ y^2 = x^3 + (a+1)x + a, defined over F_q ? ellap(E) %8 = -3
If the curve is defined over a more general number field than Q,
the maximal ideal p
must be explicitly given in idealprimedec
format. If p
is above 2
or 3
, the function currently assumes (without
checking) that the given model is locally minimal at p
. There is no
restriction at other primes.
? K = nfinit(a^2+1); E = ellinit([1+a,0,1,0,0], K); ? fa = idealfactor(K, E.disc) %2 = [ [5, [-2, 1]~, 1, 1, [2, -1; 1, 2]] 1]
[[13, [5, 1]~, 1, 1, [-5, -1; 1, -5]] 2] ? ellap(E, fa[1,1]) %3 = -1 \\ non-split multiplicative reduction ? ellap(E, fa[2,1]) %4 = 1 \\ split multiplicative reduction ? P17 = idealprimedec(K,17)[1]; ? ellap(E, P17) %6 = 6 \\ good reduction ? E2 = ellchangecurve(E, [17,0,0,0]); ? ellap(E2, P17) %8 = 6 \\ same, starting from a non-miminal model
? P3 = idealprimedec(K,3)[1]; ? E3 = ellchangecurve(E, [3,0,0,0]); ? ellap(E, P3) \\ OK: E is minimal at P3 %11 = -2 ? ellap(E3, P3) \\ junk: E3 is not minimal at P3 | 3 %12 = 0
@3Algorithms used. If E/
F_q
has CM by a principal imaginary
quadratic order we use a fast explicit formula (involving essentially
Kronecker symbols and Cornacchia's algorithm), in O(
log q)^2
.
Otherwise, we use Shanks-Mestre's baby-step/giant-step method, which runs in
time ~{O}(q^{1/4})
using ~{O}(q^{1/4})
storage, hence becomes
unreasonable when q
has about 30 digits. Above this range, the SEA
algorithm becomes available, heuristically in ~{O}(
log q)^4
, and
primes of the order of 200 digits become feasible. In small
characteristic we use Mestre's (p = 2), Kohel's (p = 3,5,7,13), Satoh-Harley
(all in ~{O}(p^{2} n^2)
) or Kedlaya's (in ~{O}(p n^3)
)
algorithms.
The library syntax is GEN
ellap(GEN E, GEN p = NULL)
.
(E,
z1,
z2)
Deprecated alias for ellheight(E,P,Q)
.
The library syntax is GEN
bilhell(GEN E, GEN z1, GEN z2, long prec)
.
(E,{p})
Let E
be an ell
structure as output by ellinit
, defined over
Q or a finite field F_q
. The argument p
is best left omitted if the
curve is defined over a finite field, and must be a prime number otherwise.
This function computes the order of the group E(
F_q)
(as would be
computed by ellgroup
).
When the characteristic of the finite field is large, the availability of
the seadata
package will speed the computation.
If the curve is defined over Q, p
must be explicitly given and the
function computes the cardinality of the reduction over F_p
; the
equation need not be minimal at p
, but a minimal model will be more
efficient. The reduction is allowed to be singular, and we return the order
of the group of non-singular points in this case.
The library syntax is GEN
ellcard(GEN E, GEN p = NULL)
.
Also available is GEN
ellcard(GEN E, GEN p)
where p
is not
NULL
.
(E,v)
Changes the data for the elliptic curve E
by changing the coordinates using the vector v = [u,r,s,t]
, i.e. if x'
and y'
are the new coordinates, then x = u^2x'+r
, y = u^3y'+su^2x'+t
.
E
must be an ell
structure as output by ellinit
. The special
case v = 1
is also used instead of [1,0,0,0]
to denote the
trivial coordinate change.
The library syntax is GEN
ellchangecurve(GEN E, GEN v)
.
(x,v)
Changes the coordinates of the point or
vector of points x
using the vector v = [u,r,s,t]
, i.e. if x'
and
y'
are the new coordinates, then x = u^2x'+r
, y = u^3y'+su^2x'+t
(see also
ellchangecurve
).
? E0 = ellinit([1,1]); P0 = [0,1]; v = [1,2,3,4]; ? E = ellchangecurve(E0, v); ? P = ellchangepoint(P0,v) %3 = [-2, 3] ? ellisoncurve(E, P) %4 = 1 ? ellchangepointinv(P,v) %5 = [0, 1]
The library syntax is GEN
ellchangepoint(GEN x, GEN v)
.
The reciprocal function GEN
ellchangepointinv(GEN x, GEN ch)
inverts the coordinate change.
(x,v)
Changes the coordinates of the point or vector of points x
using
the inverse of the isomorphism attached to v = [u,r,s,t]
,
i.e. if x'
and y'
are the old coordinates, then x = u^2x'+r
,
y = u^3y'+su^2x'+t
(inverse of ellchangepoint
).
? E0 = ellinit([1,1]); P0 = [0,1]; v = [1,2,3,4]; ? E = ellchangecurve(E0, v); ? P = ellchangepoint(P0,v) %3 = [-2, 3] ? ellisoncurve(E, P) %4 = 1 ? ellchangepointinv(P,v) %5 = [0, 1] \\ we get back P0
The library syntax is GEN
ellchangepointinv(GEN x, GEN v)
.
(
name)
Converts an elliptic curve name, as found in the elldata
database,
from a string to a triplet [
conductor,
isogeny class,
index]
. It will also convert a triplet back to a curve name.
Examples:
? ellconvertname("123b1") %1 = [123, 1, 1] ? ellconvertname(%) %2 = "123b1"
The library syntax is GEN
ellconvertname(GEN name)
.
(E,n,{v = 'x})
n
-division polynomial f_n
for the curve E
in the
variable v
. In standard notation, for any affine point P = (X,Y)
on the
curve, we have
[n]P = (
phi_n(P)
psi_n(P) :
omega_n(P) :
psi_n(P)^3)
for some polynomials phi_n,
omega_n,
psi_n
in
Z[a_1,a_2,a_3,a_4,a_6][X,Y]
. We have f_n(X) =
psi_n(X)
for n
odd, and
f_n(X) =
psi_n(X,Y) (2Y + a_1X+a_3)
for n
even. We have
f_1 = 1, f_2 = 4X^3 + b_2X^2 + 2b_4 X + b_6, f_3 = 3 X^4 + b_2 X^3 + 3b_4 X^2 + 3 b_6 X + b8,
f_4 = f_2(2X^6 + b_2 X^5 + 5b_4 X^4 + 10 b_6 X^3 + 10 b_8 X^2 +
(b_2b_8-b_4b_6)X + (b_8b_4 - b_6^2)),...
For n >= 2
, the roots of f_n
are the X
-coordinates of points in E[n]
.
The library syntax is GEN
elldivpol(GEN E, long n, long v = -1)
where v
is a variable number.
(w,k,{
flag = 0})
k
being an even positive integer, computes the numerical value of the
Eisenstein series of weight k
at the lattice w
, as given by
ellperiods
, namely
(2i
pi/
omega_2)^k
(1 + 2/
zeta(1-k)
sum_{n >= 1} n^{k-1}q^n / (1-q^n)),
where q =
exp (2i
pi tau)
and tau :=
omega_1/
omega_2
belongs to the
complex upper half-plane. It is also possible to directly input w =
[
omega_1,
omega_2]
, or an elliptic curve E
as given by ellinit
.
? w = ellperiods([1,I]); ? elleisnum(w, 4) %2 = 2268.8726415508062275167367584190557607 ? elleisnum(w, 6) %3 = -3.977978632282564763 E-33 ? E = ellinit([1, 0]); ? elleisnum(E, 4, 1) %5 = -47.999999999999999999999999999999999998
When flag is non-zero and k = 4
or 6, returns the elliptic invariants g_2
or g_3
, such that
y^2 = 4x^3 - g_2 x - g_3
is a Weierstrass equation for E
.
The library syntax is GEN
elleisnum(GEN w, long k, long flag, long prec)
.
(w)
Returns the quasi-periods [
eta_1,
eta_2]
attached to the lattice basis w = [
omega_1,
omega_2]
.
Alternatively, w can be an elliptic curve E
as output by
ellinit
, in which case, the quasi periods attached to the period
lattice basis E.omega
(namely, E.eta
) are returned.
? elleta([1, I]) %1 = [3.141592653589793238462643383, 9.424777960769379715387930149*I]
The library syntax is GEN
elleta(GEN w, long prec)
.
ellformaldifferential(E, {n =
seriesprecision}, {t = 'x})
Let omega := dx / (2y+a_1x+a_3)
be the invariant differential form
attached to the model E
of some elliptic curve (ellinit
form),
and eta := x(t)
omega. Return n
terms (seriesprecision
by default)
of f(t),g(t)
two power series in the formal parameter t = -x/y
such that
omega = f(t) dt
, eta = g(t) dt
:
f(t) = 1+a_1 t + (a_1^2 + a_2) t^2 +..., g(t) = t^{-2} +...
? E = ellinit([-1,1/4]); [f,g] = ellformaldifferential(E,7,'t); ? f %2 = 1 - 2*t^4 + 3/4*t^6 + O(t^7) ? g %3 = t^-2 - t^2 + 1/2*t^4 + O(t^5)
The library syntax is GEN
ellformaldifferential(GEN E, long precdl, long n = -1)
where n
is a variable number.
ellformalexp(E, {n =
seriesprecision}, {z = 'x})
The elliptic formal exponential Exp
attached to E
is the
isomorphism from the formal additive law to the formal group of E
. It is
normalized so as to be the inverse of the elliptic logarithm (see
ellformallog
): Exp o L =
Id . Return n
terms of this
power series:
? E=ellinit([-1,1/4]); Exp = ellformalexp(E,10,'z) %1 = z + 2/5*z^5 - 3/28*z^7 + 2/15*z^9 + O(z^11) ? L = ellformallog(E,10,'t); ? subst(Exp,z,L) %3 = t + O(t^11)
The library syntax is GEN
ellformalexp(GEN E, long precdl, long n = -1)
where n
is a variable number.
ellformallog(E, {n =
seriesprecision}, {v = 'x})
The formal elliptic logarithm is a series L
in t K[[t]]
such that d L =
omega = dx / (2y + a_1x + a_3)
, the canonical invariant
differential attached to the model E
. It gives an isomorphism
from the formal group of E
to the additive formal group.
? E = ellinit([-1,1/4]); L = ellformallog(E, 9, 't) %1 = t - 2/5*t^5 + 3/28*t^7 + 2/3*t^9 + O(t^10) ? [f,g] = ellformaldifferential(E,8,'t); ? L' - f %3 = O(t^8)
The library syntax is GEN
ellformallog(GEN E, long precdl, long n = -1)
where n
is a variable number.
ellformalpoint(E, {n =
seriesprecision}, {v = 'x})
If E
is an elliptic curve, return the coordinates x(t), y(t)
in the
formal group of the elliptic curve E
in the formal parameter t = -x/y
at oo
:
x = t^{-2} -a_1 t^{-1} - a_2 - a_3 t +...
y = - t^{-3} -a_1 t^{-2} - a_2t^{-1} -a_3 +...
Return n
terms (seriesprecision
by default) of these two power
series, whose coefficients are in Z[a_1,a_2,a_3,a_4,a_6]
.
? E = ellinit([0,0,1,-1,0]); [x,y] = ellformalpoint(E,8,'t); ? x %2 = t^-2 - t + t^2 - t^4 + 2*t^5 + O(t^6) ? y %3 = -t^-3 + 1 - t + t^3 - 2*t^4 + O(t^5) ? E = ellinit([0,1/2]); ellformalpoint(E,7) %4 = [x^-2 - 1/2*x^4 + O(x^5), -x^-3 + 1/2*x^3 + O(x^4)]
The library syntax is GEN
ellformalpoint(GEN E, long precdl, long n = -1)
where n
is a variable number.
ellformalw(E, {n =
seriesprecision}, {t = 'x})
Return the formal power series w
attached to the elliptic curve E
,
in the variable t
:
w(t) = t^3 + a_1 t^4 + (a_2 + a_1^2) t^5 +...+ O(t^{n+3}),
which is the formal expansion of -1/y
in the formal parameter t := -x/y
at oo
(take n = seriesprecision
if n
is omitted). The
coefficients of w
belong to Z[a_1,a_2,a_3,a_4,a_6]
.
? E=ellinit([3,2,-4,-2,5]); ellformalw(E, 5, 't) %1 = t^3 + 3*t^4 + 11*t^5 + 35*t^6 + 101*t^7 + O(t^8)
The library syntax is GEN
ellformalw(GEN E, long precdl, long n = -1)
where n
is a variable number.
(P)
Given a genus 1
plane curve, defined by the affine equation f(x,y) = 0
,
return the coefficients [a_1,a_2,a_3,a_4,a_6]
of a Weierstrass equation
for its Jacobian. This allows to recover a Weierstrass model for an elliptic
curve given by a general plane cubic or by a binary quartic or biquadratic
model. The function implements the f :--->f^*
formulae of Artin, Tate
and Villegas (Advances in Math. 198 (2005), pp. 366--382).
In the example below, the function is used to convert between twisted Edwards coordinates and Weierstrass coordinates.
? e = ellfromeqn(a*x^2+y^2 - (1+d*x^2*y^2)) %1 = [0, -a - d, 0, -4*d*a, 4*d*a^2 + 4*d^2*a] ? E = ellinit(ellfromeqn(y^2-x^2 - 1 +(121665/121666*x^2*y^2)),2^255-19); ? isprime(ellcard(E) / 8) %3 = 1
The elliptic curve attached to the sum of two cubes is given by
? ellfromeqn(x^3+y^3 - a) %1 = [0, 0, -9*a, 0, -27*a^2]
@3Congruent number problem:.
Let n
be an integer, if a^2+b^2 = c^2
and a b = 2 n
,
then by substituting b
by 2 n/a
in the first equation,
we get ((a^2+(2 n/a)^2)-c^2) a^2 = 0
.
We set x = a
, y = a c
.
? En = ellfromeqn((x^2 + (2*n/x)^2 - (y/x)^2)*x^2) %1 = [0, 0, 0, -16*n^2, 0]
For example 23
is congruent since the curve has a point of infinite order,
namely:
? ellheegner( ellinit(subst(En, n, 23)) ) %2 = [168100/289, 68053440/4913]
The library syntax is GEN
ellfromeqn(GEN P)
.
(j)
Returns the coefficients [a_1,a_2,a_3,a_4,a_6]
of a fixed elliptic curve
with j
-invariant j
.
The library syntax is GEN
ellfromj(GEN j)
.
(E)
If E
is an elliptic curve over the rationals, return a Z-basis of the
free part of the Mordell-Weil group attached to E
. This relies on
the elldata
database being installed and referencing the curve, and so
is only available for curves over Z of small conductors.
If E
is an elliptic curve over a finite field F_q
as output by
ellinit
, return a minimal set of generators for the group E(
F_q)
.
The library syntax is GEN
ellgenerators(GEN E)
.
(E)
Let E
be an ell
structure as output by ellinit
attached
to an elliptic curve defined over a number field. This function calculates
the arithmetic conductor and the global Tamagawa number c
.
The result [N,v,c,F,L]
is slightly different if E
is defined
over Q (domain D = 1
in ellinit
) or over a number field
(domain D
is a number field structure, including nfinit(x)
representing Q !):
@3* N
is the arithmetic conductor of the curve,
@3* v
is an obsolete field, left in place for backward compatibility.
If E
is defined over Q, v
gives the coordinate change for E
to the
standard minimal integral model (ellminimalmodel
provides it in a
cheaper way); if E
is defined over another number field, v
gives a
coordinate change to an integral model (ellintegral
model provides it
in a cheaper way).
@3* c
is the product of the local Tamagawa numbers c_p
, a quantity
which enters in the Birch and Swinnerton-Dyer conjecture,
@3* F
is the factorization of N
,
@3* L
is a vector, whose i
-th entry contains the local data
at the i
-th prime ideal divisor of N
, i.e.
L[i] = elllocalred(E,F[i,1])
. If E
is defined over Q, the local
coordinate change has been deleted and replaced by a 0; if E
is defined
over another number field the local coordinate change to a local minimal
model is given relative to the integral model afforded by v
(so either
start from an integral model so that v
be trivial, or apply v
first).
The library syntax is GEN
ellglobalred(GEN E)
.
(E,{p},{
flag})
Let E
be an ell
structure as output by ellinit
, defined over
Q or a finite field F_q
. The argument p
is best left omitted if the
curve is defined over a finite field, and must be a prime number otherwise.
This function computes the structure of the group E(
F_q) ~
Z/d_1
Z
x
Z/d_2
Z, with d_2 | d_1
.
If the curve is defined over Q, p
must be explicitly given and the
function computes the structure of the reduction over F_p
; the
equation need not be minimal at p
, but a minimal model will be more
efficient. The reduction is allowed to be singular, and we return the
structure of the (cyclic) group of non-singular points in this case.
If the flag is 0
(default), return [d_1]
or [d_1, d_2]
, if d_2 > 1
.
If the flag is 1
, return a triple [h,
cyc,
gen]
, where
h
is the curve cardinality, cyc gives the group structure as a
product of cyclic groups (as per flag = 0
). More precisely, if d_2 > 1
,
the output is [d_1d_2, [d_1,d_2],[P,Q]]
where P
is
of order d_1
and [P,Q]
generates the curve.
\misctitle{Caution} It is not guaranteed that Q
has order d_2
, which in
the worst case requires an expensive discrete log computation. Only that
ellweilpairing(E, P, Q, d1)
has order d_2
.
? E = ellinit([0,1]); \\ y^2 = x^3 + 0.x + 1, defined over Q ? ellgroup(E, 7) %2 = [6, 2] \\ Z/6 x Z/2, non-cyclic ? E = ellinit([0,1] * Mod(1,11)); \\ defined over F_11 ? ellgroup(E) \\ no need to repeat 11 %4 = [12] ? ellgroup(E, 11) \\ ... but it also works %5 = [12] ? ellgroup(E, 13) \\ ouch, inconsistent input! *** at top-level: ellgroup(E,13) *** ^-------------- *** ellgroup: inconsistent moduli in Rg_to_Fp: 11 13 ? ellgroup(E, 7, 1) %6 = [12, [6, 2], [[Mod(2, 7), Mod(4, 7)], [Mod(4, 7), Mod(4, 7)]]]
If E
is defined over Q, we allow singular reduction and in this case we
return the structure of the group of non-singular points, satisfying
#E_{ns}(
F_p) = p - a_p
.
? E = ellinit([0,5]); ? ellgroup(E, 5, 1) %2 = [5, [5], [[Mod(4, 5), Mod(2, 5)]]] ? ellap(E, 5) %3 = 0 \\ additive reduction at 5 ? E = ellinit([0,-1,0,35,0]); ? ellgroup(E, 5, 1) %5 = [4, [4], [[Mod(2, 5), Mod(2, 5)]]] ? ellap(E, 5) %6 = 1 \\ split multiplicative reduction at 5 ? ellgroup(E, 7, 1) %7 = [8, [8], [[Mod(3, 7), Mod(5, 7)]]] ? ellap(E, 7) %8 = -1 \\ non-split multiplicative reduction at 7
The library syntax is GEN
ellgroup0(GEN E, GEN p = NULL, long flag)
.
Also available is GEN
ellgroup(GEN E, GEN p)
, corresponding
to flag = 0.
(E)
Let E
be an elliptic curve over the rationals, assumed to be of
(analytic) rank 1
. This returns a non-torsion rational point on the curve,
whose canonical height is equal to the product of the elliptic regulator by the
analytic Sha.
This uses the Heegner point method, described in Cohen GTM 239; the complexity is proportional to the product of the square root of the conductor and the height of the point (thus, it is preferable to apply it to strong Weil curves).
? E = ellinit([-157^2,0]); ? u = ellheegner(E); print(u[1], "\n", u[2]) 69648970982596494254458225/166136231668185267540804 538962435089604615078004307258785218335/67716816556077455999228495435742408 ? ellheegner(ellinit([0,1])) \\ E has rank 0 ! *** at top-level: ellheegner(E=ellinit *** ^-------------------- *** ellheegner: The curve has even analytic rank.
The library syntax is GEN
ellheegner(GEN E)
.
(E,P,{Q})
Global Néron-Tate height h(P)
of the point P
on the elliptic curve
E/
Q, using the normalization in Cremona's Algorithms for modular
elliptic curves. E
must be an ell
as output by ellinit
; it
needs not be given by a minimal model although the computation will be faster
if it is.
If the argument Q
is present, computes the value of the bilinear
form (h(P+Q)-h(P-Q)) / 4
.
The library syntax is GEN
ellheight0(GEN E, GEN P, GEN Q = NULL, long prec)
.
Also available is GEN
ellheight(GEN E, GEN P, long prec)
(Q
omitted).
(E,x)
x
being a vector of points, this
function outputs the Gram matrix of x
with respect to the Néron-Tate
height, in other words, the (i,j)
component of the matrix is equal to
ellbil(E,x[i],x[j])
. The rank of this matrix, at least in some
approximate sense, gives the rank of the set of points, and if x
is a
basis of the Mordell-Weil group of E
, its determinant is equal to
the regulator of E
. Note our height normalization follows Cremona's
Algorithms for modular elliptic curves: this matrix should be divided
by 2 to be in accordance with, e.g., Silverman's normalizations.
The library syntax is GEN
ellheightmatrix(GEN E, GEN x, long prec)
.
(E)
Look up the elliptic curve E
, defined by an arbitrary model over Q,
in the elldata
database.
Return [[N, M, G], C]
where N
is the curve name in Cremona's
elliptic curve database, M
is the minimal model, G
is a Z-basis of
the free part of the Mordell-Weil group E(
Q)
and C
is the
change of coordinates change, suitable for ellchangecurve
.
The library syntax is GEN
ellidentify(GEN E)
.
(x,{D = 1})
Initialize an ell
structure, attached to the elliptic curve E
.
E
is either
@3* a 5
-component vector [a_1,a_2,a_3,a_4,a_6]
defining the elliptic
curve with Weierstrass equation
Y^2 + a_1 XY + a_3 Y = X^3 + a_2 X^2 + a_4 X + a_6,
@3* a 2
-component vector [a_4,a_6]
defining the elliptic
curve with short Weierstrass equation
Y^2 = X^3 + a_4 X + a_6,
@3* a character string in Cremona's notation, e.g. "11a1"
, in which
case the curve is retrieved from the elldata
database if available.
The optional argument D
describes the domain over which the curve is
defined:
@3* the t_INT
1
(default): the field of rational numbers Q.
@3* a t_INT
p
, where p
is a prime number: the prime finite field
F_p
.
@3* an t_INTMOD
Mod(a, p)
, where p
is a prime number: the
prime finite field F_p
.
@3* a t_FFELT
, as returned by ffgen
: the corresponding finite
field F_q
.
@3* a t_PADIC
, O(p^n)
: the field Q_p
, where p
-adic quantities
will be computed to a relative accuracy of n
digits. We advise to input a
model defined over Q for such curves. In any case, if you input an
approximate model with t_PADIC
coefficients, it will be replaced by a lift
to Q (an exact model ``close'' to the one that was input) and all quantities
will then be computed in terms of this lifted model, at the given accuracy.
@3* a t_REAL
x
: the field C of complex numbers, where floating
point quantities are by default computed to a relative accuracy of
precision
(x)
. If no such argument is given, the value of
realprecision
at the time ellinit
is called will be used.
@3* a number field K
, given by a nf
or bnf
structure; a
bnf
is required for ellminimalmodel
.
@3* a prime ideal p, given by a prid
structure; valid if
x
is a curve defined over a number field K
and the equation is integral
and minimal at p.
This argument D
is indicative: the curve coefficients are checked for
compatibility, possibly changing D
; for instance if D = 1
and
an t_INTMOD
is found. If inconsistencies are detected, an error is
raised:
? ellinit([1 + O(5), 1], O(7)); *** at top-level: ellinit([1+O(5),1],O *** ^-------------------- *** ellinit: inconsistent moduli in ellinit: 7 != 5
@3If the curve coefficients are too general to fit any of the above domain categories, only basic operations, such as point addition, will be supported later.
If the curve (seen over the domain D
) is singular, fail and return an
empty vector []
.
? E = ellinit([0,0,0,0,1]); \\ y^2 = x^3 + 1, over Q ? E = ellinit([0,1]); \\ the same curve, short form ? E = ellinit("36a1"); \\ sill the same curve, Cremona's notations ? E = ellinit([0,1], 2) \\ over F2: singular curve %4 = [] ? E = ellinit(['a4,'a6] * Mod(1,5)); \\ over F_5[a4,a6], basic support !
The result of ellinit
is an ell structure. It contains at least
the following information in its components:
a_1,a_2,a_3,a_4,a_6,b_2,b_4,b_6,b_8,c_4,c_6,
Delta,j.
All are accessible via member functions. In particular, the discriminant is
E.disc
, and the j
-invariant is E.j
.
? E = ellinit([a4, a6]); ? E.disc %2 = -64*a4^3 - 432*a6^2 ? E.j %3 = -6912*a4^3/(-4*a4^3 - 27*a6^2)
Further components contain domain-specific data, which are in general dynamic: only computed when needed, and then cached in the structure.
? E = ellinit([2,3], 10^60+7); \\ E over F_p, p large ? ellap(E) time = 4,440 ms. %2 = -1376268269510579884904540406082 ? ellcard(E); \\ now instantaneous ! time = 0 ms. ? ellgenerators(E); time = 5,965 ms. ? ellgenerators(E); \\ second time instantaneous time = 0 ms.
See the description of member functions related to elliptic curves at the beginning of this section.
The library syntax is GEN
ellinit(GEN x, GEN D = NULL, long prec)
.
(E,{&v})
Let E
be an ell
structure over a number field K
. This function
returns an integral model. If v
is present, sets v = [u,0,0,0]
to the
corresponding change of variable: the return value is identical to that of
ellchangecurve(E, v)
.
The library syntax is GEN
ellintegralmodel(GEN E, GEN *v = NULL)
.
(E,P,n,{&Q}))
Given E/K
a number field and P
in E(K)
return 1
if P = [n]R
for some R
in E(K)
and set Q
to one such R
;
and return 0
otherwise. The integer n >= 0
may be given as
ellxn(E,n)
, if many points need to be tested.
? K = nfinit(polcyclo(11,t)); ? E = ellinit([0,-1,1,0,0], K); ? P = [0,0]; ? ellorder(E,P) %4 = 5 ? ellisdivisible(E,P,5, &Q) %5 = 1 ? lift(Q) %6 = [-t^7-t^6-t^5-t^4+1, -t^9-2*t^8-2*t^7-3*t^6-3*t^5-2*t^4-2*t^3-t^2-1] ? ellorder(E, Q) %7 = 25
@3The algebraic complexity of the underlying algorithm is in
O(n^4)
, so it is advisable to first factor n
, then use a chain of checks
attached to the prime divisors of n
: the function will do it itself unless
n
is given in ellxn
form.
The library syntax is long
ellisdivisible(GEN E, GEN P, GEN n, GEN *Q) = NULL)
.
ellisogeny(E, G, {
only_image = 0}, {x = 'x}, {y = 'y})
Given an elliptic curve E
, a finite subgroup G
of E
is given either
as a generating point P
(for a cyclic G
) or as a polynomial whose roots
vanish on the x
-coordinates of the non-zero elements of G
(general case
and more efficient if available). This function returns the
[a_1,a_2,a_3,a_4,a_6]
invariants of the quotient elliptic curve E/G
and
(if only_image is zero (the default)) a vector of rational
functions [f, g, h]
such that the isogeny E \to E/G
is given by (x,y)
:--->(f(x)/h(x)^2, g(x,y)/h(x)^3)
.
? E = ellinit([0,1]); ? elltors(E) %2 = [6, [6], [[2, 3]]] ? ellisogeny(E, [2,3], 1) \\ Weierstrass model for E/<P> %3 = [0, 0, 0, -135, -594] ? ellisogeny(E,[-1,0]) %4 = [[0,0,0,-15,22], [x^3+2*x^2+4*x+3, y*x^3+3*y*x^2-2*y, x+1]]
The library syntax is GEN
ellisogeny(GEN E, GEN G, long only_image, long x = -1, long y = -1)
where x
, y
are variable numbers.
(f, g)
Given an isogeny of elliptic curves f:E'\to E
(being the result of a call
to ellisogeny
), apply f
to g
:
@3* if g
is a point P
in the domain of f
, return the image f(P)
;
@3* if g:E''\to E'
is a compatible isogeny, return the composite
isogeny f o g: E''\to E
.
? one = ffgen(101, 't)^0; ? E = ellinit([6, 53, 85, 32, 34] * one); ? P = [84, 71] * one; ? ellorder(E, P) %4 = 5 ? [F, f] = ellisogeny(E, P); \\ f: E->F = E/<P> ? ellisogenyapply(f, P) %6 = [0] ? F = ellinit(F); ? Q = [89, 44] * one; ? ellorder(F, Q) %9 = 2 ? [G, g] = ellisogeny(F, Q); \\ g: F->G = F/<Q> ? gof = ellisogenyapply(g, f); \\ gof: E -> G
The library syntax is GEN
ellisogenyapply(GEN f, GEN g)
.
ellisomat(E, {
fl = 0})
Given an elliptic curve E
defined over Q, compute representatives of the
isomorphism classes of elliptic curves Q-isogenous to E
. The function
returns a vector [L,M]
where L
is a list of triples [E_i, f_i, g_i]
,
where E_i
is an elliptic curve in [a_4,a_6]
form, f_i: E \to E_i
is a rational isogeny, g_i: E_i \to E
is the dual isogeny of f_i
,
and M
is the matrix such that M_{i,j}
is the degree of the isogeny between
E_i
and E_j
. Furthermore the first curve E_1
is isomorphic to E
by f_1
. If the flag fl = 1
, the f_i
and g_i
are not computed,
which saves time, and L
is the list of the curves E_i
.
? E = ellinit("14a1"); ? [L,M] = ellisomat(E); ? LE = apply(x->x[1], L) \\ list of curves %3 = [[215/48,-5291/864],[-675/16,6831/32],[-8185/48,-742643/864], [-1705/48,-57707/864],[-13635/16,306207/32],[-131065/48,-47449331/864]] ? L[2][2] \\ isogeny f_2 %4 = [x^3+3/4*x^2+19/2*x-311/12, 1/2*x^4+(y+1)*x^3+(y-4)*x^2+(-9*y+23)*x+(55*y+55/2),x+1/3] ? L[2][3] \\ dual isogeny g_2 %5 = [1/9*x^3-1/4*x^2-141/16*x+5613/64, -1/18*x^4+(1/27*y-1/3)*x^3+(-1/12*y+87/16)*x^2+(49/16*y-48)*x +(-3601/64*y+16947/512),x-3/4] ? apply(E->ellidentify(ellinit(E))[1][1], LE) %6 = ["14a1","14a4","14a3","14a2","14a6","14a5"] ? M %7 = [1 3 3 2 6 6]
[3 1 9 6 2 18]
[3 9 1 6 18 2]
[2 6 6 1 3 3]
[6 2 18 3 1 9]
[6 18 2 3 9 1]
The library syntax is GEN
ellisomat(GEN E, long fl)
.
(E,z)
Gives 1 (i.e. true) if the point z
is on the elliptic curve E
, 0
otherwise. If E
or z
have imprecise coefficients, an attempt is made to
take this into account, i.e. an imprecise equality is checked, not a precise
one. It is allowed for z
to be a vector of points in which case a vector
(of the same type) is returned.
The library syntax is GEN
ellisoncurve(GEN E, GEN z)
.
Also available is int
oncurve(GEN E, GEN z)
which does not
accept vectors of points.
(E,{p})
Return 1 if the elliptic curve E
defined over a number field
or a finite field is supersingular at p
, and 0
otherwise.
If the curve is defined over a number field, p
must be explicitly given,
and must be a prime number, resp. a maximal ideal, if the curve is defined
over Q, resp. a general number field: we return 1
if and only if E
has supersingular good reduction at p
.
Alternatively, E
can be given by its j
-invariant in a finite field. In
this case p
must be omitted.
? g = ffprimroot(ffgen(7^5)) %1 = x^3 + 2*x^2 + 3*x + 1 ? [g^n | n <- [1 .. 7^5 - 1], ellissupersingular(g^n)] %2 = [6]
? K = nfinit(y^3-2); P = idealprimedec(K, 2)[1]; ? E = ellinit([y,1], K); ? ellissupersingular(E, P) %5 = 1
The library syntax is GEN
ellissupersingular(GEN E, GEN p = NULL)
.
Also available is
int
elljissupersingular(GEN j)
where j
is a j
-invariant of a curve
over a finite field.
(x)
Elliptic j
-invariant. x
must be a complex number
with positive imaginary part, or convertible into a power series or a
p
-adic number with positive valuation.
The library syntax is GEN
jell(GEN x, long prec)
.
(E,p)
Calculates the Kodaira type of the local fiber of the elliptic curve
E
at p
. E
must be an ell
structure as output by
ellinit
, over Q (p
a rational prime) or a number field K
(p
a maximal ideal given by a prid
structure), and is assumed to have all
its coefficients a_i
integral.
The result is a 4-component vector [f,kod,v,c]
. Here f
is the exponent of
p
in the arithmetic conductor of E
, and kod
is the Kodaira type which
is coded as follows:
1 means good reduction (type I_0
), 2, 3 and 4 mean types II, III and IV
respectively, 4+
nu with nu > 0
means type I_
nu;
finally the opposite values -1
, -2
, etc. refer to the starred types
I_0^*
, II^*
, etc. The third component v
is itself a vector [u,r,s,t]
giving the coordinate changes done during the local reduction;
u = 1
if and only if the given equation was already minimal at p
.
Finally, the last component c
is the local Tamagawa number c_p
.
The library syntax is GEN
elllocalred(GEN E, GEN p)
.
(E,P,G,{o})
Given two points P
and G
on the elliptic curve E/
F_q
, returns the
discrete logarithm of P
in base G
, i.e. the smallest non-negative
integer n
such that P = [n]G
.
See znlog
for the limitations of the underlying discrete log algorithms.
If present, o
represents the order of G
, see Label se:DLfun;
the preferred format for this parameter is [N, factor(N)]
, where N
is the order of G
.
If no o
is given, assume that G
generates the curve.
The function also assumes that P
is a multiple of G
.
? a = ffgen(ffinit(2,8),'a); ? E = ellinit([a,1,0,0,1]); \\ over F_{2^8} ? x = a^3; y = ellordinate(E,x)[1]; ? P = [x,y]; G = ellmul(E, P, 113); ? ord = [242, factor(242)]; \\ P generates a group of order 242. Initialize. ? ellorder(E, G, ord) %4 = 242 ? e = elllog(E, P, G, ord) %5 = 15 ? ellmul(E,G,e) == P %6 = 1
The library syntax is GEN
elllog(GEN E, GEN P, GEN G, GEN o = NULL)
.
(E,s,{A = 1})
This function is deprecated, use lfun(E,s)
instead.
E
being an elliptic curve, given by an arbitrary model over Q as output
by ellinit
, this function computes the value of the L
-series of E
at
the (complex) point s
. This function uses an O(N^{1/2})
algorithm, where
N
is the conductor.
The optional parameter A
fixes a cutoff point for the integral and is best
left omitted; the result must be independent of A
, up to
realprecision
, so this allows to check the function's accuracy.
The library syntax is GEN
elllseries(GEN E, GEN s, GEN A = NULL, long prec)
.
(E,{&v})
Let E
be an ell
structure over a number field K
. This function
determines whether E
admits a global minimal integral model. If so, it
returns it and sets v = [u,r,s,t]
to the corresponding change of variable:
the return value is identical to that of ellchangecurve(E, v)
.
Else return the (non-principal) Weierstrass class of E
, i.e. the class of
prod p^{(v_{
p}{
Delta} -
delta_{
p}) / 12}
where
Delta = E.disc
is the model's discriminant and
p ^
delta_{
p}
is the local minimal discriminant.
This function requires either that E
be defined
over the rational field Q (with domain D = 1
in ellinit
),
in which case a global minimal model always exists, or over a number
field given by a bnf structure. The Weierstrass class is given in
bnfisprincipal
format, i.e. in terms of the K.gen
generators.
The resulting model has integral coefficients and is everywhere minimal, the
coefficients a_1
and a_3
are reduced modulo 2
(in terms of the fixed
integral basis K.zk
) and a_2
is reduced modulo 3
. Over Q, we
further require that a_1
and a_3
be 0
or 1
, that a_2
be 0
or +-
1
and that u > 0
in the change of variable: both the model and the change
of variable v
are then unique.
? e = ellinit([6,6,12,55,233]); \\ over Q ? E = ellminimalmodel(e, &v); ? E[1..5] %3 = [0, 0, 0, 1, 1] ? v %4 = [2, -5, -3, 9]
? K = bnfinit(a^2-65); \\ over a non-principal number field ? K.cyc %2 = [2] ? u = Mod(8+a, K.pol); ? E = ellinit([1,40*u+1,0,25*u^2,0], K); ? ellminimalmodel(E) \\ no global minimal model exists over Z_K %6 = [1]~
The library syntax is GEN
ellminimalmodel(GEN E, GEN *v = NULL)
.
(E, {
flag = 0})
Let E
be an elliptic curve defined over Q, return
a discriminant D
such that the twist of E
by D
is minimal among all
possible quadratic twists, i.e. if flag = 0
, its minimal model has minimal
discriminant, or if flag = 1
, it has minimal conductor.
In the example below, we find a curve with j
-invariant 3
and minimal
conductor.
? E=ellminimalmodel(ellinit(ellfromj(3))); ? ellglobalred(E)[1] %2 = 357075 ? D = ellminimaltwist(E,1) %3 = -15 ? E2=ellminimalmodel(ellinit(elltwist(E,D))); ? ellglobalred(E2)[1] %5 = 14283
The library syntax is GEN
ellminimaltwist0(GEN E, long flag)
.
Also available are
GEN
ellminimaltwist(E)
for flag = 0
, and
GEN
ellminimaltwistcond(E)
for flag = 1
.
(e)
e
being an elliptic curve defined over Q output by ellinit
,
compute the modular degree of e
divided by the square of
the Manin constant. Return [D, err]
, where D
is a rational number and
err is exponent of the truncation error.
The library syntax is GEN
ellmoddegree(GEN e, long bitprec)
.
(N,{x},{y})
Given a prime N < 500
, return a vector [P,t]
where P(x,y)
is a modular equation of level N
, i.e. a bivariate polynomial with integer
coefficients; t
indicates the type of this equation: either
canonical (t = 0
) or Atkin (t = 1
). This function requires
the seadata
package and its only use is to give access to the package
contents. See polmodular
for a more general and more flexible function.
Let j
be the j
-invariant function. The polynomial P
satisfies
the functional equation,
P(f,j) = P(f | W_N, j | W_N) = 0
for some modular function f = f_N
(hand-picked for each fixed N
to
minimize its size, see below), where W_N(
tau) = -1 / (N
tau)
is the
Atkin-Lehner involution. These two equations allow to compute the values of
the classical modular polynomial Phi_N
, such that Phi_N(j(
tau),
j(N
tau)) = 0
, while being much smaller than the latter. More precisely, we
have j(W_N(
tau)) = j(N
tau)
; the function f
is invariant under
Gamma_0(N)
and also satisfies
@3* for Atkin type: f | W_N = f
;
@3* for canonical type: let s = 12/
gcd (12,N-1)
, then
f | W_N = N^s / f
. In this case, f
has a simple definition:
f(
tau) = N^s (
eta(N
tau) /
eta(
tau) )^{2 s}
,
where eta is Dedekind's eta function.
The following GP function returns values of the classical modular polynomial
by eliminating f_N(
tau)
in the above functional equation,
for N <= 31
or N\in{41,47,59,71}
.
classicaleqn(N, X='X, Y='Y)= { my([P,t] = ellmodulareqn(N), Q, d); if (poldegree(P,'y) > 2, error("level unavailable in classicaleqn")); if (t == 0, \\ Canonical my(s = 12/gcd(12,N-1)); Q = 'x^(N+1) * substvec(P,['x,'y],[N^s/'x,Y]); d = N^(s*(2*N+1)) * (-1)^(N+1); , \\ Atkin Q = subst(P,'y,Y); d = (X-Y)^(N+1)); polresultant(subst(P,'y,X), Q) / d; }
The library syntax is GEN
ellmodulareqn(long N, long x = -1, long y = -1)
where x
, y
are variable numbers.
(E,z,n)
Computes [n]z
, where z
is a point on the elliptic curve E
. The
exponent n
is in Z, or may be a complex quadratic integer if the curve E
has complex multiplication by n
(if not, an error message is issued).
? Ei = ellinit([1,0]); z = [0,0]; ? ellmul(Ei, z, 10) %2 = [0] \\ unsurprising: z has order 2 ? ellmul(Ei, z, I) %3 = [0, 0] \\ Ei has complex multiplication by Z[i] ? ellmul(Ei, z, quadgen(-4)) %4 = [0, 0] \\ an alternative syntax for the same query ? Ej = ellinit([0,1]); z = [-1,0]; ? ellmul(Ej, z, I) *** at top-level: ellmul(Ej,z,I) *** ^-------------- *** ellmul: not a complex multiplication in ellmul. ? ellmul(Ej, z, 1+quadgen(-3)) %6 = [1 - w, 0]
The simple-minded algorithm for the CM case assumes that we are in
characteristic 0
, and that the quadratic order to which n
belongs has
small discriminant.
The library syntax is GEN
ellmul(GEN E, GEN z, GEN n)
.
(E,z)
Opposite of the point z
on elliptic curve E
.
The library syntax is GEN
ellneg(GEN E, GEN z)
.
(E,P)
Given an elliptic curve E/
Q (more precisely, a model defined over Q
of a curve) and a rational point P \in E(
Q)
, returns the pair [R,n]
,
where n
is the least positive integer such that R := [n]P
has good
reduction at every prime. More precisely, its image in a minimal model is
everywhere non-singular.
? e = ellinit("57a1"); P = [2,-2]; ? ellnonsingularmultiple(e, P) %2 = [[1, -1], 2] ? e = ellinit("396b2"); P = [35, -198]; ? [R,n] = ellnonsingularmultiple(e, P); ? n %5 = 12
The library syntax is GEN
ellnonsingularmultiple(GEN E, GEN P)
.
(E,z,{o})
Gives the order of the point z
on the elliptic
curve E
, defined over a finite field or a number field.
Return (the impossible value) zero if the point has infinite order.
? E = ellinit([-157^2,0]); \\ the "157-is-congruent" curve ? P = [2,2]; ellorder(E, P) %2 = 2 ? P = ellheegner(E); ellorder(E, P) \\ infinite order %3 = 0 ? K = nfinit(polcyclo(11,t)); E=ellinit("11a3", K); T = elltors(E); ? ellorder(E, T.gen[1]) %5 = 25 ? E = ellinit(ellfromj(ffgen(5^10))); ? ellcard(E) %7 = 9762580 ? P = random(E); ellorder(E, P) %8 = 4881290 ? p = 2^160+7; E = ellinit([1,2], p); ? N = ellcard(E) %9 = 1461501637330902918203686560289225285992592471152 ? o = [N, factor(N)]; ? for(i=1,100, ellorder(E,random(E))) time = 260 ms.
The parameter o
, is now mostly useless, and kept for backward
compatibility. If present, it represents a non-zero multiple of the order
of z
, see Label se:DLfun; the preferred format for this parameter is
[ord, factor(ord)]
, where ord
is the cardinality of the curve.
It is no longer needed since PARI is now able to compute it over large
finite fields (was restricted to small prime fields at the time this feature
was introduced), and caches the result in E
so that it is computed
and factored only once. Modifying the last example, we see that including
this extra parameter provides no improvement:
? o = [N, factor(N)]; ? for(i=1,100, ellorder(E,random(E),o)) time = 260 ms.
The library syntax is GEN
ellorder(GEN E, GEN z, GEN o = NULL)
.
The obsolete form GEN
orderell(GEN e, GEN z)
should no longer be
used.
(E,x)
Gives a 0, 1 or 2-component vector containing
the y
-coordinates of the points of the curve E
having x
as
x
-coordinate.
The library syntax is GEN
ellordinate(GEN E, GEN x, long prec)
.
(E, p, n, {s = 0}, {r = 0}, {D = 1})
Returns the value (or r
-th derivative) on a character chi^s
of
Z_p^*
of the p
-adic L
-function of the elliptic curve E/
Q, twisted by
D
, given modulo p^n
.
@3Characters. The set of continuous characters of
Gal(
Q(
mu_{p^{ oo }})/
Q)
is identified to Z_p^*
via the
cyclotomic character chi with values in \overline{
Q_p}^*
. Denote by
tau:
Z_p^*\to
Z_p^*
the Teichmüller character, with values
in the (p-1)
-th roots of 1
for p != 2
, and {-1,1}
for p = 2
;
finally, let
<
chi>=
chi tau^{-1}
, with values in 1 + 2p
Z_p
.
In GP, the continuous character of
Gal(
Q(
mu_{p^{ oo }})/
Q)
given by <
chi>^{s_1}
tau^{s_2}
is represented by the pair of integers s = (s_1,s_2)
, with s_1
\in
Z_p
and s_2 mod p-1
for p > 2
, (resp. mod 2
for p = 2
); s
may be also an integer, representing (s,s)
or chi^s
.
@3The p
-adic L
function.
The p
-adic L
function L_p
is defined on the set of continuous
characters of Gal(
Q(
mu_{p^{ oo }})/
Q)
, as int_{
Z_p^*}
chi^s d
mu for a certain p
-adic distribution mu on Z_p^*
. The
derivative is given by
L_p^{(r)}(E,
chi^s) =
int_{
Z_p^*}
log _p^r(a)
chi^s(a) d
mu(a).
More precisely:
@3* When E
has good supersingular reduction, L_p
takes its
values in Q_p \otimes H^1_{dR}(E/
Q)
and satisfies
(1-p^{-1} F)^{-2} L_p(E,
chi^0) = (L(E,1) /
Omega).
omega
where F
is the Frobenius, L(E,1)
is the value of the complex L
function at 1
, omega is the Néron differential
and Omega the attached period on E(
R)
. Here, chi^0
represents
the trivial character.
The function returns the components of L_p^{(r)}(E,
chi^s)
in
the basis (
omega, F(
omega))
.
@3* When E
has ordinary good reduction, this method only defines
the projection of L_p(E,
chi^s)
on the alpha-eigenspace,
where alpha is the unit eigenvalue for F
. This is what the function
returns. We have
(1-
alpha^{-1})^{-2} L_{p,
alpha}(E,
chi^0) = L(E,1) /
Omega.
Two supersingular examples:
? cxL(e) = bestappr( ellL1(e) / e.omega[1] );
? e = ellinit("17a1"); p=3; \\ supersingular, a3 = 0 ? L = ellpadicL(e,p,4); ? F = [0,-p;1,ellap(e,p)]; \\ Frobenius matrix in the basis (omega,F(omega)) ? (1-p^(-1)*F)^-2 * L / cxL(e) %5 = [1 + O(3^5), O(3^5)]~ \\ [1,0]~
? e = ellinit("116a1"); p=3; \\ supersingular, a3 != 0~ ? L = ellpadicL(e,p,4); ? F = [0,-p; 1,ellap(e,p)]; ? (1-p^(-1)*F)^-2*L~ / cxL(e) %9 = [1 + O(3^4), O(3^5)]~
Good ordinary reduction:
? e = ellinit("17a1"); p=5; ap = ellap(e,p) %1 = -2 \\ ordinary ? L = ellpadicL(e,p,4) %2 = 4 + 3*5 + 4*5^2 + 2*5^3 + O(5^4) ? al = padicappr(x^2 - ap*x + p, ap + O(p^7))[1]; ? (1-al^(-1))^(-2) * L / cxL(e) %4 = 1 + O(5^4)
Twist and Teichmüller:
? e = ellinit("17a1"); p=5; \\ ordinary \\ 2nd derivative at tau^1, twist by -7 ? ellpadicL(e, p, 4, [0,1], 2, -7) %2 = 2*5^2 + 5^3 + O(5^4)
This function is a special case of mspadicL
, and it also appears
as the first term of mspadicseries
:
? e = ellinit("17a1"); p=5; ? L = ellpadicL(e,p,4) %2 = 4 + 3*5 + 4*5^2 + 2*5^3 + O(5^4) ? [M,phi] = msfromell(e, 1); ? Mp = mspadicinit(M, p, 4); ? mu = mspadicmoments(Mp, phi); ? mspadicL(mu) %6 = 4 + 3*5 + 4*5^2 + 2*5^3 + 2*5^4 + 5^5 + O(5^6) ? mspadicseries(mu) %7 = (4 + 3*5 + 4*5^2 + 2*5^3 + 2*5^4 + 5^5 + O(5^6)) + (3 + 3*5 + 5^2 + 5^3 + O(5^4))*x + (2 + 3*5 + 5^2 + O(5^3))*x^2 + (3 + 4*5 + 4*5^2 + O(5^3))*x^3 + (3 + 2*5 + O(5^2))*x^4 + O(x^5)
@3These are more cumbersome than ellpadicL
but allow to
compute at different characters, or successive derivatives, or to
twist by a quadratic character essentially for the cost of a single call to
ellpadicL
due to precomputations.
The library syntax is GEN
ellpadicL(GEN E, GEN p, long n, GEN s = NULL, long r, GEN D = NULL)
.
(E,p,n)
If p > 2
is a prime and E
is a elliptic curve on Q with good
reduction at p
, return the matrix of the Frobenius endomorphism varphi on
the crystalline module D_p(E) =
Q_p \otimes H^1_{dR}(E/
Q)
with respect to
the basis of the given model (
omega,
eta = x
omega)
, where
omega = dx/(2 y+a_1 x+a_3)
is the invariant differential.
The characteristic polynomial of varphi is x^2 - a_p x + p
.
The matrix is computed to absolute p
-adic precision p^n
.
? E = ellinit([1,-1,1,0,0]); ? F = ellpadicfrobenius(E,5,3); ? lift(F) %3 = [120 29]
[ 55 5] ? charpoly(F) %4 = x^2 + O(5^3)*x + (5 + O(5^3)) ? ellap(E, 5) %5 = 0
The library syntax is GEN
ellpadicfrobenius(GEN E, long p, long n)
.
(E, p,n, P,{Q})
Cyclotomic p
-adic height of the rational point P
on the elliptic curve
E
(defined over Q), given to n
p
-adic digits.
If the argument Q
is present, computes the value of the bilinear
form (h(P+Q)-h(P-Q)) / 4
.
Let D_{dR}(E) := H^1_{dR}(E) \otimes_
Q Q_p
be the Q_p
vector space
spanned by omega
(invariant differential dx/(2y+a_1x+a3)
related to the given model) and
eta = x
omega. Then the cyclotomic p
-adic height associates to
P\in E(
Q)
an element f
omega + g
eta in D_{dR}
.
This routine returns the vector [f, g]
to n
p
-adic digits.
If P\in E(
Q)
is in the kernel of reduction mod p
and if its reduction
at all finite places is non singular, then g = -(
log _E P)^2
, where
log _E
is the logarithm for the formal group of E
at p
.
If furthermore the model is of the form Y^2 = X^3 + a X + b
and P = (x,y)
,
then
f =
log _p(denominator(x)) - 2
log _p(
sigma(P))
where sigma(P)
is given by ellsigma
(E,P)
.
Recall (Advanced topics in the arithmetic of elliptic curves, Theorem 3.2) that the local height function over the complex numbers is of the form
lambda(z) = -
log (|E.disc|) / 6 +
Re (z
eta(z)) - 2
log (
sigma(z).
(N.B. our normalization for local and global heights is twice that of Silverman's).
? E = ellinit([1,-1,1,0,0]); P = [0,0]; ? ellpadicheight(E,5,4, P) %2 = [3*5 + 5^2 + 2*5^3 + O(5^4), 5^2 + 4*5^4 + O(5^6)] ? E = ellinit("11a1"); P = [5,5]; \\ torsion point ? ellpadicheight(E,19,6, P) %4 = O(19^6) ? E = ellinit([0,0,1,-4,2]); P = [-2,1]; ? ellpadicheight(E,3,5, P) %6 = [2*3^2 + 2*3^3 + 3^4 + O(3^5), 2*3^2 + 3^4 + 2*3^5 + 3^6 + O(3^7)] ? ellpadicheight(E,3,5, P, elladd(E,P,P))
One can replace the parameter p
prime by a vector [p,[a,b]]
, in which
case the routine returns the p
-adic number af + bg
.
When E
has good ordinary reduction at p
, the ``canonical''
p
-adic height is given by
s2 = ellpadics2(E,p,n); ellpadicheight(E, [p,[1,-s2]], n, P)
@3Since s_2
does not depend on P
, it is preferable to
compute it only once:
? E = ellinit("5077a1"); p = 5; n = 7; ? s2 = ellpadics2(E,p,n); ? M = ellpadicheightmatrix(E,[p,[1,-s2]], n, E.gen); ? matdet(M) \\ p-adic regulator %4 = 5 + 5^2 + 4*5^3 + 2*5^4 + 2*5^5 + 5^6 + O(5^7)
The library syntax is GEN
ellpadicheight0(GEN E, GEN p, long n, GEN P, GEN Q = NULL)
.
(E,p,n,v)
v
being a vector of points, this function outputs the Gram matrix of
v
with respect to the cyclotomic p
-adic height, given to n
p
-adic
digits; in other words, the (i,j)
component of the matrix is equal to
ellpadicheight
(E,p,n, v[i],v[j]) = [f,g]
.
See ellpadicheight
; in particular one can replace the parameter p
prime by a vector [p,[a,b]]
, in which case the routine returns the matrix
containing the p
-adic numbers af + bg
.
The library syntax is GEN
ellpadicheightmatrix(GEN E, GEN p, long n, GEN v)
.
(E,p,n,P)
Given E
defined over K =
Q or Q_p
and P = [x,y]
on E(K)
in the
kernel of reduction mod p
, let t(P) = -x/y
be the formal group
parameter; this function returns L(t)
, where L
denotes the formal
logarithm (mapping the formal group of E
to the additive formal group)
attached to the canonical invariant differential:
dL = dx/(2y + a_1x + a_3)
.
The library syntax is GEN
ellpadiclog(GEN E, GEN p, long n, GEN P)
.
(E,p,n)
If p > 2
is a prime and E/
Q is a elliptic curve with ordinary good
reduction at p
, returns the slope of the unit eigenvector
of ellpadicfrobenius(E,p,n)
, i.e. the action of Frobenius varphi on
the crystalline module D_p(E) =
Q_p \otimes H^1_{dR}(E/
Q)
in the basis of
the given model (
omega,
eta = x
omega)
, where omega is the invariant
differential dx/(2 y+a_1 x+a_3)
. In other words, eta + s_2
omega
is an eigenvector for the unit eigenvalue of varphi.
This slope is the unique c \in 3^{-1}
Z_p
such that the odd solution
sigma(t) = t + O(t^2)
of
- d((1)/(
sigma) (d
sigma)/(
omega))
= (x(t) + c)
omega
is in t
Z_p[[t]]
.
It is equal to b_2/12 - E_2/12
where E_2
is the value of the Katz
p
-adic Eisenstein series of weight 2 on (E,
omega)
. This is
used to construct a canonical p
-adic height when E
has good ordinary
reduction at p
as follows
s2 = ellpadics2(E,p,n); h(E,p,n, P, s2) = ellpadicheight(E, [p,[1,-s2]],n, P);
@3Since s_2
does not depend on the point P
, we compute it
only once.
The library syntax is GEN
ellpadics2(GEN E, GEN p, long n)
.
(w, {
flag = 0})
Let w
describe a complex period lattice (w = [w_1,w_2]
or an ellinit
structure). Returns normalized periods [W_1,W_2]
generating
the same lattice such that tau := W_1/W_2
has positive imaginary part
and lies in the standard fundamental domain for SL_2(
Z)
.
If flag = 1
, the function returns [[W_1,W_2], [
eta_1,
eta_2]]
, where
eta_1
and eta_2
are the quasi-periods attached to
[W_1,W_2]
, satisfying eta_1 W_2 -
eta_2 W_1 = 2 i
pi.
The output of this function is meant to be used as the first argument given to ellwp, ellzeta, ellsigma or elleisnum. Quasi-periods are needed by ellzeta and ellsigma only.
The library syntax is GEN
ellperiods(GEN w, long flag, long prec)
.
(E,P)
If E/
C ~
C/
Lambda is a complex elliptic curve (Lambda =
E.omega
),
computes a complex number z
, well-defined modulo the lattice Lambda,
corresponding to the point P
; i.e. such that
P = [
wp_
Lambda(z),
wp'_
Lambda(z)]
satisfies the equation
y^2 = 4x^3 - g_2 x - g_3,
where g_2
, g_3
are the elliptic invariants.
If E
is defined over R and P\in E(
R)
, we have more precisely, 0 <=
Re (t) < w1
and 0 <=
Im (t) <
Im (w2)
, where (w1,w2)
are the real and
complex periods of E
.
? E = ellinit([0,1]); P = [2,3]; ? z = ellpointtoz(E, P) %2 = 3.5054552633136356529375476976257353387 ? ellwp(E, z) %3 = 2.0000000000000000000000000000000000000 ? ellztopoint(E, z) - P %4 = [6.372367644529809109 E-58, 7.646841173435770930 E-57] ? ellpointtoz(E, [0]) \\ the point at infinity %5 = 0
If E/
Q_p
has multiplicative reduction, then E/\bar{
Q_p}
is analytically
isomorphic to \bar{
Q}_p^*/q^
Z (Tate curve) for some p
-adic integer q
.
The behaviour is then as follows:
@3* If the reduction is split (E.tate[2]
is a t_PADIC
), we have
an isomorphism phi: E(
Q_p) ~
Q_p^*/q^
Z and the function returns
phi(P)\in
Q_p
.
@3* If the reduction is not split (E.tate[2]
is a
t_POLMOD
), we only have an isomorphism phi: E(K) ~ K^*/q^
Z over
the unramified quadratic extension K/
Q_p
. In this case, the output
phi(P)\in K
is a t_POLMOD
.
? E = ellinit([0,-1,1,0,0], O(11^5)); P = [0,0]; ? [u2,u,q] = E.tate; type(u) \\ split multiplicative reduction %2 = "t_PADIC" ? ellmul(E, P, 5) \\ P has order 5 %3 = [0] ? z = ellpointtoz(E, [0,0]) %4 = 3 + 11^2 + 2*11^3 + 3*11^4 + O(11^5) ? z^5 %5 = 1 + O(11^5) ? E = ellinit(ellfromj(1/4), O(2^6)); x=1/2; y=ellordinate(E,x)[1]; ? z = ellpointtoz(E,[x,y]); \\ t_POLMOD of t_POL with t_PADIC coeffs ? liftint(z) \\ lift all p-adics %8 = Mod(8*u + 7, u^2 + 437)
The library syntax is GEN
zell(GEN E, GEN P, long prec)
.
(E,z,n)
Deprecated alias for ellmul
.
The library syntax is GEN
ellmul(GEN E, GEN z, GEN n)
.
(E,{p})
E
being an ell
structure over Q as output by ellinit
,
this function computes the local root number of its L
-series at the place
p
(at the infinite place if p = 0
). If p
is omitted, return the global
root number. Note that the global root number is the sign of the functional
equation and conjecturally is the parity of the rank of the
Mordell-Weil group. The equation for E
needs not be minimal at p
,
but if the model is already minimal the function will run faster.
The library syntax is long
ellrootno(GEN E, GEN p = NULL)
.
ellsea(E,{
tors = 0})
Let E
be an ell structure as output by ellinit
, defined over
a finite field F_q
. This function computes the order of the group
E(
F_q)
using the SEA algorithm and the tors
argument allows to
speed up a search for curves having almost prime order.
@3* If the characteristic is too small (p <= 7
) the generic algorithm
ellcard
is used instead and the tors
argument is ignored.
@3* When tors
is set to a non-zero value, the function returns 0
as soon as it detects that the order has a small prime factor not dividing
tors
; SEA considers modular polynomials of increasing prime degree
ell and we return 0
as soon as we hit an ell (coprime to tors
)
dividing #E(
F_q)
.
In particular, you should set tors
to 1
if you want a curve with
prime order, to 2
if you want to allow a cofacteur which is a power of two
(e.g. for Edwards's curves), etc.
The availability of the seadata
package will speed up the computation,
and is strongly recommended.
The following function returns a curve of prime order over F_p
.
cryptocurve(p) = { while(1, my(E, N, j = Mod(random(p), p)); E = ellinit(ellfromj(j)); N = ellsea(E, 1); if(!N, continue); if (isprime(N), return(E)); \\ try the quadratic twist for free if (isprime(2*p+2 - N), return(ellinit(elltwist(E)))); ); } ? p = randomprime([2^255, 2^256]); ? E = cryptocurve(p); \\ insist on prime order %2 = 47,447ms
@3The same example without early abort (using ellsea(E,1)
instead of ellsea(E)
) runs for about 5 minutes before finding a
suitable curve.
The library syntax is GEN
ellsea(GEN E, ulong tors)
.
(N)
This function finds all curves in the elldata
database satisfying
the constraint defined by the argument N
:
@3* if N
is a character string, it selects a given curve, e.g.
"11a1"
, or curves in the given isogeny class, e.g. "11a"
, or
curves with given conductor, e.g. "11"
;
@3* if N
is a vector of integers, it encodes the same constraints
as the character string above, according to the ellconvertname
correspondance, e.g. [11,0,1]
for "11a1"
, [11,0]
for
"11a"
and [11]
for "11"
;
@3* if N
is an integer, curves with conductor N
are selected.
If N
codes a full curve name, for instance "11a1"
or [11,0,1]
,
the output format is [N, [a_1,a_2,a_3,a_4,a_6], G]
where
[a_1,a_2,a_3,a_4,a_6]
are the coefficients of the Weierstrass equation of
the curve and G
is a Z-basis of the free part of the
Mordell-Weil group attached to the curve.
? ellsearch("11a3") %1 = ["11a3", [0, -1, 1, 0, 0], []] ? ellsearch([11,0,3]) %2 = ["11a3", [0, -1, 1, 0, 0], []]
If N
is not a full curve name, then the output is a vector of all matching
curves in the above format:
? ellsearch("11a") %1 = [["11a1", [0, -1, 1, -10, -20], []], ["11a2", [0, -1, 1, -7820, -263580], []], ["11a3", [0, -1, 1, 0, 0], []]] ? ellsearch("11b") %2 = []
The library syntax is GEN
ellsearch(GEN N)
.
Also available is GEN
ellsearchcurve(GEN N)
that only
accepts complete curve names (as t_STR
).
(L,{z = 'x},{
flag = 0})
Computes the value at z
of the Weierstrass sigma function attached to
the lattice L
as given by ellperiods
(,1)
: including quasi-periods
is useful, otherwise there are recomputed from scratch for each new z
.
sigma(z, L) = z
prod_{
omega\in L^*} (1 -
(z)/(
omega))e^{(z)/(
omega) + (z^2)/(2
omega^2)}.
It is also possible to directly input L = [
omega_1,
omega_2]
,
or an elliptic curve E
as given by ellinit
(L = E.omega
).
? w = ellperiods([1,I], 1); ? ellsigma(w, 1/2) %2 = 0.47494937998792065033250463632798296855 ? E = ellinit([1,0]); ? ellsigma(E) \\ at 'x, implicitly at default seriesprecision %4 = x + 1/60*x^5 - 1/10080*x^9 - 23/259459200*x^13 + O(x^17)
If flag = 1
, computes an arbitrary determination of log (
sigma(z))
.
The library syntax is GEN
ellsigma(GEN L, GEN z = NULL, long flag, long prec)
.
(E,
z1,
z2)
Difference of the points z1
and z2
on the
elliptic curve corresponding to E
.
The library syntax is GEN
ellsub(GEN E, GEN z1, GEN z2)
.
elltaniyama(E, {d =
seriesprecision})
Computes the modular parametrization of the elliptic curve E/
Q,
where E
is an ell
structure as output by ellinit
. This returns
a two-component vector [u,v]
of power series, given to d
significant
terms (seriesprecision
by default), characterized by the following two
properties. First the point (u,v)
satisfies the equation of the elliptic
curve. Second, let N
be the conductor of E
and Phi: X_0(N)\to E
be a modular parametrization; the pullback by Phi of the
Néron differential du/(2v+a_1u+a_3)
is equal to 2i
pi
f(z)dz
, a holomorphic differential form. The variable used in the power
series for u
and v
is x
, which is implicitly understood to be equal to
exp (2i
pi z)
.
The algorithm assumes that E
is a strong Weil curve
and that the Manin constant is equal to 1: in fact, f(x) =
sum_{n > 0}
ellan(E, n) x^n
.
The library syntax is GEN
elltaniyama(GEN E, long precdl)
.
(E, P, Q, m)
Computes the Tate pairing of the two points P
and Q
on the elliptic
curve E
. The point P
must be of m
-torsion.
The library syntax is GEN
elltatepairing(GEN E, GEN P, GEN Q, GEN m)
.
(E)
If E
is an elliptic curve defined over a number field or a finite field,
outputs the torsion subgroup of E
as a 3-component vector [t,v1,v2]
,
where t
is the order of the torsion group, v1
gives the structure
of the torsion group as a product of cyclic groups (sorted by decreasing
order), and v2
gives generators for these cyclic groups. E
must be an
ell
structure as output by ellinit
.
? E = ellinit([-1,0]); ? elltors(E) %1 = [4, [2, 2], [[0, 0], [1, 0]]]
Here, the torsion subgroup is isomorphic to Z/2
Z x
Z/2
Z, with
generators [0,0]
and [1,0]
.
The library syntax is GEN
elltors(GEN E)
.
(E,{P})
Returns the coefficients [a_1,a_2,a_3,a_4,a_6]
of the twist of the
elliptic curve E
by the quadratic extension of the coefficient ring
defined by P
(when P
is a polynomial) or quadpoly(P)
when P
is an
integer. If E
is defined over a finite field, then P
can be omitted,
in which case a random model of the unique non-trivial twist is returned.
If E
is defined over a number field, the model should be replaced by a
minimal model (if one exists).
Example: Twist by discriminant -3
:
? elltwist(ellinit([0,a2,0,a4,a6]),-3) %1 = [0,-3*a2,0,9*a4,-27*a6]
Twist by the Artin-Shreier extension given by x^2+x+T
in
characteristic 2
:
? lift(elltwist(ellinit([a1,a2,a3,a4,a6]*Mod(1,2)),x^2+x+T)) %1 = [a1,a2+a1^2*T,a3,a4,a6+a3^2*T]
Twist of an elliptic curve defined over a finite field:
? E=ellinit([1,7]*Mod(1,19));lift(elltwist(E)) %1 = [0,0,0,11,12]
The library syntax is GEN
elltwist(GEN E, GEN P = NULL)
.
(E, P, Q, m)
Computes the Weil pairing of the two points of m
-torsion P
and Q
on the elliptic curve E
.
The library syntax is GEN
ellweilpairing(GEN E, GEN P, GEN Q, GEN m)
.
(w,{z = 'x},{
flag = 0})
Computes the value at z
of the Weierstrass wp function attached to
the lattice w
as given by ellperiods
. It is also possible to
directly input w = [
omega_1,
omega_2]
, or an elliptic curve E
as given
by ellinit
(w = E.omega
).
? w = ellperiods([1,I]); ? ellwp(w, 1/2) %2 = 6.8751858180203728274900957798105571978 ? E = ellinit([1,1]); ? ellwp(E, 1/2) %4 = 3.9413112427016474646048282462709151389
@3One can also compute the series expansion around z = 0
:
? E = ellinit([1,0]); ? ellwp(E) \\ 'x implicitly at default seriesprecision %5 = x^-2 - 1/5*x^2 + 1/75*x^6 - 2/4875*x^10 + O(x^14) ? ellwp(E, x + O(x^12)) \\ explicit precision %6 = x^-2 - 1/5*x^2 + 1/75*x^6 + O(x^9)
Optional flag means 0 (default): compute only wp(z)
, 1: compute
[
wp(z),
wp'(z)]
.
The library syntax is GEN
ellwp0(GEN w, GEN z = NULL, long flag, long prec)
.
For flag = 0
, we also have
GEN
ellwp(GEN w, GEN z, long prec)
, and
GEN
ellwpseries(GEN E, long v, long precdl)
for the power series in
variable v
.
(E,n,{v = 'x})
In standard notation, for any affine point P = (v,w)
on the
curve E
, we have
[n]P = (
phi_n(P)
psi_n(P) :
omega_n(P) :
psi_n(P)^3)
for some polynomials phi_n,
omega_n,
psi_n
in
Z[a_1,a_2,a_3,a_4,a_6][v,w]
. This function returns
[
phi_n(P),
psi_n(P)^2]
, which give the numerator and denominator of
the abcissa of [n]P
and depend only on v
.
The library syntax is GEN
ellxn(GEN E, long n, long v = -1)
where v
is a variable number.
(w,{z = 'x})
Computes the value at z
of the Weierstrass zeta function attached to
the lattice w
as given by ellperiods
(,1)
: including quasi-periods
is useful, otherwise there are recomputed from scratch for each new z
.
zeta(z, L) = (1)/(z) + z^2
sum_{
omega\in L^*}
(1)/(
omega^2(z-
omega)).
It is also possible to directly input w = [
omega_1,
omega_2]
,
or an elliptic curve E
as given by ellinit
(w = E.omega
).
The quasi-periods of zeta, such that
zeta(z + a
omega_1 + b
omega_2) =
zeta(z) + a
eta_1 + b
eta_2
for integers a
and b
are obtained as eta_i = 2
zeta(
omega_i/2)
.
Or using directly elleta
.
? w = ellperiods([1,I],1); ? ellzeta(w, 1/2) %2 = 1.5707963267948966192313216916397514421 ? E = ellinit([1,0]); ? ellzeta(E, E.omega[1]/2) %4 = 0.84721308479397908660649912348219163647
@3One can also compute the series expansion around z = 0
(the quasi-periods are useless in this case):
? E = ellinit([0,1]); ? ellzeta(E) \\ at 'x, implicitly at default seriesprecision %4 = x^-1 + 1/35*x^5 - 1/7007*x^11 + O(x^15) ? ellzeta(E, x + O(x^20)) \\ explicit precision %5 = x^-1 + 1/35*x^5 - 1/7007*x^11 + 1/1440257*x^17 + O(x^18)
The library syntax is GEN
ellzeta(GEN w, GEN z = NULL, long prec)
.
(E,z)
E
being an ell as output by
ellinit
, computes the coordinates [x,y]
on the curve E
corresponding to the complex number z
. Hence this is the inverse function
of ellpointtoz
. In other words, if the curve is put in Weierstrass
form y^2 = 4x^3 - g_2x - g_3
, [x,y]
represents the Weierstrass
wp-function-function> and its derivative. More
precisely, we have
x =
wp(z) - b_2/12, y =
wp'(z) - (a_1 x + a_3)/2.
If z
is in the lattice defining E
over C, the result is the point at
infinity [0]
.
The library syntax is GEN
pointell(GEN E, GEN z, long prec)
.
(
PQ,{p})
Let PQ
be a polynomial P
, resp. a vector [P,Q]
of polynomials, with
rational coefficients.
Determines the reduction at p > 2
of the (proper, smooth) genus 2
curve C/
Q, defined by the hyperelliptic equation y^2 = P(x)
, resp.
y^2 + Q(x)*y = P(x)
.
(The special fiber X_p
of the minimal regular model X
of C
over Z.)
If p
is omitted, determines the reduction type for all (odd) prime
divisors of the discriminant.
@3This function was rewritten from an implementation of Liu's
algorithm by Cohen and Liu (1994), genus2reduction-0.3
, see
liu/G2R/>http://www.math.u-bordeaux.fr/~liu/G2R/.
@3CAVEAT. The function interface may change: for the
time being, it returns [N,
FaN, T, V]
where N
is either the local conductor at p
or the
global conductor, FaN is its factorization, y^2 = T
defines a
minimal model over Z[1/2]
and V
describes the reduction type at the
various considered p
. Unfortunately, the program is not complete for
p = 2
, and we may return the odd part of the conductor only: this is the
case if the factorization includes the (impossible) term 2^{-1}
; if the
factorization contains another power of 2
, then this is the exact local
conductor at 2
and N
is the global conductor.
? default(debuglevel, 1); ? genus2red(x^6 + 3*x^3 + 63, 3) (potential) stable reduction: [1, []] reduction at p: [III{9}] page 184, [3, 3], f = 10 %1 = [59049, Mat([3, 10]), x^6 + 3*x^3 + 63, [3, [1, []], ["[III{9}] page 184", [3, 3]]]] ? [N, FaN, T, V] = genus2red(x^3-x^2-1, x^2-x); \\ X_1(13), global reduction p = 13 (potential) stable reduction: [5, [Mod(0, 13), Mod(0, 13)]] reduction at p: [I{0}-II-0] page 159, [], f = 2 ? N %3 = 169 ? FaN %4 = Mat([13, 2]) \\ in particular, good reduction at 2 ! ? T %5 = x^6 + 58*x^5 + 1401*x^4 + 18038*x^3 + 130546*x^2 + 503516*x + 808561 ? V %6 = [[13, [5, [Mod(0, 13), Mod(0, 13)]], ["[I{0}-II-0] page 159", []]]]
We now first describe the format of the vector V = V_p
in the case where
p
was specified (local reduction at p
): it is a triple [p,
stable,
red]
. The component stable = [
type,
vecj]
contains
information about the stable reduction after a field extension;
depending on types, the stable reduction is
@3* 1: smooth (i.e. the curve has potentially good reduction). The
Jacobian J(C)
has potentially good reduction.
@3* 2: an elliptic curve E
with an ordinary double point; vecj
contains j
mod p
, the modular invariant of E
. The (potential)
semi-abelian reduction of J(C)
is the extension of an elliptic curve (with
modular invariant j
mod p
) by a torus.
@3* 3: a projective line with two ordinary double points. The Jacobian
J(C)
has potentially multiplicative reduction.
@3* 4: the union of two projective lines crossing transversally at three
points. The Jacobian J(C)
has potentially multiplicative reduction.
@3* 5: the union of two elliptic curves E_1
and E_2
intersecting
transversally at one point; vecj contains their modular invariants
j_1
and j_2
, which may live in a quadratic extension of F_p
and need
not be distinct. The Jacobian J(C)
has potentially good reduction,
isomorphic to the product of the reductions of E_1
and E_2
.
@3* 6: the union of an elliptic curve E
and a projective line which has
an ordinary double point, and these two components intersect transversally
at one point; vecj contains j
mod p
, the modular invariant of E
.
The (potential) semi-abelian reduction of J(C)
is the extension of an
elliptic curve (with modular invariant j
mod p
) by a torus.
@3* 7: as in type 6, but the two components are both singular. The
Jacobian J(C)
has potentially multiplicative reduction.
The component red = [
NUtype,
neron]
contains two data
concerning the reduction at p
without any ramified field extension.
The NUtype is a t_STR
describing the reduction at p
of C
,
following Namikawa-Ueno, The complete classification of fibers in
pencils of curves of genus two, Manuscripta Math., vol. 9, (1973), pages
143-186. The reduction symbol is followed by the corresponding page number
or page range in this article.
The second datum neron is the group of connected components (over an
algebraic closure of F_p
) of the Néron model of J(C)
, given as a
finite abelian group (vector of elementary divisors).
If p = 2
, the red component may be omitted altogether (and
replaced by []
, in the case where the program could not compute it.
When p
was not specified, V
is the vector of all V_p
, for all
considered p
.
@3Notes about Namikawa-Ueno types.
@3* A lower index is denoted between braces: for instance,
[I{2}-II-5]
means [I_2-II-5]
.
@3* If K
and K'
are Kodaira symbols for singular fibers of elliptic
curves, then [K-K'-m]
and [K'-K-m]
are the same.
We define a total ordering on Kodaira symbol by fixing I < I* <
II < II*,...
. If the reduction type is the same, we order by
the number of components, e.g. I_2 < I_4
, etc.
Then we normalize our output so that K <= K'
.
@3* [K-K'--1]
is [K-K'-
alpha]
in the notation of
Namikawa-Ueno.
@3* The figure [2I_0-m]
in Namikawa-Ueno, page 159, must be denoted
by [2I_0-(m+1)]
.
The library syntax is GEN
genus2red(GEN PQ, GEN p = NULL)
.
(X)
X
being a non-singular hyperelliptic curve defined over a finite field,
return the characteristic polynomial of the Frobenius automorphism.
X
can be given either by a squarefree polynomial P
such that
X: y^2 = P(x)
or by a vector [P,Q]
such that
X: y^2 + Q(x) y = P(x)
and Q^2+4 P
is squarefree.
The library syntax is GEN
hyperellcharpoly(GEN X)
.
(Q,p,n)
Let X
be the curve defined by y^2 = Q(x)
, where Q
is a polynomial of
degree d
over Q and p >= d
a prime such that X
has good reduction
at p
return the matrix of the Frobenius endomorphism varphi on the
crystalline module D_p(X) =
Q_p \otimes H^1_{dR}(X/
Q)
with respect to the
basis of the given model (
omega, x
omega,...,x^{g-1}
omega)
, where
omega = dx/(2 y)
is the invariant differential, where g
is the genus of
X
(either d = 2 g+1
or d = 2 g+2
). The characteristic polynomial of
varphi is the numerator of the zeta-function of the reduction of the curve
X
modulo p
. The matrix is computed to absolute p
-adic precision p^n
.
The library syntax is GEN
hyperellpadicfrobenius(GEN Q, ulong p, long n)
.
L
-functionsThis section describes routines related to L
-functions. We first introduce
the basic concept and notations, then explain how to represent them in GP.
Let Gamma_
R(s) =
pi^{-s/2}
Gamma(s/2)
, where Gamma is Euler's gamma
function. Given d >= 1
and a d
-tuple A = [
alpha_1,...,
alpha_d]
of
complex numbers, we let gamma_A(s) =
prod_{
alpha \in A}
Gamma_
R(s +
alpha)
.
Given a sequence a = (a_n)_{n >= 1}
of complex numbers (such that a_1 = 1
),
a positive conductor N \in
Z, and a gamma factor
gamma_A
as above, we consider the Dirichlet series
L(a,s) =
sum_{n >= 1} a_n n^{-s}
and the attached completed function
Lambda(a,s) = N^{s/2}
gamma_A(s).L(a,s).
Such a datum defines an L
-function if it satisfies the three
following assumptions:
@3* [Convergence] The a_n = O_
epsilon(n^{k_1+
epsilon})
have polynomial
growth, equivalently L(s)
converges absolutely in some right half-plane
Re (s) > k_1 + 1
.
@3* [Analytic continuation] L(s)
has a meromorphic continuation to the
whole complex plane with finitely many poles.
@3* [Functional equation] There exist an integer k
, a complex number
epsilon (usually of modulus 1
), and an attached sequence a^*
defining both an L
-function L(a^*,s)
satisfying the above two assumptions
and a completed function Lambda(a^*,s) = N^{s/2}
gamma_A(s).
L(a^*,s)
, such that
Lambda(a,k-s) =
epsilon Lambda(a^*,s)
for all regular points.
More often than not in number theory we have a^ *= \overline{a}
(which
forces |
epsilon |= 1
), but this needs not be the case. If a
is a real
sequence and a = a^*
, we say that L
is self-dual. We do not assume
that the a_n
are multiplicative, nor equivalently that L(s)
has an Euler
product.
@3Remark.
Of course, a
determines the L
-function, but the (redundant) datum a,a^*,
A, N, k,
epsilon describes the situation in a form more suitable for fast
computations; knowing the polar part r
of Lambda(s)
(a rational function
such that Lambda-r
is holomorphic) is also useful. A subset of these,
including only finitely many a_n
-values will still completely determine L
(in suitable families), and we provide routines to try and compute missing
invariants from whatever information is available.
@3Important Caveat.
We currently assume that we can take the growth exponent k_1 = (k-1)/2
if
L
is entire and k_1 = k-1
otherwise, and that the implied constants in
the O_
epsilon are small. This may be changed and made user-configurable
in future versions but the essential point remains that it is impossible to
return proven results in such a generic framework, without more detailed
information about the L
function. The intended use of the L
-function
package is not to prove theorems, but to experiment and formulate
conjectures, so all numerical results should be taken with a grain of salt.
One can always increase realbitprecision
and recompute: the difference
estimates the actual absolute error in the original output.
@3Note. The requested precision has a major impact on runtimes.
Because of this, most L
-function routines, in particular lfun
itself,
specify the requested precision in bits, not in decimal digits.
This is transparent for the user once realprecision
or
realbitprecision
are set. We advise to manipulate precision via
realbitprecision
as it allows finer granularity: realprecision
increases by increments of 64 bits, i.e. 19 decimal digits at a time.
Given an L
-function as above, we define an attached theta function
via Mellin inversion: for any positive real t > 0
, we let
theta(a,t) := (1)/(2
pi i)
int_{
Re (s) = c} t^{-s}
Lambda(s) ds
where c
is any positive real number c > k_1+1
such that c +
Re (a) > 0
for all a\in A
. In fact, we have
theta(a,t) =
sum_{n >= 1} a_n K(nt/N^{1/2})
where K(t) := (1)/(2
pi i)
int_{
Re (s) = c} t^{-s}
gamma_A(s) ds.
Note that this function is analytic and actually makes sense for complex t
,
such that Re (t^{2/d}) > 0
, i.e. in a cone containing the positive real
half-line. The functional equation for Lambda translates into
theta(a,1/t) -
epsilon t^k
theta(a^*,t) = P_
Lambda(t),
where P_
Lambda is an explicit polynomial in t
and log t
given by the
Taylor development of the polar part of Lambda: there are no log 's if
all poles are simple, and P = 0
if Lambda is entire. The values
theta(t)
are generally easier to compute than the L(s)
, and this
functional equation provides a fast way to guess possible values for
missing invariants in the L
-function definition.
L
and theta functionsWe have 3 levels of description:
@3* an Lmath
is an arbitrary description of the underlying
mathematical situation (to which e.g., we associate the a_p
as traces of
Frobenius elements); this is done via constructors to be described in the
subsections below.
@3* an Ldata
is a computational description of situation, containing
the complete datum (a,a^*,A,k,N,
epsilon,r
). Where a
and a^*
describe
the coefficients (given n,b
we must be able to compute [a_1,...,a_n]
with bit accuracy b
), A
describes the Euler factor, the (classical) weight
is k
, N
is the conductor, and r
describes the polar part of L(s)
.
This is obtained via the function lfuncreate
. N.B. For motivic
L
-functions, the motivic weight w
is w = k-1
; but we also support
non-motivic L
-functions.
@3Design problem. All components of an Ldata
should be given
exactly since the accuracy to which they must be computed is not bounded a
priori; but this is not always possible, in particular for epsilon and r
.
@3* an Linit
contains an Ldata
and everything needed for fast
numerical computations. It specifies the functions to be considered
(either L^{(j)}(s)
or theta^{(j)}(t)
for derivatives of order j <=
m
, where m
is now fixed) and specifies a domain which limits
the range of arguments (t
or s
, respectively to certain cones and
rectangular regions) and the output accuracy. This is obtained via the
functions lfuninit
or lfunthetainit
.
All the functions which are specific to L
or theta functions share the
prefix lfun
. They take as first argument either an Lmath
, an
Ldata
, or an Linit
. If a single value is to be computed,
this makes no difference, but when many values are needed (e.g. for plots or
when searching for zeros), one should first construct an Linit
attached to the search range and use it in all subsequent calls.
If you attempt to use an Linit
outside the range for which it was
initialized, a warning is issued, because the initialization is
performed again, a major inefficiency:
? Z = lfuncreate(1); \\ Riemann zeta ? L = lfuninit(Z, [1/2, 0, 100]); \\ zeta(1/2+it), |t| < 100 ? lfun(L, 1/2) \\ OK, within domain %3 = -1.4603545088095868128894991525152980125 ? lfun(L, 0) \\ not on critical strip ! *** lfun: Warning: lfuninit: insufficient initialization. %4 = -0.50000000000000000000000000000000000000 ? lfun(L, 1/2, 1) \\ attempt first derivative ! *** lfun: Warning: lfuninit: insufficient initialization. %5 = -3.9226461392091517274715314467145995137
For many L
-functions, passing from Lmath
to an Ldata
is
inexpensive: in that case one may use lfuninit
directly from the
Lmath
even when evaluations in different domains are needed. The
above example could equally have skipped the lfuncreate
:
? L = lfuninit(1, [1/2, 0, 100]); \\ zeta(1/2+it), |t| < 100
@3In fact, when computing a single value, you can even skip
lfuninit
:
? L = lfun(1, 1/2, 1); \\ zeta'(1/2) ? L = lfun(1, 1+x+O(x^5)); \\ first 5 terms of Taylor development at 1
@3Both give the desired results with no warning.
@3Complexity.
The implementation requires O(N(|t|+1))^{1/2}
coefficients a_n
to evaluate L
of conductor N
at s =
sigma + i t
.
We now describe the available high-level constructors, for built-in L
functions.
L
-functionsGiven a Dirichlet character chi:(
Z/N
Z)^*\to
C, we let
L(
chi, s) =
sum_{n >= 1}
chi(n) n^{-s}.
Only primitive characters are supported. Given a fundamental discriminant
D
, the function L((D/.), s)
, for the quadratic Kronecker symbol, is encoded
by the t_INT
D
. This includes Riemann zeta function via the special
case D = 1
.
More general characters can be represented in a variety of ways:
@3* via Conrey notation (see znconreychar
): chi_N(m,.)
is given as the t_INTMOD
Mod(m,N)
.
@3* via a bid structure describing the abelian group (
Z/N
Z)^*
,
where the character is given in terms of the bid generators:
? bid = idealstar(,100,2); \\ (Z/100Z)^* ? bid.cyc \\ ~ Z/20 . g1 + Z/2 . g2 for some generators g1 and g2 %2 = [20, 2] ? bid.gen %3 = [77, 51] ? chi = [a, b] \\ maps g1 to e(a/20) and g2 to e(b/2); e(x) = exp(2ipi x)
More generally, let (
Z/N
Z)^ *= \oplus (
Z/d_i
Z) g_i
be given via a
bid structure G
(G.cyc
gives the d_i
and G.gen
the
g_i
). A character chi on G
is given by a row vector
v = [a_1,...,a_n]
such that chi(
prod g_i^{n_i}) =
exp (2
pi i
sum a_i
n_i / d_i)
. The pair [
bid, v]
encodes the primitive character
attached to chi.
@3* in fact, this construction [
bid, m]
describing a character
is more general: m
is also allowed to be a Conrey index as seen above,
or a Conrey logarithm (see znconreylog
), and the latter format is
actually the fastest one.
@3* it is also possible to view Dirichlet characters as Hecke characters
over K =
Q (see below), for a modulus [N, [1]]
but this is both more
complicated and less efficient.
L
-functionsThe Dedekind zeta function of a number field K =
Q[X]/(T)
is encoded
either by the defining polynomial T
, or any absolute number fields
structure (preferably at least a bnf).
Given a finite order Hecke character chi: Cl_f(K)\to
C, we let
L(
chi, s) =
sum_{A \subset O_K}
chi(A) (N_{K/
Q}A)^{-s}.
Let Cl_f(K) = \oplus (
Z/d_i
Z) g_i
given by a bnr structure with
generators: the d_i
are given by K.cyc
and the g_i
by K.gen
.
A character chi on the ray class group is given by a row vector
v = [a_1,...,a_n]
such that chi(
prod g_i^{n_i}) =
exp (2
pi i
sum
a_i n_i / d_i)
. The pair [
bnr, v]
encodes the primitive
character attached to chi.
? K = bnfinit(x^2-60); ? Cf = bnrinit(K, [7, [1,1]], 1); \\ f = 7 oo_1 oo_2 ? Cf.cyc %3 = [6, 2, 2] ? Cf.gen %4 = [[2, 1; 0, 1], [22, 9; 0, 1], [-6, 7]~] ? lfuncreate([Cf, [1,0,0]]); \\ F<chi>(g_1) = F<zeta>_6, F<chi>(g_2) = F<chi>(g_3) = 1
@3Dirichlet characters on (
Z/N
Z)^*
are a special case,
where K =
Q:
? Q = bnfinit(x); ? Cf = bnrinit(Q, [100, [1]]); \\ for odd characters on (Z/100Z)*
For even characters, replace by bnrinit(K, N)
. Note that the simpler
direct construction in the previous section will be more efficient.
L
functionsGiven a Galois number field N/
Q with group G = galoisinit(N)
,
a representation rho of G
over the cyclotomic field Q(
zeta_n)
is specified by the matrices giving the images of G.gen
by rho.
The corresponding Artin L
function is created using lfunartin
.
P = quadhilbert(-47); \\ degree 5, Galois group D_5 N = nfinit(nfsplitting(P)); \\ Galois closure G = galoisinit(N); [s,t] = G.gen; \\ order 5 and 2 L = lfunartin(N,G, [[a,0;0,a^-1],[0,1;1,0]], 5); \\ irr. degree 2
@3In the above, the polynomial variable (here a
) represents
zeta_5 :=
exp (2i
pi/5)
and the two matrices give the images of
s
and t
. Here, priority of a
must be lower than the priority
of x
.
L
-functions of algebraic varietiesL
-function of elliptic curves over number fields are supported.
? E = ellinit([1,1]); ? L = lfuncreate(E); \\ L-function of E/Q ? E2 = ellinit([1,a], nfinit(a^2-2)); ? L2 = lfuncreate(E2); \\ L-function of E/Q(sqrt(2))
L
-function of hyperelliptic genus-2
curve can be created with
lfungenus2
. To create the L
function of the curve
y^2+(x^3+x^2+1)y = x^2+x
:
? L = lfungenus2([x^2+x, x^3+x^2+1]);
Currently, the model needs to be minimal at 2
, and if the conductor is even,
its valuation at 2
might be incorrect (a warning is issued).
An eta quotient is created by applying lfunetaquo
to a matrix with
2 columns [m, r_m]
representing
f(
tau) :=
prod_m
eta(m
tau)^{r_m}.
It is currently assumed that f
is a self-dual cuspidal form on
Gamma_0(N)
for some N
.
For instance, the L
-function sum tau(n) n^{-s}
attached to Ramanujan's Delta function is encoded as follows
? L = lfunetaquo(Mat([1,24])); ? lfunan(L, 100) \\ first 100 values of tau(n)
More general modular forms defined by modular symbols will be added later.
When no direct constructor is available, you can still input an L
function
directly by supplying [a, a^*,A, k, N,
epsilon, r]
to lfuncreate
(see ??lfuncreate
for details).
It is strongly suggested to first check consistency of the created
L
-function:
? L = lfuncreate([a, as, A, k, N, eps, r]); ? lfuncheckfeq(L) \\ check functional equation
(L,s,{D = 0})
Compute the L-function value L(s)
, or if D
is set, the
derivative of order D
at s
. The parameter
L
is either an Lmath, an Ldata (created by lfuncreate
, or an
Linit (created by lfuninit
), preferrably the latter if many values
are to be computed.
The argument s
is also allowed to be a power series; for instance, if s =
alpha + x + O(x^n)
, the function returns the Taylor expansion of order n
around alpha. The result is given with absolute error less than 2^{-B}
,
where B = realbitprecision
.
@3Caveat. The requested precision has a major impact on runtimes.
It is advised to manipulate precision via realbitprecision
as
explained above instead of realprecision
as the latter allows less
granularity: realprecision
increases by increments of 64 bits, i.e. 19
decimal digits at a time.
? lfun(x^2+1, 2) \\ Lmath: Dedekind zeta for Q(i) at 2 %1 = 1.5067030099229850308865650481820713960
? L = lfuncreate(ellinit("5077a1")); \\ Ldata: Hasse-Weil zeta function ? lfun(L, 1+x+O(x^4)) \\ zero of order 3 at the central point %3 = 0.E-58 - 5.[...] E-40*x + 9.[...] E-40*x^2 + 1.7318[...]*x^3 + O(x^4)
\\ Linit: zeta(1/2+it), |t| < 100, and derivative ? L = lfuninit(1, [100], 1); ? T = lfunzeros(L, [1,25]); %5 = [14.134725[...], 21.022039[...]] ? z = 1/2 + I*T[1]; ? abs( lfun(L, z) ) %7 = 8.7066865533412207420780392991125136196 E-39 ? abs( lfun(L, z, 1) ) %8 = 0.79316043335650611601389756527435211412 \\ simple zero
The library syntax is GEN
lfun0(GEN L, GEN s, long D, long bitprec)
.
lfunabelianrelinit(
bnfL,
bnfK,
polrel,
sdom,{
der = 0})
Returns the Linit
structure attached to the Dedekind zeta function
of the number field L
(see lfuninit
), given a subfield K
such that
L/K
is abelian.
Here polrel
defines L
over K
, as usual with the priority of the
variable of bnfK
lower than that of polrel
.
sdom
and der
are as in lfuninit
.
? D = -47; K = bnfinit(y^2-D); ? rel = quadhilbert(D); T = rnfequation(K.pol, rel); \\ degree 10 ? L = lfunabelianrelinit(T,K,rel, [2,0,0]); \\ at 2 time = 84 ms. ? lfun(L, 2) %4 = 1.0154213394402443929880666894468182650 ? lfun(T, 2) \\ using parisize > 300MB time = 652 ms. %5 = 1.0154213394402443929880666894468182656
@3As the example shows, using the (abelian) relative structure is more efficient than a direct computation. The difference becomes drastic as the absolute degree increases while the subfield degree remains constant.
The library syntax is GEN
lfunabelianrelinit(GEN bnfL, GEN bnfK, GEN polrel, GEN sdom, long der, long bitprec)
.
(L,n)
Compute the first n
terms of the Dirichlet series attached to the
L
-function given by L
(Lmath
, Ldata
or Linit
).
? lfunan(1, 10) \\ Riemann zeta %1 = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] ? lfunan(5, 10) \\ Dirichlet L-function for kronecker(5,.) %2 = [1, -1, -1, 1, 0, 1, -1, -1, 1, 0]
The library syntax is GEN
lfunan(GEN L, long n, long prec)
.
(
nf,
gal,M,n)
Returns the Ldata
structure attached to the
Artin L
-function attached to the representation rho of the Galois group
of the extension K/
Q, defined over the cyclotomic field Q(
zeta_n)
,
where nf is the nfinit structure attached to K
,
gal is the galoisinit structure attached to K/
Q, and M
is
the vector of the image of the generators gal.gen
by rho.
The elements of M
are matrices with polynomial entries, whose variable
is understood as the complex number exp (2 i
pi/n)
.
In the following example we build the Artin L
-functions attached to the two
irreducible degree 2
representations of the dihedral group D_{10}
defined
over Q(
zeta_5)
, for the extension H/
Q where H
is the Hilbert class
field of Q(
sqrt {-47})
.
We show numerically some identities involving Dedekind zeta functions and
Hecke L
series.
? P = quadhilbert(-47); ? N = nfinit(nfsplitting(P)); ? G = galoisinit(N); ? L1 = lfunartin(N,G, [[a,0;0,a^-1],[0,1;1,0]], 5); ? L2 = lfunartin(N,G, [[a^2,0;0,a^-2],[0,1;1,0]], 5); ? s = 1 + x + O(x^4); ? lfun(1,s)*lfun(-47,s)*lfun(L1,s)^2*lfun(L2,s)^2 - lfun(N,s) %6 ~ 0 ? lfun(1,s)*lfun(L1,s)*lfun(L2,s) - lfun(P,s) %7 ~ 0 ? bnr = bnrinit(bnfinit(x^2+47),1,1); ? lfun([bnr,[1]], s) - lfun(L1, s) %9 ~ 0 ? lfun([bnr,[1]], s) - lfun(L1, s) %10 ~ 0
The first identity is the factorisation of the regular representation of
D_{10}
, the second the factorisation of the natural representation of
D_{10}\subset S_5
, the next two are the expressions of the degree 2
representations as induced from degree 1
representations.
The library syntax is GEN
lfunartin(GEN nf, GEN gal, GEN M, long n)
.
(L,{t})
Given the data attached to an L
-function (Lmath
, Ldata
or Linit
), check whether the functional equation is satisfied.
This is most useful for an Ldata
constructed ``by hand'', via
lfuncreate
, to detect mistakes.
If the function has poles, the polar part must be specified. The routine
returns a bit accuracy b
such that |w - ^{w}| < 2^{b}
, where w
is
the root number contained in data
, and ^{w}
is a computed value
derived from \overline{
theta}(t)
and theta(1/t)
at some t != 0
and
the assumed functional equation. Of course, a large negative value of the
order of realbitprecision
is expected.
If t
is given, it should be close to the unit disc for efficiency and
such that \overline{
theta}(t) != 0
. We then check the functional
equation at that t
.
? \pb 128 \\ 128 bits of accuracy ? default(realbitprecision) %1 = 128 ? L = lfuncreate(1); \\ Riemann zeta ? lfuncheckfeq(L) %3 = -124
@3i.e. the given data is consistent to within 4 bits for the
particular check consisting of estimating the root number from all other
given quantities. Checking away from the unit disc will either fail with
a precision error, or give disappointing results (if theta(1/t)
is
large it will be computed with a large absolute error)
? lfuncheckfeq(L, 2+I) %4 = -115 ? lfuncheckfeq(L,10) *** at top-level: lfuncheckfeq(L,10) *** ^------------------ *** lfuncheckfeq: precision too low in lfuncheckfeq.
The library syntax is long
lfuncheckfeq(GEN L, GEN t = NULL, long bitprec)
.
lfunconductor(L,{
ab = [1,{10000}]},{
flag = 0})
Compute the conductor of the given L
-function
(if the structure contains a conductor, it is ignored);
ab = [a,b]
is the interval where we expect to find the conductor;
it may be given as a single scalar b
, in which case we look in [1,b]
.
Increasing ab
slows down the program but gives better accuracy for the
result.
If flag
is 0
(default), give either the conductor found as an
integer, or a vector (possibly empty) of conductors found. If flag
is
1
, same but give the computed floating point approximations to the
conductors found, without rounding to integers. It flag
is 2
, give
all the conductors found, even those far from integers.
@3Caveat. This is a heuristic program and the result is not proven in any way:
? L = lfuncreate(857); \\ Dirichlet L function for kronecker(857,.) ? \p19 realprecision = 19 significant digits ? lfunconductor(L) %2 = [17, 857] ? lfunconductor(L,,1) \\ don't round %3 = [16.99999999999999999, 857.0000000000000000]
? \p38 realprecision = 38 significant digits ? lfunconductor(L) %4 = 857
@3Note. This program should only be used if the primes dividing the
conductor are unknown, which is rare. If they are known, a direct
search through possible prime exponents using lfuncheckfeq
will
be more efficient and rigorous:
? E = ellinit([0,0,0,4,0]); /* Elliptic curve y^2 = x^3+4x */ ? E.disc \\ |disc E| = 2^12 %2 = -4096 \\ create Ldata by hand. Guess that root number is 1 and conductor N ? L(N) = lfuncreate([n->ellan(E,n), 0, [0,1], 1, N, 1]); ? fordiv(E.disc, d, print(d,": ",lfuncheckfeq(L(d)))) 1: 0 2: 0 4: -1 8: -2 16: -3 32: -127 64: -3 128: -2 256: -2 512: -1 1024: -1 2048: 0 4096: 0 ? lfunconductor(L(1)) \\ lfunconductor ignores conductor = 1 in Ldata ! %5 = 32
@3The above code assumed that root number was 1
;
had we set it to -1
, none of the lfuncheckfeq
values would have been
acceptable:
? L2(N) = lfuncreate([n->ellan(E,n), 0, [0,1], 1, N, -1]); ? [ lfuncheckfeq(L2(d)) | d<-divisors(E.disc) ] %7 = [0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, -1, -1]
The library syntax is GEN
lfunconductor(GEN L, GEN ab = NULL, long 10000], long bitprec)
.
lfuncost(L,{
sdom},{
der = 0})
Estimate the cost of running
lfuninit(L,sdom,der)
at current bit precision. Returns [t,b]
, to
indicate that t
coefficients a_n
will be computed, as well as t
values of
gammamellininv
, all at bit accuracy b
.
A subsequent call to lfun
at s
evaluates a polynomial of degree t
at exp (h s)
for some real parameter h
, at the same bit accuracy b
.
If L
is already an Linit
, then sdom and der are ignored
and are best left omitted; the bit accuracy is also inferred from L
: in
short we get an estimate of the cost of using that particular Linit
.
? \pb 128 ? lfuncost(1, [100]) \\ for zeta(1/2+I*t), |t| < 100 %1 = [7, 242] \\ 7 coefficients, 242 bits ? lfuncost(1, [1/2, 100]) \\ for zeta(s) in the critical strip, |Im s| < 100 %2 = [7, 246] \\ now 246 bits ? lfuncost(1, [100], 10) \\ for zeta(1/2+I*t), |t| < 100 %3 = [8, 263] \\ 10th derivative increases the cost by a small amount ? lfuncost(1, [10^5]) %3 = [158, 113438] \\ larger imaginary part: huge accuracy increase
? L = lfuncreate(polcyclo(5)); \\ Dedekind zeta for Q(zeta_5) ? lfuncost(L, [100]) \\ at s = 1/2+I*t), |t| < 100 %5 = [11457, 582] ? lfuncost(L, [200]) \\ twice higher %6 = [36294, 1035] ? lfuncost(L, [10^4]) \\ much higher: very costly ! %7 = [70256473, 45452] ? \pb 256 ? lfuncost(L, [100]); \\ doubling bit accuracy %8 = [17080, 710]
@3In fact, some L
functions can be factorized algebraically
by the lfuninit
call, e.g. the Dedekind zeta function of abelian
fields, leading to much faster evaluations than the above upper bounds.
In that case, the function returns a vector of costs as above for each
individual function in the product actually evaluated:
? L = lfuncreate(polcyclo(5)); \\ Dedekind zeta for Q(zeta_5) ? lfuncost(L, [100]) \\ a priori cost %2 = [11457, 582] ? L = lfuninit(L, [100]); \\ actually perform all initializations ? lfuncost(L) %4 = [[16, 242], [16, 242], [7, 242]]
@3The Dedekind function of this abelian quartic field
is the product of four Dirichlet L
-functions attached to the trivial
character, a non-trivial real character and two complex conjugate
characters. The non-trivial characters happen to have the same conductor
(hence same evaluation costs), and correspond to two evaluations only
since the two conjugate characters are evaluated simultaneously.
For a total of three L
-functions evaluations, which explains the three
components above. Note that the actual cost is much lower than the a priori
cost in this case.
The library syntax is GEN
lfuncost0(GEN L, GEN sdom = NULL, long der, long bitprec)
.
Also available is
GEN
lfuncost(GEN L, GEN dom, long der, long bitprec)
when L
is not an Linit
; the return value is a t_VECSMALL
in this case.
(
obj)
This low-level routine creates Ldata
structures, needed by
lfun functions, describing an L
-function and its functional equation.
You are urged to use a high-level constructor when one is available,
and this function accepts them, see ??lfun
:
? L = lfuncreate(1); \\ Riemann zeta ? L = lfuncreate(5); \\ Dirichlet L-function for quadratic character (5/.) ? L = lfuncreate(x^2+1); \\ Dedekind zeta for Q(i) ? L = lfuncreate(ellinit([0,1])); \\ L-function of E/Q: y^2=x^3+1
@3One can then use, e.g., Lfun(L,s)
to directly
evaluate the respective L
-functions at s
, or lfuninit(L, [c,w,h]
to initialize computations in the rectangular box Re (s-c) <= w
,
Im (s) <= h
.
We now describe the low-level interface, used to input non-builtin
L
-functions. The input is now a 6
or 7
component vector
V = [a,astar,Vga,k,N,eps,poles]
, whose components are as follows:
@3* V[1] = a
encodes the Dirichlet series coefficients. The
preferred format is a closure of arity 1: n- > vector(n,i,a(i))
giving
the vector of the first n
coefficients. The closure is allowed to return
a vector of more than n
coefficients (only the first n
will be
considered) or even less than n
, in which case loss of accuracy will occur
and a warning that #an
is less than expected is issued. This
allows to precompute and store a fixed large number of Dirichlet
coefficients in a vector v
and use the closure n- > v
, which
does not depend on n
. As a shorthand for this latter case, you can input
the vector v
itself instead of the closure.
A second format is limited to multiplicative L
functions affording an
Euler product. It is a closure of arity 2 (p,d)- > L(p)
giving the local
factor L_p
at p
as a rational function, to be evaluated at p^{-s}
as in
direuler
; d
is set to the floor of log _p(n)
, where n
is the
total number of Dirichlet coefficients (a_1,...,a_n)
that will be
computed in this way. This parameter d
allows to compute only part of L_p
when p
is large and L_p
expensive to compute, but it can of course be
ignored by the closure.
Finally one can describe separately the generic Dirichlet coefficients
and the bad local factors by setting dir = [an, [p_1,L^{-1}_{p_1}],
...,[p_k,L^{-1}_{p_k}]]
, where an
describes the generic coefficients
in one of the two formats above, except that coefficients a_n
with
p_i | n
for some i <= k
will be ignored. The subsequent pairs [p,
L_p^{-1}]
give the bad primes and corresponding inverse local
factors.
@3* V[2] = astar
is the Dirichlet series coefficients of the dual
function, encoded as a
above. The sentinel values 0
and 1
may
be used for the special cases where a = a^*
and a = \overline{a^*}
,
respectively.
@3* V[3] = Vga
is the vector of alpha_j
such that the gamma
factor of the L
-function is equal to
gamma_A(s) =
prod_{1 <= j <= d}
Gamma_{
R}(s+
alpha_j),
where Gamma_{
R}(s) =
pi^{-s/2}
Gamma(s/2)
.
This same syntax is used in the gammamellininv
functions.
In particular the length d
of Vga
is the degree of the L
-function.
In the present implementation, the alpha_j
are assumed to be exact
rational numbers. However when calling theta functions with complex
(as opposed to real) arguments, determination problems occur which may
give wrong results when the alpha_j
are not integral.
@3* V[4] = k
is a positive integer k
. The functional equation relates
values at s
and k-s
. For instance, for an Artin L
-series such as a
Dedekind zeta function we have k = 1
, for an elliptic curve k = 2
, and
for a modular form, k
is its weight. For motivic L
-functions, the
motivic weight w
is w = k-1
.
@3* V[5] = N
is the conductor, an integer N >= 1
, such that
Lambda(s) = N^{s/2}
gamma_A(s)L(s)
with gamma_A(s)
as above.
@3* V[6] = eps
is the root number varepsilon, i.e., the
complex number (usually of modulus 1
) such that
Lambda(a, k-s) =
varepsilon Lambda(a^*, s)
.
@3* The last optional component V[7] = poles
encodes the poles of the
L
or Lambda-functions, and is omitted if they have no poles.
A polar part is given by a list of 2
-component vectors
[
beta,P_{
beta}(x)]
, where
beta is a pole and the power series P_{
beta}(x)
describes
the attached polar part, such that L(s) - P_
beta(s-
beta)
is holomorphic
in a neighbourhood of beta. For instance P_
beta = r/x+O(1)
for a
simple pole at beta or r_1/x^2+r_2/x+O(1)
for a double pole.
The type of the list describing the polar part allows to distinguish between
L
and Lambda: a t_VEC
is attached to L
, and a t_COL
is attached to Lambda.
The latter is mandatory unless a = \overline{a^*}
(coded by astar
equal to 0
or 1
): otherwise, the poles of L^*
cannot be infered from
the poles of L
! (Whereas the functional equation allows to deduce
the polar part of Lambda^*
from the polar part of Lambda.)
The special coding poles = r
a complex scalar is available in this
case, to describe a L
function with at most a single simple pole at s =
k
and residue r
. (This is the usual situation, for instance for Dedekind
zeta functions.) This value r
can be set to 0
if unknown, and it will be
computed.
The library syntax is GEN
lfuncreate(GEN obj)
.
(
L1,
L2)
Creates the Ldata
structure (without initialization) corresponding
to the quotient of the Dirichlet series L_1
and L_2
given by
L1
and L2
. Assume that v_z(L_1) >= v_z(L_2)
at all
complex numbers z
: the construction may not create new poles, nor increase
the order of existing ones.
The library syntax is GEN
lfundiv(GEN L1, GEN L2, long bitprec)
.
(M)
Returns the Ldata
structure attached to the L
function
attached to the modular form
z:--->
prod_{i = 1}^n
eta(M_{i,1} z)^{M_{i,2}}
It is currently assumed that f
is a self-dual cuspidal form on
Gamma_0(N)
for some N
.
For instance, the L
-function sum tau(n) n^{-s}
attached to Ramanujan's Delta function is encoded as follows
? L = lfunetaquo(Mat([1,24])); ? lfunan(L, 100) \\ first 100 values of tau(n)
The library syntax is GEN
lfunetaquo(GEN M)
.
(F)
Returns the Ldata
structure attached to the L
function
attached to the genus-2 curve defined by y^2 = F(x)
or
y^2+Q(x) y = P(x)
if F = [P,Q]
.
Currently, the model needs to be minimal at 2, and if the conductor
is even, its valuation at 2
might be incorrect (a warning is issued).
The library syntax is GEN
lfungenus2(GEN F)
.
(L,t)
Variant of the Hardy Z
-function given by L
, used for
plotting or locating zeros of L(k/2+it)
on the critical line.
The precise definition is as
follows: if as usual k/2
is the center of the critical strip, d
is the
degree, alpha_j
the entries of Vga
giving the gamma factors,
and varepsilon the root number, then if we set
s = k/2+it =
rho e^{i
theta}
and
E = (d(k/2-1)+
sum_{1 <= j <= d}
alpha_j)/2
, the computed function at t
is
equal to
Z(t) =
varepsilon^{-1/2}
Lambda(s).|s|^{-E}e^{dt
theta/2} ,
which is a real function of t
for self-dual Lambda,
vanishing exactly when L(k/2+it)
does on the critical line. The
normalizing factor |s|^{-E}e^{dt
theta/2}
compensates the
exponential decrease of gamma_A(s)
as t\to oo
so that
Z(t) ~ 1
.
? T = 100; \\ maximal height ? L = lfuninit(1, [T]); \\ initialize for zeta(1/2+it), |t|<T ? \p19 \\ no need for large accuracy ? ploth(t = 0, T, lfunhardy(L,t))
@3Using lfuninit
is critical for this particular
applications since thousands of values are computed. Make sure to initialize
up to the maximal t
needed: otherwise expect to see many warnings for
unsufficient initialization and suffer major slowdowns.
The library syntax is GEN
lfunhardy(GEN L, GEN t, long bitprec)
.
lfuninit(L,
sdom,{
der = 0})
Initalization function for all functions linked to the
computation of the L
-function L(s)
encoded by L
, where
s
belongs to the rectangular domain sdom = [
center,w,h]
centered on the real axis, |
Re (s)-
center| <= w
, |
Im (s)| <= h
,
where all three components of sdom
are real and w
, h
are
non-negative. der
is the maximum order of derivation that will be used.
The subdomain [k/2, 0, h]
on the critical line (up to height h
)
can be encoded as [h]
for brevity. The subdomain [k/2, w, h]
centered on the critical line can be encoded as [w, h]
for brevity.
The argument L
is an Lmath
, an Ldata
or an Linit
. See
??Ldata
and ??lfuncreate
for how to create it.
The height h
of the domain is a crucial parameter: if you only
need L(s)
for real s
, set h
to 0.
The running time is roughly proportional to
(B / d+
pi h/4)^{d/2+3}N^{1/2},
where B
is the default bit accuracy, d
is the degree of the
L
-function, and N
is the conductor (the exponent d/2+3
is reduced
to d/2+2
when d = 1
and d = 2
). There is also a dependency on w
,
which is less crucial, but make sure to use the smallest rectangular
domain that you need.
? L0 = lfuncreate(1); \\ Riemann zeta ? L = lfuninit(L0, [1/2, 0, 100]); \\ for zeta(1/2+it), |t| < 100 ? lfun(L, 1/2 + I) ? L = lfuninit(L0, [100]); \\ same as above !
The library syntax is GEN
lfuninit0(GEN L, GEN sdom, long der, long bitprec)
.
(L,s,{D = 0})
Compute the completed L
-function Lambda(s) = N^{s/2}
gamma(s)L(s)
,
or if D
is set, the derivative of order D
at s
.
The parameter L
is either an Lmath
, an Ldata
(created by
lfuncreate
, or an Linit
(created by lfuninit
), preferrably the
latter if many values are to be computed.
The result is given with absolute error less than 2^{-B}|
gamma(s)N^{s/2}|
,
where B = realbitprecision
.
The library syntax is GEN
lfunlambda0(GEN L, GEN s, long D, long bitprec)
.
(L)
Returns [valeven,valodd,omminus,omplus]
,
where valeven
(resp., valodd
) is the vector of even (resp., odd)
periods of the modular form given by L
, and omminus
and
omplus
the corresponding real numbers omega^-
and omega^+
normalized in a noncanonical way. For the moment, only for modular forms of even weight.
The library syntax is GEN
lfunmfspec(GEN L, long bitprec)
.
(
L1,
L2)
Creates the Ldata
structure (without initialization) corresponding
to the product of the Dirichlet series given by L1
and
L2
.
The library syntax is GEN
lfunmul(GEN L1, GEN L2, long bitprec)
.
(L, {m = -1})
Computes the order of the possible zero of the L
-function at the
center k/2
of the critical strip; return 0
if L(k/2)
does not vanish.
If m
is given and has a non-negative value, assumes the order is at most m
.
Otherwise, the algorithm chooses a sensible default:
@3* if the L
argument is an Linit
, assume that a multiple zero at
s = k / 2
has order less than or equal to the maximal allowed derivation
order.
@3* else assume the order is less than 4
.
You may explicitly increase this value using optional argument m
; this
overrides the default value above. (Possibly forcing a recomputation
of the Linit
.)
The library syntax is long
lfunorderzero(GEN L, long m, long bitprec)
.
(Q)
Returns the Ldata
structure attached to the Theta function
of the lattice attached to the definite positive quadratic form Q
.
? L = lfunqf(matid(2)); ? lfunqf(L,2) %2 = 6.0268120396919401235462601927282855839 ? lfun(x^2+1,2)*4 %3 = 6.0268120396919401235462601927282855839
The library syntax is GEN
lfunqf(GEN Q, long prec)
.
(
data)
Given the Ldata
attached to an L
-function (or the output of
lfunthetainit
), compute the root number and the residues.
The output is a 3-component vector [r,R,w]
, where r
is the
residue of L(s)
at the unique pole, R
is the residue of Lambda(s)
,
and w
is the root number. In the present implementation,
@3* either the polar part must be completely known (and is then arbitrary): the function determines the root number,
? L = lfunmul(1,1); \\ zeta^2 ? [r,R,w] = lfunrootres(L); ? r \\ single pole at 1, double %3 = [[1, 1.[...]*x^-2 + 1.1544[...]*x^-1 + O(x^0)]] ? w %4 = 1 ? R \\ double pole at 0 and 1 %5 = [[1,[...]], [0,[...]]
@3* or at most a single pole is allowed: the function computes both
the root number and the residue (0
if no pole).
The library syntax is GEN
lfunrootres(GEN data, long bitprec)
.
(
data,t,{m = 0})
Compute the value of the m
-th derivative
at t
of the theta function attached to the L
-function given by data
.
data
can be either the standard L
-function data, or the output of
lfunthetainit
.
The theta function is defined by the formula
Theta(t) =
sum_{n >= 1}a(n)K(nt/
sqrt (N))
, where a(n)
are the coefficients
of the Dirichlet series, N
is the conductor, and K
is the inverse Mellin
transform of the gamma product defined by the Vga
component.
Its Mellin transform is equal to Lambda(s)-P(s)
, where Lambda(s)
is the completed L
-function and the rational function P(s)
its polar part.
In particular, if the L
-function is the L
-function of a modular form
f(
tau) =
sum_{n >= 0}a(n)q^n
with q =
exp (2
pi i
tau)
, we have
Theta(t) = 2(f(it/
sqrt {N})-a(0))
. Note that an easy theorem on modular
forms implies that a(0)
can be recovered by the formula a(0) = -L(f,0)
.
The library syntax is GEN
lfuntheta(GEN data, GEN t, long m, long bitprec)
.
lfunthetacost(L,{
tdom},{m = 0})
This function estimates the cost of running
lfunthetainit(L,tdom,m)
at current bit precision. Returns the number of
coefficients a_n
that would be computed. This also estimates the
cost of a subsequent evaluation lfuntheta
, which must compute
that many values of gammamellininv
at the current bit precision.
If L
is already an Linit
, then tdom and m
are ignored
and are best left omitted: we get an estimate of the cost of using that
particular Linit
.
? \pb 1000 ? L = lfuncreate(1); \\ Riemann zeta ? lfunthetacost(L); \\ cost for theta(t), t real >= 1 %1 = 15 ? lfunthetacost(L, 1 + I); \\ cost for theta(1+I). Domain error ! *** at top-level: lfunthetacost(1,1+I) *** ^-------------------- *** lfunthetacost: domain error in lfunthetaneed: arg t > 0.785 ? lfunthetacost(L, 1 + I/2) \\ for theta(1+I/2). %2 = 23 ? lfunthetacost(L, 1 + I/2, 10) \\ for theta^((10))(1+I/2). %3 = 24 ? lfunthetacost(L, [2, 1/10]) \\ cost for theta(t), |t| >= 2, |arg(t)| < 1/10 %4 = 8
? L = lfuncreate( ellinit([1,1]) ); ? lfunthetacost(L) \\ for t >= 1 %6 = 2471
The library syntax is long
lfunthetacost0(GEN L, GEN tdom = NULL, long m, long bitprec)
.
lfunthetainit(L,{
tdom},{m = 0})
Initalization function for evaluating the m
-th derivative of theta
functions with argument t
in domain tdom. By default (tdom
omitted), t
is real, t >= 1
. Otherwise, tdom may be
@3* a positive real scalar rho: t
is real, t >=
rho.
@3* a non-real complex number: compute at this particular t
; this
allows to compute theta(z)
for any complex z
satisfying |z| >= |t|
and |
arg z| <= |
arg t|
; we must have |2
arg z / d| <
pi/2
, where
d
is the degree of the Gamma factor.
@3* a pair [
rho,
alpha]
: assume that |t| >=
rho and |
arg t| <=
alpha; we must have |2
alpha / d| <
pi/2
, where d
is the degree of
the Gamma factor.
? \p500 ? L = lfuncreate(1); \\ Riemann zeta ? t = 1+I/2; ? lfuntheta(L, t); \\ direct computation time = 30 ms. ? T = lfunthetainit(L, 1+I/2); time = 30 ms. ? lfuntheta(T, t); \\ instantaneous
@3The T
structure would allow to quickly compute theta(z)
for any z
in the cone delimited by t
as explained above. On the other hand
? lfuntheta(T,I) *** at top-level: lfuntheta(T,I) *** ^-------------- *** lfuntheta: domain error in lfunthetaneed: arg t > 0.785398163397448
The initialization is equivalent to
? lfunthetainit(L, [abs(t), arg(t)])
The library syntax is GEN
lfunthetainit(GEN L, GEN tdom = NULL, long m, long bitprec)
.
lfunzeros(L,
lim,{
divz = 8})
lim
being either a positive upper limit or a non-empty real
interval inside [0,+ oo [
, computes an
ordered list of zeros of L(s)
on the critical line up to the given
upper limit or in the given interval. Use a naive algorithm which may miss
some zeros: it assumes that two consecutive zeros at height T >= 1
differ at least by 2
pi/
omega, where
omega := divz.(d
log (T/2
pi) +d+ 2
log (N/(
pi/2)^d)).
To use a finer search mesh, set divz to some integral value larger than the default ( = 8).
? lfunzeros(1, 30) \\ zeros of Rieman zeta up to height 30 %1 = [14.134[...], 21.022[...], 25.010[...]] ? #lfunzeros(1, [100,110]) \\ count zeros with 100 <= Im(s) <= 110 %2 = 4
@3The algorithm also assumes that all zeros are simple except
possibly on the real axis at s = k/2
and that there are no poles in the
search interval. (The possible zero at s = k/2
is repeated according to
its multiplicity.)
Should you pass an Linit
argument to the function, beware that the
algorithm needs at least
L = lfuninit(Ldata, T+1)
@3where T
is the upper bound of the interval defined by
lim
: this allows to detect zeros near T
. Make sure that your
Linit
domain contains this one. The algorithm assumes
that a multiple zero at s = k / 2
has order less than or equal to
the maximal derivation order allowed by the Linit
. You may increase
that value in the Linit
but this is costly: only do it for zeros
of low height or in lfunorderzero
instead.
The library syntax is GEN
lfunzeros(GEN L, GEN lim, long divz, long bitprec)
.
Let Delta := Div^0(
P^1(
Q))
be the abelian group of divisors of
degree 0
on the rational projective line. The standard GL(2,
Q)
action on P^1(
Q)
via homographies naturally extends to Delta. Given
@3* G
a finite index subgroup of SL(2,
Z)
,
@3* a field F
and a finite dimensional representation V/F
of
GL(2,
Q)
,
@3we consider the space of modular symbols M :=
Hom _G(
Delta, V)
. This finite dimensional F
-vector
space is a G
-module, canonically isomorphic to H^1_c(X(G), V)
,
and allows to compute modular forms for G
.
Currently, we only support the groups Gamma_0(N)
(N > 1
an integer)
and the representations V_k =
Q[X,Y]_{k-2}
(k >= 2
an integer) over
Q. We represent a space of modular symbols by an ms structure,
created by the function msinit
. It encodes basic data attached to the
space: chosen Z[G]
-generators (g_i)
for Delta (and relations among
those) and an F
-basis of M
. A modular symbol s
is thus given either in
terms of this fixed basis, or as a collection of values s(g_i)
satisfying certain relations.
A subspace of M
(e.g. the cuspidal or Eisenstein subspaces, the new or
old modular symbols, etc.) is given by a structure allowing quick projection
and restriction of linear operators; its first component is a matrix whose
columns form an F
-basis of the subspace.
(M,Q,{H})
Let M
be a full modular symbol space of level N
,
as given by msinit
, let Q | N
, (Q,N/Q) = 1
,
and let H
be a subspace stable under the Atkin-Lehner involution w_Q
.
Return the matrix of w_Q
acting on H
(M
if omitted).
? M = msinit(36,2); \\ M_2(Gamma_0(36)) ? w = msatkinlehner(M,4); w^2 == 1 %2 = 1 ? #w \\ involution acts on a 13-dimensional space %3 = 13 ? M = msinit(36,2, -1); \\ M_2(Gamma_0(36))^- ? w = msatkinlehner(M,4); w^2 == 1 %5 = 1 ? #w %6 = 4
The library syntax is GEN
msatkinlehner(GEN M, long Q, GEN H = NULL)
.
(M, {
flag = 0})
M
being a full modular symbol space, as given by msinit
,
return its cuspidal part S
. If flag = 1
, return
[S,E]
its decomposition into cuspidal and Eisenstein parts.
A subspace is given by a structure allowing quick projection and restriction of linear operators; its first component is a matrix with integer coefficients whose columns form a Q-basis of the subspace.
? M = msinit(2,8, 1); \\ M_8(Gamma_0(2))^+ ? [S,E] = mscuspidal(M, 1); ? E[1] \\ 2-dimensional %3 = [0 -10]
[0 -15]
[0 -3]
[1 0]
? S[1] \\ 1-dimensional %4 = [ 3]
[30]
[ 6]
[-8]
The library syntax is GEN
mscuspidal(GEN M, long flag)
.
(M)
M
being a full modular symbol space, as given by msinit
,
return its Eisenstein subspace.
A subspace is given by a structure allowing quick projection and
restriction of linear operators; its first component is
a matrix with integer coefficients whose columns form a Q-basis of
the subspace.
This is the same basis as given by the second component of
mscuspidal
(M, 1)
.
? M = msinit(2,8, 1); \\ M_8(Gamma_0(2))^+ ? E = mseisenstein(M); ? E[1] \\ 2-dimensional %3 = [0 -10]
[0 -15]
[0 -3]
[1 0]
? E == mscuspidal(M,1)[2] %4 = 1
The library syntax is GEN
mseisenstein(GEN M)
.
(M,s,{p})
Let Delta := Div^0(
P^1 (
Q))
.
Let M
be a full modular symbol space, as given by msinit
,
let s
be a modular symbol from M
, i.e. an element
of Hom _G(
Delta, V)
, and let p = [a,b] \in
Delta be a path between
two elements in P^1(
Q)
, return s(p)\in V
. The path extremities a
and
b
may be given as t_INT
, t_FRAC
or oo = (1:0)
.
The symbol s
is either
@3* a t_COL
coding an element of a modular symbol subspace in terms of
the fixed basis of Hom _G(
Delta,V)
chosen in M
; if M
was
initialized with a non-zero sign (+
or -
), then either the
basis for the full symbol space or the +-
-part can be used (the dimension
being used to distinguish the two).
@3* a t_VEC
(v_i)
of elements of V
, where the v_i = s(g_i)
give
the image of the generators g_i
of Delta, see mspathgens
.
We assume that s
is a proper symbol, i.e. that the v_i
satisfy
the mspathgens
relations.
If p
is omitted, convert the symbol s
to the second form: a vector of
the s(g_i)
.
? M = msinit(2,8,1); \\ M_8(Gamma_0(2))^+ ? g = mspathgens(M)[1] %2 = [[+oo, 0], [0, 1]] ? N = msnew(M)[1]; #N \\ Q-basis of new subspace, dimension 1 %3 = 1 ? s = N[,1] \\ t_COL representation %4 = [-3, 6, -8]~ ? S = mseval(M, s) \\ t_VEC representation %5 = [64*x^6-272*x^4+136*x^2-8, 384*x^5+960*x^4+192*x^3-672*x^2-432*x-72] ? mseval(M,s, g[1]) %6 = 64*x^6 - 272*x^4 + 136*x^2 - 8 ? mseval(M,S, g[1]) %7 = 64*x^6 - 272*x^4 + 136*x^2 - 8
@3Note that the symbol should have values in
V =
Q[x,y]_{k-2}
, we return the de-homogenized values corresponding to y
= 1
instead.
The library syntax is GEN
mseval(GEN M, GEN s, GEN p = NULL)
.
(M, c)
Returns the modular symbol attached to the cusp
c
, where M
is a modular symbol space of level N
, attached to
G =
Gamma_0(N)
. The cusp c
in P^1(
Q)/G
can be given either as oo
( = (1:0)
), as a rational number a/b
( = (a:b)
). The attached symbol maps the path [b] - [a] \in
Div^0 (
P^1(
Q))
to E_c(b) - E_c(a)
, where E_c(r)
is
0
when r != c
and X^{k-2} |
gamma_r
otherwise, where
gamma_r.r = (1:0)
. These symbol span the Eisenstein subspace
of M
.
? M = msinit(2,8); \\ M_8(Gamma_0(2)) ? E = mseisenstein(M); ? E[1] \\ two-dimensional %3 = [0 -10]
[0 -15]
[0 -3]
[1 0]
? s = msfromcusp(M,oo) %4 = [0, 0, 0, 1]~ ? mseval(M, s) %5 = [1, 0] ? s = msfromcusp(M,1) %6 = [-5/16, -15/32, -3/32, 0]~ ? mseval(M,s) %7 = [-x^6, -6*x^5 - 15*x^4 - 20*x^3 - 15*x^2 - 6*x - 1]
In case M
was initialized with a non-zero sign, the symbol is given
in terms of the fixed basis of the whole symbol space, not the +
or -
part (to which it need not belong).
? M = msinit(2,8, 1); \\ M_8(Gamma_0(2))^+ ? E = mseisenstein(M); ? E[1] \\ still two-dimensional, in a smaller space %3 = [ 0 -10]
[ 0 3]
[-1 0]
? s = msfromcusp(M,oo) \\ in terms of the basis for M_8(Gamma_0(2)) ! %4 = [0, 0, 0, 1]~ ? mseval(M, s) \\ same symbol as before %5 = [1, 0]
The library syntax is GEN
msfromcusp(GEN M, GEN c)
.
msfromell(E, {
sign = 0})
Let E/
Q be an elliptic curve of conductor N
. For varepsilon =
+-1
, we define the (cuspidal, new) modular symbol x^
varepsilon in
H^1_c(X_0(N),
Q)^
varepsilon attached to
E
. For all primes p
not dividing N
we have
T_p(x^
varepsilon) = a_p x^
varepsilon, where a_p = p+1-#E(
F_p)
.
Let Omega^ += E.omega[1]
be the real period of E
(integration of the Néron differential dx/(2y+a_1x+a3)
on the connected
component of E(
R)
, i.e. the generator of H_1(E,
Z)^+
) normalized by
Omega^+ > 0
. Let i
Omega^-
the integral on a generator of H_1(E,
Z)^-
with
Omega^- \in
R_{ > 0}
. If c_ oo
is the number of connected
components of E(
R)
, Omega^-
is equal to
(-2/c_ oo ) x imag(E.omega[2])
.
The complex modular symbol is defined by
F:
delta \to 2i
pi int_{
delta} f(z) dz
The modular symbols x^
varepsilon are normalized so that
F = x^+
Omega^+ + x^- i
Omega^-
.
In particular, we have
x^+([0]-[ oo ]) = L(E,1) /
Omega^+,
which defines x^{+-}
unless L(E,1) = 0
.
Furthermore, for all fundamental discriminants D
such
that varepsilon.D > 0
, we also have
sum_{0 <= a < |D|} (D|a) x^
varepsilon([a/|D|]-[ oo ])
= L(E,(D|.),1) /
Omega^{
varepsilon},
where (D|.)
is the Kronecker symbol.
The period Omega^-
is also 2/c_ oo x
the real period
of the twist E^{(-4)} = elltwist(E,-4)
.
This function returns the pair [M, x]
, where M
is
msinit
(N,2)
and x
is x^{
sign}
as above when sign =
+-1
, and x = [x^+,x^-]
when sign is 0
.
The modular symbols x^+-
are given as a t_COL
(in terms
of the fixed basis of Hom _G(
Delta,
Q)
chosen in M
).
? E=ellinit([0,-1,1,-10,-20]); \\ X_0(11) ? [M,xp]= msfromell(E,1); ? xp %3 = [1/5, -1/2, -1/2]~ ? [M,x]= msfromell(E); ? x \\ both x^+ and x^- %5 = [[1/5, -1/2, -1/2]~, [0, 1/2, -1/2]~] ? p = 23; (mshecke(M,p) - ellap(E,p))*x[1] %6 = [0, 0, 0]~ \\ true at all primes, including p = 11; same for x[2]
The library syntax is GEN
msfromell(GEN E, long sign)
.
(M, v, {H})
Given a msinit M
and a vector v
of pairs [p, P]
(where p
is prime
and P
is a polynomial with integer coefficients), return a basis of all
modular symbols such that P(T_p)(s) = 0
. If H
is present, it must
be a Hecke-stable subspace and we restrict to s \in H
. When T_p
has
a rational eigenvalue and P(x) = x-a_p
has degree 1
, we also accept the
integer a_p
instead of P
.
? E = ellinit([0,-1,1,-10,-20]) \\11a1 ? ellap(E,2) %2 = -2 ? ellap(E,3) %3 = -1 ? M = msinit(11,2); ? S = msfromhecke(M, [[2,-2],[3,-1]]) %5 = [ 1 1]
[-5 0]
[ 0 -5] ? mshecke(M, 2, S) %6 = [-2 0]
[ 0 -2]
? M = msinit(23,4); ? S = msfromhecke(M, [[5, x^4-14*x^3-244*x^2+4832*x-19904]]); ? factor( charpoly(mshecke(M,5,S)) ) %9 = [x^4 - 14*x^3 - 244*x^2 + 4832*x - 19904 2]
The library syntax is GEN
msfromhecke(GEN M, GEN v, GEN H = NULL)
.
(M)
M
being a full modular symbol space, as given by msinit
, return
its level N
.
The library syntax is long
msgetlevel(GEN M)
.
(M)
M
being a full modular symbol space, as given by msinit
, return
its sign: +-1
or 0 (unset).
? M = msinit(11,4, 1); ? msgetsign(M) %2 = 1 ? M = msinit(11,4); ? msgetsign(M) %4 = 0
The library syntax is long
msgetsign(GEN M)
.
(M)
M
being a full modular symbol space, as given by msinit
, return
its weight k
.
? M = msinit(11,4); ? msgetweight(M) %2 = 4
The library syntax is long
msgetweight(GEN M)
.
(M,p,{H})
M
being a full modular symbol space, as given by msinit
,
p
being a prime number, and H
being a Hecke-stable subspace (M
if
omitted) return the matrix of T_p
acting on H
(U_p
if p
divides N
). Result is undefined if H
is not stable
by T_p
(resp. U_p
).
? M = msinit(11,2); \\ M_2(Gamma_0(11)) ? T2 = mshecke(M,2) %2 = [3 0 0]
[1 -2 0]
[1 0 -2] ? M = msinit(11,2, 1); \\ M_2(Gamma_0(11))^+ ? T2 = mshecke(M,2) %4 = [ 3 0]
[-1 -2]
? N = msnew(M)[1] \\ Q-basis of new cuspidal subspace %5 = [-2]
[-5]
? p = 1009; mshecke(M, p, N) \\ action of T_1009 on N %6 = [-10] ? ellap(ellinit("11a1"), p) %7 = -10
The library syntax is GEN
mshecke(GEN M, long p, GEN H = NULL)
.
msinit(G, V, {
sign = 0})
Given G
a finite index subgroup of SL(2,
Z)
and a finite dimensional representation V
of GL(2,
Q)
, creates a
space of modular symbols, the G
-module Hom _G(Div^0(
P^1
(
Q)), V)
. This is canonically isomorphic to H^1_c(X(G), V)
, and allows to
compute modular forms for G
. If sign is present and non-zero, it
must be +-1
and we consider the subspace defined by Ker (
sigma -
sign)
, where sigma is induced by [-1,0;0,1]
. Currently the
only supported groups are the Gamma_0(N)
, coded by the integer N > 1
.
The only supported representation is V_k =
Q[X,Y]_{k-2}
, coded by the
integer k >= 2
.
The library syntax is GEN
msinit(GEN G, GEN V, long sign)
.
(M,s)
M
being a full modular symbol space, as given by msinit
,
check whether s
is a modular symbol attached to M
.
? M = msinit(7,8, 1); \\ M_8(Gamma_0(7))^+ ? N = msnew(M)[1]; ? s = N[,1]; ? msissymbol(M, s) %4 = 1 ? S = mseval(M,s); ? msissymbol(M, S) %6 = 1 ? [g,R] = mspathgens(M); g %7 = [[+oo, 0], [0, 1/2], [1/2, 1]] ? #R \\ 3 relations among the generators g_i %8 = 3 ? T = S; T[3]++; \\ randomly perturb S(g_3) ? msissymbol(M, T) %10 = 0 \\ no longer satisfies the relations
The library syntax is long
msissymbol(GEN M, GEN s)
.
(M)
M
being a full modular symbol space, as given by msinit
,
return the new part of its cuspidal subspace. A subspace is given by
a structure allowing quick projection and restriction of linear operators;
its first component is a matrix with integer coefficients whose columns form
a Q-basis of the subspace.
? M = msinit(11,8, 1); \\ M_8(Gamma_0(11))^+ ? N = msnew(M); ? #N[1] \\ 6-dimensional %3 = 6
The library syntax is GEN
msnew(GEN M)
.
(
Mp,
PHI,
path)
Return the vectors of moments of the p
-adic distribution attached
to the path path
by the overconvergent modular symbol PHI
.
? M = msinit(3,6,1); ? Mp= mspadicinit(M,5,10); ? phi = [5,-3,-1]~; ? msissymbol(M,phi) %4 = 1 ? PHI = mstooms(Mp,phi); ? ME = msomseval(Mp,PHI,[oo, 0]);
The library syntax is GEN
msomseval(GEN Mp, GEN PHI, GEN path)
.
(
mu, {s = 0}, {r = 0})
Returns the value (or r
-th derivative)
on a character chi^s
of Z_p^*
of the p
-adic L
-function
attached to mu
.
Let Phi be the p
-adic distribution-valued overconvergent symbol
attached to a modular symbol phi for Gamma_0(N)
(eigenvector for
T_N(p)
for the eigenvalue a_p
). Then L_p(
Phi,
chi^s) = L_p(
mu,s)
is the
p
-adic L
function defined by
L_p(
Phi,
chi^s) =
int_{
Z_p^*}
chi^s(z) d
mu(z)
where mu is the distribution on Z_p^*
defined by the restriction of
Phi([ oo ]-[0])
to Z_p^*
. The r
-th derivative is taken in
direction <
chi>
:
L_p^{(r)}(
Phi,
chi^s) =
int_{
Z_p^*}
chi^s(z) (
log z)^r d
mu(z).
In the argument list,
@3* mu
is as returned by mspadicmoments
(distributions
attached to Phi by restriction to discs a + p^
nuZ_p
, (a,p) = 1
).
@3* s = [s_1,s_2]
with s_1 \in
Z \subset
Z_p
and s_2 mod p-1
or
s_2 mod 2
for p = 2
, encoding the p
-adic character chi^s := <
chi >^{s_1}
tau^{s_2}
; here chi is the cyclotomic character from
Gal(
Q_p(
mu_{p^ oo })/
Q_p)
to Z_p^*
, and tau is the
Teichmüller character (for p > 2
and the character of order 2 on
(
Z/4
Z)^*
if p = 2
); for convenience, the character [s,s]
can also be
represented by the integer s
.
When a_p
is a p
-adic unit, L_p
takes its values in Q_p
.
When a_p
is not a unit, it takes its values in the
two-dimensional Q_p
-vector space D_{cris}(M(
phi))
where M(
phi)
is
the ``motive'' attached to phi, and we return the two p
-adic components
with respect to some fixed Q_p
-basis.
? M = msinit(3,6,1); phi=[5, -3, -1]~; ? msissymbol(M,phi) %2 = 1 ? Mp = mspadicinit(M, 5, 4); ? mu = mspadicmoments(Mp, phi); \\ no twist \\ End of initializations
? mspadicL(mu,0) \\ L_p(chi^0) %5 = 5 + 2*5^2 + 2*5^3 + 2*5^4 + ... ? mspadicL(mu,1) \\ L_p(chi), zero for parity reasons %6 = [O(5^13)]~ ? mspadicL(mu,2) \\ L_p(chi^2) %7 = 3 + 4*5 + 4*5^2 + 3*5^5 + ... ? mspadicL(mu,[0,2]) \\ L_p(tau^2) %8 = 3 + 5 + 2*5^2 + 2*5^3 + ... ? mspadicL(mu, [1,0]) \\ L_p(<chi>) %9 = 3*5 + 2*5^2 + 5^3 + 2*5^7 + 5^8 + 5^10 + 2*5^11 + O(5^13) ? mspadicL(mu,0,1) \\ L_p'(chi^0) %10 = 2*5 + 4*5^2 + 3*5^3 + ... ? mspadicL(mu, 2, 1) \\ L_p'(chi^2) %11 = 4*5 + 3*5^2 + 5^3 + 5^4 + ...
Now several quadratic twists: mstooms
is indicated.
? PHI = mstooms(Mp,phi); ? mu = mspadicmoments(Mp, PHI, 12); \\ twist by 12 ? mspadicL(mu) %14 = 5 + 5^2 + 5^3 + 2*5^4 + ... ? mu = mspadicmoments(Mp, PHI, 8); \\ twist by 8 ? mspadicL(mu) %16 = 2 + 3*5 + 3*5^2 + 2*5^4 + ... ? mu = mspadicmoments(Mp, PHI, -3); \\ twist by -3 < 0 ? mspadicL(mu) %18 = O(5^13) \\ always 0, phi is in the + part and D < 0
One can locate interesting symbols of level N
and weight k
with
msnew
and mssplit
. Note that instead of a symbol, one can
input a 1-dimensional Hecke-subspace from mssplit
: the function will
automatically use the underlying basis vector.
? M=msinit(5,4,1); \\ M_4(Gamma_0(5))^+ ? L = mssplit(M, msnew(M)); \\ list of irreducible Hecke-subspaces ? phi = L[1]; \\ one Galois orbit of newforms ? #phi[1] \\... this one is rational %4 = 1 ? Mp = mspadicinit(M, 3, 4); ? mu = mspadicmoments(Mp, phi); ? mspadicL(mu) %7 = 1 + 3 + 3^3 + 3^4 + 2*3^5 + 3^6 + O(3^9)
? M = msinit(11,8, 1); \\ M_8(Gamma_0(11))^+ ? Mp = mspadicinit(M, 3, 4); ? L = mssplit(M, msnew(M)); ? phi = L[1]; #phi[1] \\ ... this one is two-dimensional %11 = 2 ? mu = mspadicmoments(Mp, phi); *** at top-level: mu=mspadicmoments(Mp,ph *** ^-------------------- *** mspadicmoments: incorrect type in mstooms [dim_Q (eigenspace) > 1]
The library syntax is GEN
mspadicL(GEN mu, GEN s = NULL, long r)
.
(M, p, n, {
flag})
M
being a full modular symbol space, as given by msinit
, and p
a prime, initialize technical data needed to compute with overconvergent
modular symbols, modulo p^n
. If flag is unset, allow
all symbols; else initialize only for a restricted range of symbols
depending on flag: if flag = 0
restrict to ordinary symbols, else
restrict to symbols phi such that T_p(
phi) = a_p
phi,
with v_p(a_p) >=
flag, which is faster as flag increases.
(The fastest initialization is obtained for flag = 0
where we only allow
ordinary symbols.) For supersingular eigensymbols, such that p | a_p
, we
must further assume that p
does not divide the level.
? E = ellinit("11a1"); ? [M,phi] = msfromell(E,1); ? ellap(E,3) %3 = -1 ? Mp = mspadicinit(M, 3, 10, 0); \\ commit to ordinary symbols ? PHI = mstooms(Mp,phi);
If we restrict the range of allowed symbols with flag (for faster
initialization), exceptions will occur if v_p(a_p)
violates this bound:
? E = ellinit("15a1"); ? [M,phi] = msfromell(E,1); ? ellap(E,7) %3 = 0 ? Mp = mspadicinit(M,7,5,0); \\ restrict to ordinary symbols ? PHI = mstooms(Mp,phi) *** at top-level: PHI=mstooms(Mp,phi) *** ^--------------- *** mstooms: incorrect type in mstooms [v_p(ap) > mspadicinit flag] (t_VEC). ? Mp = mspadicinit(M,7,5); \\ no restriction ? PHI = mstooms(Mp,phi);
@3This function uses O(N^2(n+k)^2p)
memory, where N
is the
level of M
.
The library syntax is GEN
mspadicinit(GEN M, long p, long n, long flag)
.
(
Mp,
PHI, {D = 1})
Given Mp
from mspadicinit
, an overconvergent
eigensymbol PHI
from mstooms
and a fundamental discriminant
D
coprime to p
,
let PHI^D
denote the twisted symbol. This function computes
the distribution mu = PHI^D([0] - oo ]) |
Z_p^*
restricted
to Z_p^*
. More precisely, it returns
the moments of the p-1
distributions PHI^D([0]-[ oo ])
| (a + p
Z_p)
, 0 < a < p
.
We also allow PHI
to be given as a classical
symbol, which is then lifted to an overconvergent symbol by mstooms
;
but this is wasteful if more than one twist is later needed.
The returned data mu (p
-adic distributions attached to PHI
)
can then be used in mspadicL
or mspadicseries
.
This precomputation allows to quickly compute derivatives of different
orders or values at different characters.
? M = msinit(3,6, 1); ? phi = [5,-3,-1]~; ? msissymbol(M, phi) %3 = 1 ? p = 5; mshecke(M,p) * phi \\ eigenvector of T_5, a_5 = 6 %4 = [30, -18, -6]~ ? Mp = mspadicinit(M, p, 10, 0); \\ restrict to ordinary symbols, mod p^10 ? PHI = mstooms(Mp, phi); ? mu = mspadicmoments(Mp, PHI); ? mspadicL(mu) %8 = 5 + 2*5^2 + 2*5^3 + ... ? mu = mspadicmoments(Mp, PHI, 12); \\ twist by 12 ? mspadicL(mu) %10 = 5 + 5^2 + 5^3 + 2*5^4 + ...
The library syntax is GEN
mspadicmoments(GEN Mp, GEN PHI, long D)
.
(
mu, {i = 0})
Let Phi be the p
-adic distribution-valued overconvergent symbol
attached to a modular symbol phi for Gamma_0(N)
(eigenvector for
T_N(p)
for the eigenvalue a_p
).
If mu is the distribution on Z_p^*
defined by the restriction of
Phi([ oo ]-[0])
to Z_p^*
, let
^{L}_p(
mu,
tau^{i})(x)
=
int_{
Z_p^*}
tau^i(t) (1+x)^{
log _p(t)/
log _p(u)}d
mu(t)
Here, tau is the Teichmüller character and u
is a specific
multiplicative generator of 1+2p
Z_p
. (Namely 1+p
if p > 2
or 5
if p = 2
.) To explain
the formula, let G_ oo := Gal(
Q(
mu_{p^{ oo }})/
Q)
,
let chi:G_ oo \to
Z_p^*
be the cyclotomic character (isomorphism)
and gamma the element of G_ oo
such that chi(
gamma) = u
;
then
chi(
gamma)^{
log _p(t)/
log _p(u)} = <t >
.
The p
-padic precision of individual terms is maximal given the precision of
the overconvergent symbol mu.
? [M,phi] = msfromell(ellinit("17a1"),1); ? Mp = mspadicinit(M, 5,7); ? mu = mspadicmoments(Mp, phi,1); \\ overconvergent symbol ? mspadicseries(mu) %4 = (4 + 3*5 + 4*5^2 + 2*5^3 + 2*5^4 + 5^5 + 4*5^6 + 3*5^7 + O(5^9)) \ + (3 + 3*5 + 5^2 + 5^3 + 2*5^4 + 5^6 + O(5^7))*x \ + (2 + 3*5 + 5^2 + 4*5^3 + 2*5^4 + O(5^5))*x^2 \ + (3 + 4*5 + 4*5^2 + O(5^3))*x^3 \ + (3 + O(5))*x^4 + O(x^5)
An example with non-zero Teichmüller:
? [M,phi] = msfromell(ellinit("11a1"),1); ? Mp = mspadicinit(M, 3,10); ? mu = mspadicmoments(Mp, phi,1); ? mspadicseries(mu, 2) %4 = (2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + 3^7 + 3^10 + 3^11 + O(3^12)) \ + (1 + 3 + 2*3^2 + 3^3 + 3^5 + 2*3^6 + 2*3^8 + O(3^9))*x \ + (1 + 2*3 + 3^4 + 2*3^5 + O(3^6))*x^2 \ + (3 + O(3^2))*x^3 + O(x^4)
Supersingular example (not checked)
? E = ellinit("17a1"); ellap(E,3) %1 = 0 ? [M,phi] = msfromell(E,1); ? Mp = mspadicinit(M, 3,7); ? mu = mspadicmoments(Mp, phi,1); ? mspadicseries(mu) %5 = [(2*3^-1 + 1 + 3 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + O(3^7)) \ + (2 + 3^3 + O(3^5))*x \ + (1 + 2*3 + O(3^2))*x^2 + O(x^3),\ (3^-1 + 1 + 3 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + O(3^7)) \ + (1 + 2*3 + 2*3^2 + 3^3 + 2*3^4 + O(3^5))*x \ + (3^-2 + 3^-1 + O(3^2))*x^2 + O(3^-2)*x^3 + O(x^4)]
Example with a twist:
? E = ellinit("11a1"); ? [M,phi] = msfromell(E,1); ? Mp = mspadicinit(M, 3,10); ? mu = mspadicmoments(Mp, phi,5); \\ twist by 5 ? L = mspadicseries(mu) %5 = (2*3^2 + 2*3^4 + 3^5 + 3^6 + 2*3^7 + 2*3^10 + O(3^12)) \ + (2*3^2 + 2*3^6 + 3^7 + 3^8 + O(3^9))*x \ + (3^3 + O(3^6))*x^2 + O(3^2)*x^3 + O(x^4) ? mspadicL(mu) %6 = [2*3^2 + 2*3^4 + 3^5 + 3^6 + 2*3^7 + 2*3^10 + O(3^12)]~ ? ellpadicL(E,3,10,,5) %7 = 2 + 2*3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^6 + 2*3^7 + O(3^10) ? mspadicseries(mu,1) \\ must be 0 %8 = O(3^12) + O(3^9)*x + O(3^6)*x^2 + O(3^2)*x^3 + O(x^4)
The library syntax is GEN
mspadicseries(GEN mu, long i)
.
(M)
Let Delta := Div^0(
P^1(
Q))
.
Let M
being a full modular symbol space, as given by msinit
,
return a set of Z[G]
-generators for Delta. The output
is [g,R]
, where g
is a minimal system of generators and R
the vector of Z[G]
-relations between the given generators. A
relation is coded by a vector of pairs [a_i,i]
with a_i\in
Z[G]
and i
the index of a generator, so that sum_i a_i g[i] = 0
.
An element [v]-[u]
in Delta is coded by the ``path'' [u,v]
,
where oo
denotes the point at infinity (1:0)
on the projective
line.
An element of Z[G]
is coded by a ``factorization matrix'': the first
column contains distinct elements of G
, and the second integers:
? M = msinit(11,8); \\ M_8(Gamma_0(11)) ? [g,R] = mspathgens(M); ? g %3 = [[+oo, 0], [0, 1/3], [1/3, 1/2]] \\ 3 paths ? #R \\ a single relation %4 = 1 ? r = R[1]; #r \\ ...involving all 3 generators %5 = 3 ? r[1] %6 = [[1, 1; [1, 1; 0, 1], -1], 1] ? r[2] %7 = [[1, 1; [7, -2; 11, -3], -1], 2] ? r[3] %8 = [[1, 1; [8, -3; 11, -4], -1], 3]
The given relation is of the form sum_i (1-
gamma_i) g_i = 0
, with
gamma_i\in
Gamma_0(11)
. There will always be a single relation involving
all generators (corresponding to a round trip along all cusps), then
relations involving a single generator (corresponding to 2
and 3
-torsion
elements in the group:
? M = msinit(2,8); \\ M_8(Gamma_0(2)) ? [g,R] = mspathgens(M); ? g %3 = [[+oo, 0], [0, 1]]
Note that the output depends only on the group G
, not on the
representation V
.
The library syntax is GEN
mspathgens(GEN M)
.
(M,p)
Let Delta := Div^0(
P^1(
Q))
.
Let M
being a full modular symbol space, as given by msinit
,
encoding fixed Z[G]
-generators (g_i)
of Delta (see mspathgens
).
A path p = [a,b]
between two elements in P^1(
Q)
corresponds to
[b]-[a]\in
Delta. The path extremities a
and b
may be given as
t_INT
, t_FRAC
or oo = (1:0)
.
Returns (p_i)
in Z[G]
such that p =
sum_i p_i g_i
.
? M = msinit(2,8); \\ M_8(Gamma_0(2)) ? [g,R] = mspathgens(M); ? g %3 = [[+oo, 0], [0, 1]] ? p = mspathlog(M, [1/2,2/3]); ? p[1] %5 = [[1, 0; 2, 1] 1]
? p[2] %6 = [[1, 0; 0, 1] 1]
[[3, -1; 4, -1] 1]
@3
Note that the output depends only on the group G
, not on the
representation V
.
The library syntax is GEN
mspathlog(GEN M, GEN p)
.
msqexpansion(M,
projH,{B =
seriesprecision})
M
being a full modular symbol space, as given by msinit
,
and projH being a projector on a Hecke-simple subspace (as given
by mssplit
), return the Fourier coefficients a_n
, n <= B
of the
corresponding normalized newform. If B
is omitted, use
seriesprecision
.
This function uses a naive O(B^2 d^3)
algorithm, where d = O(kN)
is the dimension of M_k(
Gamma_0(N))
.
? M = msinit(11,2, 1); \\ M_2(Gamma_0(11))^+ ? L = mssplit(M, msnew(M)); ? msqexpansion(M,L[1], 20) %3 = [1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4, -2, 4, 0, 2] ? ellan(ellinit("11a1"), 20) %4 = [1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4, -2, 4, 0, 2]
@3The shortcut msqexpansion(M, s, B)
is available for
a symbol s
, provided it is a Hecke eigenvector:
? E = ellinit("11a1"); ? [M,s]=msfromell(E); ? msqexpansion(M,s,10) %3 = [1, -2, -1, 2, 1, 2, -2, 0, -2, -2] ? ellan(E, 10) %4 = [1, -2, -1, 2, 1, 2, -2, 0, -2, -2]
The library syntax is GEN
msqexpansion(GEN M, GEN projH, long precdl)
.
mssplit(M,H,{
dimlim})
Let M
denote a full modular symbol space, as given by msinit
(N,k,1)
or msinit(N,k,-1)
and let H
be a Hecke-stable subspace of
msnew
(M)
. This function split H
into Hecke-simple subspaces. If
dimlim
is present and positive, restrict to subspaces of dimension
<= dimlim
. A subspace is given by a structure allowing quick
projection and restriction of linear operators; its first component is a
matrix with integer coefficients whose columns form a Q-basis of the
subspace.
? M = msinit(11,8, 1); \\ M_8(Gamma_0(11))^+ ? L = mssplit(M, msnew(M)); ? #L %3 = 2 ? f = msqexpansion(M,L[1],5); f[1].mod %4 = x^2 + 8*x - 44 ? lift(f) %5 = [1, x, -6*x - 27, -8*x - 84, 20*x - 155] ? g = msqexpansion(M,L[2],5); g[1].mod %6 = x^4 - 558*x^2 + 140*x + 51744
@3To a Hecke-simple subspace corresponds an orbit of (normalized) newforms, defined over a number field. In the above example, we printed the polynomials defining the said fields, as well as the first 5 Fourier coefficients (at the infinite cusp) of one such form.
The library syntax is GEN
mssplit(GEN M, GEN H, long dimlim)
.
(M,{H})
M
being a full modular symbol space, as given by msinit
,
return the matrix of the *
involution, induced by complex conjugation,
acting on the (stable) subspace H
(M
if omitted).
? M = msinit(11,2); \\ M_2(Gamma_0(11)) ? w = msstar(M); ? w^2 == 1 %3 = 1
The library syntax is GEN
msstar(GEN M, GEN H = NULL)
.
(
Mp,
phi)
Given Mp
from mspadicinit
, lift the (classical) eigen symbol
phi
to a p
-adic distribution-valued overconvergent symbol in the
sense of Pollack and Stevens. More precisely, let phi belong to the space
W
of modular symbols of level N
, v_p(N) <= 1
, and weight k
which is
an eigenvector for the Hecke operator T_N(p)
for a non-zero eigenvalue
a_p
and let N_0 = lcm(N,p)
.
Under the action of T_{N_0}(p)
, phi generates a subspace W_
phi of
dimension 1
(if p | N
) or 2
(if p
does not divide N
) in the
space of modular symbols of level N_0
.
Let V_p = [p,0;0,1]
and C_p = [a_p,p^{k-1};-1,0]
.
When p
does not divide N
and a_p
is divisible by p
, mstooms
returns the lift Phi of (
phi,
phi|_k V_p)
such that
T_{N_0}(p)
Phi = C_p
Phi
When p
does not divide N
and a_p
is not divisible by p
, mstooms
returns the lift Phi of phi -
alpha^{-1}
phi|_k V_p
which is an eigenvector of T_{N_0}(p)
for the unit eigenvalue
where alpha^2 - a_p
alpha + p^{k-1} = 0
.
The resulting overconvergent eigensymbol can then be used in
mspadicmoments
, then mspadicL
or mspadicseries
.
? M = msinit(3,6, 1); p = 5; ? Tp = mshecke(M, p); factor(charpoly(Tp)) %2 = [x - 3126 2]
[ x - 6 1] ? phi = matker(Tp - 6)[,1] \\ generator of p-Eigenspace, a_p = 6 %3 = [5, -3, -1]~ ? Mp = mspadicinit(M, p, 10, 0); \\ restrict to ordinary symbols, mod p^10 ? PHI = mstooms(Mp, phi); ? mu = mspadicmoments(Mp, PHI); ? mspadicL(mu) %7 = 5 + 2*5^2 + 2*5^3 + ...
A non ordinary symbol.
? M = msinit(4,6,1); p = 3; ? Tp = mshecke(M, p); factor(charpoly(Tp)) %2 = [x - 244 3]
[ x + 12 1] ? phi = matker(Tp + 12)[,1] \\ a_p = -12 is divisible by p = 3 %3 = [-1/32, -1/4, -1/32, 1]~ ? msissymbol(M,phi) %4 = 1 ? Mp = mspadicinit(M,3,5,0); ? PHI = mstooms(Mp,phi); *** at top-level: PHI=mstooms(Mp,phi) *** ^--------------- *** mstooms: incorrect type in mstooms [v_p(ap) > mspadicinit flag] (t_VEC). ? Mp = mspadicinit(M,3,5,1); ? PHI = mstooms(Mp,phi);
The library syntax is GEN
mstooms(GEN Mp, GEN phi)
.
In this section, we describe functions related to general number fields. Functions related to quadratic number fields are found in Label se:arithmetic (Arithmetic functions).
Let K =
Q[X] / (T)
a number field, Z_K
its ring of integers, T\in
Z[X]
is monic. Three basic number field structures can be attached to K
in
GP:
@3* nf denotes a number field, i.e. a data structure output by
nfinit
. This contains the basic arithmetic data attached to the
number field: signature, maximal order (given by a basis nf.zk
),
discriminant, defining polynomial T
, etc.
@3* bnf denotes a ``Buchmann's number field'', i.e. a
data structure output by
bnfinit
. This contains
nf and the deeper invariants of the field: units U(K)
, class group
Cl (K)
, as well as technical data required to solve the two attached
discrete logarithm problems.
@3* bnr denotes a ``ray number field'', i.e. a data structure
output by
bnrinit
, corresponding to the ray class group structure of
the field, for some modulus f
. It contains a bnf, the modulus
f
, the ray class group Cl _f(K)
and data attached to
the discrete logarithm problem therein.
@3An algebraic number belonging to K =
Q[X]/(T)
is given as
@3* a t_INT
, t_FRAC
or t_POL
(implicitly modulo T
), or
@3* a t_POLMOD
(modulo T
), or
@3* a t_COL
v
of dimension N = [K:
Q]
, representing
the element in terms of the computed integral basis, as
sum(i = 1, N, v[i] * nf.zk[i])
. Note that a t_VEC
will not be recognized.
@3An ideal is given in any of the following ways:
@3* an algebraic number in one of the above forms, defining a principal ideal.
@3* a prime ideal, i.e. a 5-component vector in the format output by
idealprimedec
or idealfactor
.
@3* a t_MAT
, square and in Hermite Normal Form (or at least
upper triangular with non-negative coefficients), whose columns represent a
Z-basis of the ideal.
One may use idealhnf
to convert any ideal to the last (preferred) format.
@3* an extended ideal is a 2-component
vector [I, t]
, where I
is an ideal as above and t
is an algebraic
number, representing the ideal (t)I
. This is useful whenever idealred
is involved, implicitly working in the ideal class group, while keeping track
of principal ideals. Ideal operations suitably update the principal part
when it makes sense (in a multiplicative context), e.g. using idealmul
on [I,t]
, [J,u]
, we obtain [IJ, tu]
. When it does not make sense, the
extended part is silently discarded, e.g. using idealadd
with the above
input produces I+J
.
The ``principal part'' t
in an extended ideal may be
represented in any of the above forms, and also as a factorization
matrix (in terms of number field elements, not ideals!), possibly the empty
matrix [;]
representing 1
. In the latter case, elements stay in
factored form, or famat for factorization matrix, which
is a convenient way to avoid coefficient explosion. To recover the
conventional expanded form, try nffactorback
; but many functions
already accept famats as input, for instance ideallog
, so
expanding huge elements should never be necessary.
A finite abelian group G
in user-readable format is given by its Smith
Normal Form as a pair [h,d]
or triple [h,d,g]
.
Here h
is the cardinality of G
, (d_i)
is the vector of elementary
divisors, and (g_i)
is a vector of generators. In short,
G = \oplus_{i <= n} (
Z/d_i
Z) g_i
, with d_n | ... | d_2 | d_1
and prod d_i = h
. This information can also be retrieved as
G.no
, G.cyc
and G.gen
.
@3* a character on the abelian group
\oplus (
Z/d_j
Z) g_j
is given by a row vector chi = [a_1,...,a_n]
such that
chi(
prod g_j^{n_j}) =
exp (2
pi i
sum a_j n_j / d_j)
.
@3* given such a structure, a subgroup H
is input as a square
matrix in HNF, whose columns express generators of H
on the given generators
g_i
. Note that the determinant of that matrix is equal to the index (G:H)
.
We now have a look at data structures attached to relative extensions
of number fields L/K
, and to projective Z_K
-modules. When defining a
relative extension L/K
, the nf attached to the base field K
must be defined by a variable having a lower priority (see
Label se:priority) than the variable defining the extension. For example,
you may use the variable name y
to define the base field K
, and x
to
define the relative extension L/K
.
ZKmodules}
@3* rnf denotes a relative number field, i.e. a data structure
output by
rnfinit
, attached to the extension L/K
. The nf
attached to be base field K
is rnf.nf
.
@3* A relative matrix is an m x n
matrix whose entries are
elements of K
, in any form. Its m
columns A_j
represent elements
in K^n
.
@3* An ideal list is a row vector of fractional ideals of the number field nf.
@3* A pseudo-matrix is a 2-component row vector (A,I)
where A
is a relative m x n
matrix and I
an ideal list of length n
. If I =
{
a_1,...,
a_n}
and the columns of A
are (A_1,...,
A_n)
, this data defines the torsion-free (projective) Z_K
-module
a_1 A_1\oplus
a_n A_n
.
@3* An integral pseudo-matrix is a 3-component row vector w(A,I,J)
where A = (a_{i,j})
is an m x n
relative matrix and I =
(
b_1,...,
b_m)
, J = (
a_1,...,
a_n)
are ideal
lists, such that a_{i,j} \in
b_i
a_j^{-1}
for all i,j
. This
data defines two abstract projective Z_K
-modules
N =
a_1
omega_1\oplus...\oplus
a_n
omega_n
in K^n
,
P =
b_1
eta_1\oplus...\oplus
b_m
eta_m
in K^m
, and a
Z_K
-linear map f:N\to P
given by
f(
sum alpha_j
omega_j) =
sum_i (a_{i,j}
alpha_j)
eta_i.
This data defines the Z_K
-module M = P/f(N)
.
@3* Any projective Z_K
-module\varsidx{projective module} M
of finite type in K^m
can be given by a pseudo matrix (A,I)
.
@3* An arbitrary Z_K
modules of finite type in K^m
, with non-trivial
torsion, is given by an integral pseudo-matrix (A,I,J)
@3* The pair (A,I)
is a pseudo-basis of the module it
generates if the a_j
are non-zero, and the A_j
are K
-linearly
independent. We call n
the size of the pseudo-basis. If A
is a
relative matrix, the latter condition means it is square with non-zero
determinant; we say that it is in Hermite Normal
Form (HNF) if it is upper triangular and all the
elements of the diagonal are equal to 1.
@3* For instance, the relative integer basis rnf.zk
is a pseudo-basis
(A,I)
of Z_L
, where A = rnf.zk[1]
is a vector of elements of L
,
which are K
-linearly independent. Most rnf routines return and handle
Z_K
-modules contained in L
(e.g. Z_L
-ideals) via a pseudo-basis
(A',I')
, where A'
is a relative matrix representing a vector of elements of
L
in terms of the fixed basis rnf.zk[1]
@3* The determinant of a pseudo-basis (A,I)
is the ideal
equal to the product of the determinant of A
by all the ideals of I
. The
determinant of a pseudo-matrix is the determinant of any pseudo-basis of the
module it generates.
A modulus, in the sense of class field theory, is a divisor supported
on the non-complex places of
K
. In PARI terms, this means either an
ordinary ideal I
as above (no Archimedean component), or a pair [I,a]
,
where a
is a vector with r_1
{0,1}
-components, corresponding to the
infinite part of the divisor. More precisely, the i
-th component of a
corresponds to the real embedding attached to the i
-th real root of
K.roots
. (That ordering is not canonical, but well defined once a
defining polynomial for K
is chosen.) For instance, [1, [1,1]]
is a
modulus for a real quadratic field, allowing ramification at any of the two
places at infinity, and nowhere else.
A bid or ``big ideal'' is a structure output by idealstar
needed to compute in (
Z_K/I)^*
, where I
is a modulus in the above sense.
It is a finite abelian group as described above, supplemented by
technical data needed to solve discrete log problems.
Finally we explain how to input ray number fields (or bnr), using class
field theory. These are defined by a triple A
, B
, C
, where the
defining set [A,B,C]
can have any of the following forms:
[
bnr]
,
[
bnr,
subgroup]
,
[
bnr,
character]
,
[
bnf,
mod]
,
[
bnf,
mod,
subgroup]
. The last two forms are kept for
backward compatibility, but no longer serve any real purpose (see example
below); no newly written function will accept them.
@3* bnf is as output by bnfinit
, where units are mandatory
unless the modulus is trivial; bnr is as output by bnrinit
. This
is the ground field K
.
@3* mod is a modulus f, as described above.
@3* subgroup a subgroup of the ray class group modulo f of
K
. As described above, this is input as a square matrix expressing
generators of a subgroup of the ray class group bnr.clgp
on the
given generators.
The corresponding bnr is the subfield of the ray class field of K
modulo f, fixed by the given subgroup.
? K = bnfinit(y^2+1); ? bnr = bnrinit(K, 13) ? %.clgp %3 = [36, [12, 3]] ? bnrdisc(bnr); \\ discriminant of the full ray class field ? bnrdisc(bnr, [3,1;0,1]); \\ discriminant of cyclic cubic extension of K ? bnrconductor(bnr, [3,1]); \\ conductor of chi: g1->zeta_12^3, g2->zeta_3
We could have written directly
? bnrdisc(K, 13); ? bnrdisc(K, 13, [3,1;0,1]);
avoiding one bnrinit
, but this would actually be slower since the
bnrinit
is called internally anyway. And now twice!
All the functions which are specific to relative extensions, number fields,
Buchmann's number fields, Buchmann's number rays, share the prefix rnf
,
nf
, bnf
, bnr
respectively. They take as first argument a
number field of that precise type, respectively output by rnfinit
,
nfinit
, bnfinit
, and bnrinit
.
However, and even though it may not be specified in the descriptions of the
functions below, it is permissible, if the function expects a nf, to
use a bnf instead, which contains much more information. On the other
hand, if the function requires a bnf
, it will not launch
bnfinit
for you, which is a costly operation. Instead, it will give you
a specific error message. In short, the types
nf <= bnf <= bnr
are ordered, each function requires a minimal type to work properly, but you may always substitute a larger type.
The data types corresponding to the structures described above are rather complicated. Thus, as we already have seen it with elliptic curves, GP provides ``member functions'' to retrieve data from these structures (once they have been initialized of course). The relevant types of number fields are indicated between parentheses:
bid
(bnr ) : bid ideal structure.
bnf
(bnr, bnf ) : Buchmann's number field.
clgp
(bnr, bnf ) : classgroup. This one admits the
following three subclasses:
cyc
: cyclic decomposition
(SNF).
gen
:
generators.
no
: number of elements.
diff
(bnr, bnf, nf ) : the different ideal.
codiff
(bnr, bnf, nf ) : the codifferent
(inverse of the different in the ideal group).
disc
(bnr, bnf, nf ) : discriminant.
fu
(bnr, bnf ) : fundamental units.
index
(bnr, bnf, nf ) :
index of the power order in the ring of integers.
mod
(bnr ) : modulus.
nf
(bnr, bnf, nf ) : number field.
pol
(bnr, bnf, nf ) : defining polynomial.
r1
(bnr, bnf, nf ) : the number
of real embeddings.
r2
(bnr, bnf, nf ) : the number
of pairs of complex embeddings.
reg
(bnr, bnf ) : regulator.
roots
(bnr, bnf, nf ) : roots of the
polynomial generating the field.
sign
(bnr, bnf, nf ) : signature [r1,r2]
.
t2
(bnr, bnf, nf ) : the T_2
matrix (see
nfinit
).
tu
(bnr, bnf ) : a generator for the torsion
units.
zk
(bnr, bnf, nf ) : integral basis, i.e. a
Z-basis of the maximal order.
zkst
(bnr ) : structure of (
Z_K/m)^*
.
@3Deprecated. The following member functions are still available, but deprecated and should not be used in new scripts :
futu
(bnr, bnf, ) :
[u_1,...,u_r,w]
, (u_i)
is a vector of fundamental units,
w
generates the torsion units.
tufu
(bnr, bnf, ) :
[w,u_1,...,u_r]
, (u_i)
is a vector of fundamental units,
w
generates the torsion units.
For instance, assume that bnf = bnfinit(
pol)
, for some
polynomial. Then bnf.clgp
retrieves the class group, and
bnf.clgp.no
the class number. If we had set bnf =
nfinit(
pol)
, both would have output an error message. All these
functions are completely recursive, thus for instance
bnr.bnf.nf.zk
will yield the maximal order of bnr, which
you could get directly with a simple bnr.zk
.
Some of the functions starting with bnf
are implementations of the
sub-exponential algorithms for finding class and unit groups under GRH,
due to Hafner-McCurley, Buchmann and Cohen-Diaz-Olivier. The general
call to the functions concerning class groups of general number fields
(i.e. excluding quadclassunit
) involves a polynomial P
and a
technical vector
tech = [c_1, c_2,
nrpid ],
where the parameters are to be understood as follows:
P
is the defining polynomial for the number field, which must be in
Z[X]
, irreducible and monic. In fact, if you supply a non-monic polynomial
at this point, gp
issues a warning, then transforms your
polynomial so that it becomes monic. The nfinit
routine
will return a different result in this case: instead of res
, you get a
vector [res,Mod(a,Q)]
, where Mod(a,Q) = Mod(X,P)
gives the change
of variables. In all other routines, the variable change is simply lost.
The tech interface is obsolete and you should not tamper with these parameters. Indeed, from version 2.4.0 on,
@3* the results are always rigorous under GRH (before that version, they relied on a heuristic strengthening, hence the need for overrides).
@3* the influence of these parameters on execution time and stack size is
marginal. They can be useful to fine-tune and experiment with the
bnfinit
code, but you will be better off modifying all tuning
parameters in the C code (there are many more than just those three).
We nevertheless describe it for completeness.
The numbers c_1 <= c_2
are non-negative real numbers. By default they are
chosen so that the result is correct under GRH. For i = 1,2
, let
B_i = c_i(
log |d_K|)^2
, and denote by S(B)
the set of maximal ideals of
K
whose norm is less than B
. We want S(B_1)
to generate Cl (K)
and hope
that S(B_2)
can be proven to generate Cl (K)
.
More precisely, S(B_1)
is a factorbase used to compute a tentative
Cl (K)
by generators and relations. We then check explicitly, using
essentially bnfisprincipal
, that the elements of S(B_2)
belong to the
span of S(B_1)
. Under the assumption that S(B_2)
generates Cl (K)
, we
are done. User-supplied c_i
are only used to compute initial guesses for
the bounds B_i
, and the algorithm increases them until one can prove
under GRH that S(B_2)
generates Cl (K)
. A uniform result of Bach says
that c_2 = 12
is always suitable, but this bound is very pessimistic and a
direct algorithm due to Belabas-Diaz-Friedman is used to check the condition,
assuming GRH. The default values are c_1 = c_2 = 0
. When c_1
is equal to
0
the algorithm takes it equal to c_2
.
nrpid is the maximal number of small norm relations attached to each
ideal in the factor base. Set it to 0
to disable the search for small norm
relations. Otherwise, reasonable values are between 4 and 20. The default is
4.
@3Warning. Make sure you understand the above! By default, most of
the bnf
routines depend on the correctness of the GRH. In particular,
any of the class number, class group structure, class group generators,
regulator and fundamental units may be wrong, independently of each other.
Any result computed from such a bnf
may be wrong. The only guarantee is
that the units given generate a subgroup of finite index in the full unit
group. You must use bnfcertify
to certify the computations
unconditionally.
@3Remarks.
You do not need to supply the technical parameters (under the library you
still need to send at least an empty vector, coded as NULL
). However,
should you choose to set some of them, they must be given in the
requested order. For example, if you want to specify a given value of
nrpid, you must give some values as well for c_1
and c_2
, and provide
a vector [c_1,c_2,
nrpid]
.
Note also that you can use an nf instead of P
, which avoids
recomputing the integral basis and analogous quantities.
(
bnf,{
flag = 0})
bnf being as output by
bnfinit
, checks whether the result is correct, i.e. whether it is
possible to remove the assumption of the Generalized Riemann
Hypothesis. It is correct if and only if the answer is 1. If it is
incorrect, the program may output some error message, or loop indefinitely.
You can check its progress by increasing the debug level. The bnf
structure must contain the fundamental units:
? K = bnfinit(x^3+2^2^3+1); bnfcertify(K) *** at top-level: K=bnfinit(x^3+2^2^3+1);bnfcertify(K) *** ^------------- *** bnfcertify: missing units in bnf. ? K = bnfinit(x^3+2^2^3+1, 1); \\ include units ? bnfcertify(K) %3 = 1
If flag is present, only certify that the class group is a quotient of the one computed in bnf (much simpler in general); likewise, the computed units may form a subgroup of the full unit group. In this variant, the units are no longer needed:
? K = bnfinit(x^3+2^2^3+1); bnfcertify(K, 1) %4 = 1
The library syntax is long
bnfcertify0(GEN bnf, long flag)
.
Also available is GEN
bnfcertify(GEN bnf)
(flag = 0
).
(
bnf)
Computes a compressed version of bnf (from bnfinit
), a
``small Buchmann's number field'' (or sbnf for short) which contains
enough information to recover a full bnf vector very rapidly, but
which is much smaller and hence easy to store and print. Calling
bnfinit
on the result recovers a true bnf
, in general different
from the original. Note that an snbf is useless for almost all
purposes besides storage, and must be converted back to bnf form
before use; for instance, no nf*
, bnf*
or member function
accepts them.
An sbnf is a 12 component vector v
, as follows. Let bnf
be
the result of a full bnfinit
, complete with units. Then v[1]
is
bnf.pol
, v[2]
is the number of real embeddings bnf.sign[1]
,
v[3]
is bnf.disc
, v[4]
is bnf.zk
, v[5]
is the list of roots
bnf.roots
, v[7]
is the matrix W = bnf[1]
,
v[8]
is the matrix matalpha = bnf[2]
,
v[9]
is the prime ideal factor base bnf[5]
coded in a compact way,
and ordered according to the permutation bnf[6]
, v[10]
is the
2-component vector giving the number of roots of unity and a generator,
expressed on the integral basis, v[11]
is the list of fundamental units,
expressed on the integral basis, v[12]
is a vector containing the algebraic
numbers alpha corresponding to the columns of the matrix matalpha
,
expressed on the integral basis.
All the components are exact (integral or rational), except for the roots in
v[5]
.
The library syntax is GEN
bnfcompress(GEN bnf)
.
(
nf,m)
If m
is a module as output in the
first component of an extension given by bnrdisclist
, outputs the
true module.
? K = bnfinit(x^2+23); L = bnrdisclist(K, 10); s = L[1][2] %1 = [[Mat([8, 1]), [[0, 0, 0]]], [Mat([9, 1]), [[0, 0, 0]]]] ? bnfdecodemodule(K, s[1][1]) %2 = [2 0]
[0 1]
The library syntax is GEN
decodemodule(GEN nf, GEN m)
.
bnfinit(P,{
flag = 0},{
tech = []})
Initializes a
bnf
structure. Used in programs such as bnfisprincipal
,
bnfisunit
or bnfnarrow
. By default, the results are conditional
on the GRH, see se:GRHbnf. The result is a
10-component vector bnf.
This implements Buchmann's sub-exponential algorithm for computing the
class group, the regulator and a system of fundamental units of the
general algebraic number field K
defined by the irreducible polynomial P
with integer coefficients.
If the precision becomes insufficient, gp
does not strive to compute
the units by default (flag = 0
).
When flag = 1
, we insist on finding the fundamental units exactly. Be
warned that this can take a very long time when the coefficients of the
fundamental units on the integral basis are very large. If the fundamental
units are simply too large to be represented in this form, an error message
is issued. They could be obtained using the so-called compact representation
of algebraic numbers as a formal product of algebraic integers. The latter is
implemented internally but not publicly accessible yet.
tech is a technical vector (empty by default, see se:GRHbnf). Careful use of this parameter may speed up your computations, but it is mostly obsolete and you should leave it alone.
The components of a bnf or sbnf are technical and never used by the casual user. In fact: never access a component directly, always use a proper member function. However, for the sake of completeness and internal documentation, their description is as follows. We use the notations explained in the book by H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Maths 138, Springer-Verlag, 1993, Section 6.5, and subsection 6.5.5 in particular.
bnf[1]
contains the matrix W
, i.e. the matrix in Hermite normal
form giving relations for the class group on prime ideal generators
(
p_i)_{1 <= i <= r}
.
bnf[2]
contains the matrix B
, i.e. the matrix containing the
expressions of the prime ideal factorbase in terms of the p_i
.
It is an r x c
matrix.
bnf[3]
contains the complex logarithmic embeddings of the system of
fundamental units which has been found. It is an (r_1+r_2) x (r_1+r_2-1)
matrix.
bnf[4]
contains the matrix M''_C
of Archimedean components of the
relations of the matrix (W|B)
.
bnf[5]
contains the prime factor base, i.e. the list of prime
ideals used in finding the relations.
bnf[6]
used to contain a permutation of the prime factor base, but
has been obsoleted. It contains a dummy 0
.
bnf[7]
or bnf.nf
is equal to the number field data
nf as would be given by nfinit
.
bnf[8]
is a vector containing the classgroup bnf.clgp
as a finite abelian group, the regulator bnf.reg
, a 1
(used to
contain an obsolete ``check number''), the number of roots of unity and a
generator bnf.tu
, the fundamental units bnf.fu
.
bnf[9]
is a 3-element row vector used in bnfisprincipal
only
and obtained as follows. Let D = U W V
obtained by applying the
Smith normal form algorithm to the matrix W
( = bnf[1]
) and
let U_r
be the reduction of U
modulo D
. The first elements of the
factorbase are given (in terms of bnf.gen
) by the columns of U_r
,
with Archimedean component g_a
; let also GD_a
be the Archimedean
components of the generators of the (principal) ideals defined by the
bnf.gen[i]^bnf.cyc[i]
. Then bnf[9] = [U_r, g_a, GD_a]
.
bnf[10]
is by default unused and set equal to 0. This field is used
to store further information about the field as it becomes available, which
is rarely needed, hence would be too expensive to compute during the initial
bnfinit
call. For instance, the generators of the principal ideals
bnf.gen[i]^bnf.cyc[i]
(during a call to bnrisprincipal
), or
those corresponding to the relations in W
and B
(when the bnf
internal precision needs to be increased).
The library syntax is GEN
bnfinit0(GEN P, long flag, GEN tech = NULL, long prec)
.
Also available is GEN
Buchall(GEN P, long flag, long prec)
,
corresponding to tech = NULL
, where
flag
is either 0
(default) or nf_FORCE
(insist on finding
fundamental units). The function
GEN
Buchall_param(GEN P, double c1, double c2, long nrpid, long flag, long prec)
gives direct access to the technical parameters.
(
bnf,x)
Computes a complete system of
solutions (modulo units of positive norm) of the absolute norm equation
Norm (a) = x
,
where a
is an integer in bnf. If bnf has not been certified,
the correctness of the result depends on the validity of GRH.
See also bnfisnorm
.
The library syntax is GEN
bnfisintnorm(GEN bnf, GEN x)
.
The function GEN
bnfisintnormabs(GEN bnf, GEN a)
returns a complete system of solutions modulo units of the absolute norm
equation |
Norm (x) |= |a|
. As fast as bnfisintnorm
, but solves
the two equations Norm (x) = +- a
simultaneously.
(
bnf,x,{
flag = 1})
Tries to tell whether the
rational number x
is the norm of some element y in bnf. Returns a
vector [a,b]
where x = Norm(a)*b
. Looks for a solution which is an S
-unit,
with S
a certain set of prime ideals containing (among others) all primes
dividing x
. If bnf is known to be Galois, set flag = 0
(in
this case, x
is a norm iff b = 1
). If flag is non zero the program adds to
S
the following prime ideals, depending on the sign of flag. If flag > 0
,
the ideals of norm less than flag. And if flag < 0
the ideals dividing flag.
Assuming GRH, the answer is guaranteed (i.e. x
is a norm iff b = 1
),
if S
contains all primes less than 12
log (
disc (
Bnf))^2
, where
Bnf is the Galois closure of bnf.
See also bnfisintnorm
.
The library syntax is GEN
bnfisnorm(GEN bnf, GEN x, long flag)
.
(
bnf,x,{
flag = 1})
bnf being the
number field data output by bnfinit
, and x
being an ideal, this
function tests whether the ideal is principal or not. The result is more
complete than a simple true/false answer and solves general discrete
logarithm problem. Assume the class group is \oplus (
Z/d_i
Z)g_i
(where the generators g_i
and their orders d_i
are respectively given by
bnf.gen
and bnf.cyc
). The routine returns a row vector [e,t]
,
where e
is a vector of exponents 0 <= e_i < d_i
, and t
is a number
field element such that
x = (t)
prod_i g_i^{e_i}.
For given g_i
(i.e. for a given bnf
), the e_i
are unique,
and t
is unique modulo units.
In particular, x
is principal if and only if e
is the zero vector. Note
that the empty vector, which is returned when the class number is 1
, is
considered to be a zero vector (of dimension 0
).
? K = bnfinit(y^2+23); ? K.cyc %2 = [3] ? K.gen %3 = [[2, 0; 0, 1]] \\ a prime ideal above 2 ? P = idealprimedec(K,3)[1]; \\ a prime ideal above 3 ? v = bnfisprincipal(K, P) %5 = [[2]~, [3/4, 1/4]~] ? idealmul(K, v[2], idealfactorback(K, K.gen, v[1])) %6 = [3 0]
[0 1] ? % == idealhnf(K, P) %7 = 1
@3The binary digits of flag mean:
@3* 1
: If set, outputs [e,t]
as explained above, otherwise returns
only e
, which is much easier to compute. The following idiom only tests
whether an ideal is principal:
is_principal(bnf, x) = !bnfisprincipal(bnf,x,0);
@3* 2
: It may not be possible to recover t
, given the initial accuracy
to which the bnf
structure was computed. In that case, a warning is
printed and t
is set equal to the empty vector []~
. If this bit is
set, increase the precision and recompute needed quantities until t
can be
computed. Warning: setting this may induce lengthy computations.
The library syntax is GEN
bnfisprincipal0(GEN bnf, GEN x, long flag)
.
Instead of the above hardcoded numerical flags, one should
rather use an or-ed combination of the symbolic flags nf_GEN
(include
generators, possibly a place holder if too difficult) and nf_FORCE
(insist on finding the generators).
(
bnf,
sfu,x)
bnf being output by
bnfinit
, sfu by bnfsunit
, gives the column vector of
exponents of x
on the fundamental S
-units and the roots of unity.
If x
is not a unit, outputs an empty vector.
The library syntax is GEN
bnfissunit(GEN bnf, GEN sfu, GEN x)
.
(
bnf,x)
bnf being the number field data
output by bnfinit
and x
being an algebraic number (type integer,
rational or polmod), this outputs the decomposition of x
on the fundamental
units and the roots of unity if x
is a unit, the empty vector otherwise.
More precisely, if u_1
,...,u_r
are the fundamental units, and zeta
is the generator of the group of roots of unity (bnf.tu
), the output is
a vector [x_1,...,x_r,x_{r+1}]
such that x = u_1^{x_1}...
u_r^{x_r}.
zeta^{x_{r+1}}
. The x_i
are integers for i <= r
and is an
integer modulo the order of zeta for i = r+1
.
Note that bnf need not contain the fundamental unit explicitly:
? setrand(1); bnf = bnfinit(x^2-x-100000); ? bnf.fu *** at top-level: bnf.fu *** ^-- *** _.fu: missing units in .fu. ? u = [119836165644250789990462835950022871665178127611316131167, \ 379554884019013781006303254896369154068336082609238336]~; ? bnfisunit(bnf, u) %3 = [-1, Mod(0, 2)]~
@3The given u
is the inverse of the fundamental unit
implicitly stored in bnf. In this case, the fundamental unit was not
computed and stored in algebraic form since the default accuracy was too
low. (Re-run the command at \g1 or higher to see such diagnostics.)
The library syntax is GEN
bnfisunit(GEN bnf, GEN x)
.
(
bnf, l)
Let bnf be attached to a number field F
and let l
be
a prime number (hereafter denoted ell for typographical reasons). Return
the logarithmic ell-class group ~{Cl}_F
of F
. This is an abelian group, conjecturally finite (known to be finite
if F/
Q is abelian). The function returns if and only if
the group is indeed finite (otherwise it would run into an infinite loop).
Let S = {
p_1,...,
p_k}
be the set of ell-adic places
(maximal ideals containing ell).
The function returns [D, G(
ell), G']
, where
@3* D
is the vector of elementary divisors for ~{Cl}_F
;
@3* G(
ell)
is the vector of elementary divisors for
the (conjecturally finite) abelian group
~{
Cl }(
ell) =
{
a =
sum_{i <= k} a_i
p_i :
deg _F
a = 0},
where the p_i
are the ell-adic places of F
; this is a
subgroup of ~{
Cl }
.
@3* G'
is the vector of elementary divisors for the ell-Sylow Cl'
of the S
-class group of F
; the group ~{
Cl }
maps to Cl'
with a simple co-kernel.
The library syntax is GEN
bnflog(GEN bnf, GEN l)
.
(
nf, A, l)
Let nf be the number field data output by nfinit
,
attached to the field F
, and let l
be a prime number (hereafter
denoted ell). The
ell-adified group of id\`{e}les of F
quotiented by
the group of logarithmic units is identified to the ell-group
of logarithmic divisors \oplus
Z_
ell [
p]
, generated by the
maximal ideals of F
.
The degree map deg _F
is additive with values in Z_
ell,
defined by deg _F
p = ~{f}_{
p}
deg _
ell p
,
where the integer ~{f}
is as in bnflogef
and deg _
ell p
is log _
ell p
for p !=
ell, log _
ell (1 +
ell)
for
p =
ell != 2
and log _
ell (1 + 2^2)
for p =
ell = 2
.
Let A =
prod p^{n_{
p}}
be an ideal and let ~{A} =
sum n_
p [
p]
be the attached logarithmic divisor. Return the
exponential of the ell-adic logarithmic degree deg _F A
, which is a
natural number.
The library syntax is GEN
bnflogdegree(GEN nf, GEN A, GEN l)
.
(
nf,
pr)
Let F
be a number field represented by the nf structure,
and let pr be a prid
structure attached to the
maximal ideal p / p
. Return
[~{e}(F_
p /
Q_p), ~{f}(F_
p /
Q_p)]
the logarithmic ramification and residue degrees. Let Q_p^c/
Q_p
be the
cyclotomic Z_p
-extension, then
~{e} = [F_
p : F_
p cap Q_p^c]
~{f} = [F_
p cap Q_p^c :
Q_p]
. Note that
~{e}~{f} = e(
p/p) f(
p/p)
, where e,f
denote the
usual ramification and residue degrees.
? F = nfinit(y^6 - 3*y^5 + 5*y^3 - 3*y + 1); ? bnflogef(F, idealprimedec(F,2)[1]) %2 = [6, 1] ? bnflogef(F, idealprimedec(F,5)[1]) %3 = [1, 2]
The library syntax is GEN
bnflogef(GEN nf, GEN pr)
.
(
bnf)
bnf being as output by
bnfinit
, computes the narrow class group of bnf. The output is
a 3-component row vector v
analogous to the corresponding class group
component bnf.clgp
: the first component
is the narrow class number v.no
, the second component is a vector
containing the SNF cyclic components v.cyc
of
the narrow class group, and the third is a vector giving the generators of
the corresponding v.gen
cyclic groups. Note that this function is a
special case of bnrinit
; the bnf need not contain fundamental
units.
The library syntax is GEN
buchnarrow(GEN bnf)
.
(
bnf)
bnf being as output by
bnfinit
, this computes an r_1 x (r_1+r_2-1)
matrix having +-1
components, giving the signs of the real embeddings of the fundamental units.
The following functions compute generators for the totally positive units:
/* exponents of totally positive units generators on bnf.tufu */ tpuexpo(bnf)= { my(K, S = bnfsignunit(bnf), [m,n] = matsize(S)); \\ m = bnf.r1, n = r1+r2-1 S = matrix(m,n, i,j, if (S[i,j] < 0, 1,0)); S = concat(vectorv(m,i,1), S); \\ add sign(-1) K = matker(S * Mod(1,2)); if (K, mathnfmodid(lift(K), 2), 2*matid(n+1)) }
/* totally positive fundamental units */ tpu(bnf)= { my(ex = tpuexpo(bnf)[,2..-1]); \\ remove ex[,1], corresponds to 1 or -1 vector(#ex, i, nffactorback(bnf, bnf.tufu, ex[,i])); }
The library syntax is GEN
signunits(GEN bnf)
.
(
bnf,S)
Computes the fundamental S
-units of the
number field bnf (output by bnfinit
), where S
is a list of
prime ideals (output by idealprimedec
). The output is a vector v
with
6 components.
v[1]
gives a minimal system of (integral) generators of the S
-unit group
modulo the unit group.
v[2]
contains technical data needed by bnfissunit
.
v[3]
is an empty vector (used to give the logarithmic embeddings of the
generators in v[1]
in version 2.0.16).
v[4]
is the S
-regulator (this is the product of the regulator, the
determinant of v[2]
and the natural logarithms of the norms of the ideals
in S
).
v[5]
gives the S
-class group structure, in the usual format
(a row vector whose three components give in order the S
-class number,
the cyclic components and the generators).
v[6]
is a copy of S
.
The library syntax is GEN
bnfsunit(GEN bnf, GEN S, long prec)
.
(
bnr, {H}, {
flag = 0})
Let bnr be the number field data output by bnrinit(,,1)
and
H be a square matrix defining a congruence subgroup of the
ray class group corresponding to bnr (the trivial congruence subgroup
if omitted). This function returns, for each character chi of the ray
class group which is trivial on H
, the value at s = 1
(or s = 0
) of the
abelian L
-function attached to chi. For the value at s = 0
, the
function returns in fact for each chi a vector [r_
chi, c_
chi]
where
L(s,
chi) = c.s^r + O(s^{r + 1})
@3near 0
.
The argument flag is optional, its binary digits
mean 1: compute at s = 0
if unset or s = 1
if set, 2: compute the
primitive L
-function attached to chi if unset or the L
-function
with Euler factors at prime ideals dividing the modulus of bnr removed
if set (that is L_S(s,
chi)
, where S
is the
set of infinite places of the number field together with the finite prime
ideals dividing the modulus of bnr), 3: return also the character if
set.
K = bnfinit(x^2-229); bnr = bnrinit(K,1,1); bnrL1(bnr)
returns the order and the first non-zero term of L(s,
chi)
at s = 0
where chi runs through the characters of the class group of
K =
Q(
sqrt {229})
. Then
bnr2 = bnrinit(K,2,1); bnrL1(bnr2,,2)
returns the order and the first non-zero terms of L_S(s,
chi)
at s = 0
where chi runs through the characters of the class group of K
and S
is
the set of infinite places of K
together with the finite prime 2
. Note
that the ray class group modulo 2
is in fact the class group, so
bnrL1(bnr2,0)
returns the same answer as bnrL1(bnr,0)
.
This function will fail with the message
*** bnrL1: overflow in zeta_get_N0 [need too many primes].
@3if the approximate functional equation requires us to sum
too many terms (if the discriminant of K
is too large).
The library syntax is GEN
bnrL1(GEN bnr, GEN H = NULL, long flag, long prec)
.
(
bnr,g,{v})
Returns all characters chi on bnr.clgp
such that
chi(g_i) = e(v_i)
, where e(x) =
exp (2i
pi x)
. If v
is omitted,
returns all characters that are trivial on the g_i
. Else the vectors g
and v
must have the same length, the g_i
must be ideals in any form, and
each v_i
is a rational number whose denominator must divide the order of
g_i
in the ray class group. For convenience, the vector of the g_i
can be replaced by a matrix whose columns give their discrete logarithm,
as given by bnrisprincipal
; this allows to specify abstractly a
subgroup of the ray class group.
? bnr = bnrinit(bnfinit(x), [160,[1]], 1); /* (Z/160Z)^* */ ? bnr.cyc %2 = [8, 4, 2] ? g = bnr.gen; ? bnrchar(bnr, g, [1/2,0,0]) %4 = [[4, 0, 0]] \\ a unique character ? bnrchar(bnr, [g[1],g[3]]) \\ all characters trivial on g[1] and g[3] %5 = [[0, 1, 0], [0, 2, 0], [0, 3, 0], [0, 0, 0]] ? bnrchar(bnr, [1,0,0;0,1,0;0,0,2]) %6 = [[0, 0, 1], [0, 0, 0]] \\ characters trivial on given subgroup
The library syntax is GEN
bnrchar(GEN bnr, GEN g, GEN v = NULL)
.
(A,{B},{C})
Let A
, B
, C
define a class field L
over a ground field K
(of type [
bnr]
,
[
bnr,
subgroup]
,
or [
bnf,
modulus]
,
or [
bnf,
modulus,
subgroup]
,
Label se:CFT); this function returns the relative degree [L:K]
.
In particular if A
is a bnf (with units), and B
a modulus,
this function returns the corresponding ray class number modulo B
.
One can input the attached bid (with generators if the subgroup
C
is non trivial) for B
instead of the module itself, saving some time.
This function is faster than bnrinit
and should be used if only the
ray class number is desired. See bnrclassnolist
if you need ray class
numbers for all moduli less than some bound.
The library syntax is GEN
bnrclassno0(GEN A, GEN B = NULL, GEN C = NULL)
.
Also available is
GEN
bnrclassno(GEN bnf,GEN f)
to compute the ray class number
modulo f
.
(
bnf,
list)
bnf being as
output by bnfinit
, and list being a list of moduli (with units) as
output by ideallist
or ideallistarch
, outputs the list of the
class numbers of the corresponding ray class groups. To compute a single
class number, bnrclassno
is more efficient.
? bnf = bnfinit(x^2 - 2); ? L = ideallist(bnf, 100, 2); ? H = bnrclassnolist(bnf, L); ? H[98] %4 = [1, 3, 1] ? l = L[1][98]; ids = vector(#l, i, l[i].mod[1]) %5 = [[98, 88; 0, 1], [14, 0; 0, 7], [98, 10; 0, 1]]
The weird l[i].mod[1]
, is the first component of l[i].mod
, i.e.
the finite part of the conductor. (This is cosmetic: since by construction
the Archimedean part is trivial, I do not want to see it). This tells us that
the ray class groups modulo the ideals of norm 98 (printed as %5
) have
respectively order 1
, 3
and 1
. Indeed, we may check directly:
? bnrclassno(bnf, ids[2]) %6 = 3
The library syntax is GEN
bnrclassnolist(GEN bnf, GEN list)
.
(A,{B},{C},{
flag = 0})
Conductor f
of the subfield of a ray class field as defined by [A,B,C]
(of type [
bnr]
,
[
bnr,
subgroup]
,
[
bnf,
modulus]
or
[
bnf,
modulus,
subgroup]
,
Label se:CFT)
If flag = 0
, returns f
.
If flag = 1
, returns [f, Cl_f, H]
, where Cl_f
is the ray class group
modulo f
, as a finite abelian group; finally H
is the subgroup of Cl_f
defining the extension.
If flag = 2
, returns [f,
bnr(f), H]
, as above except Cl_f
is
replaced by a bnr
structure, as output by bnrinit(,f,1)
.
In place of a subgroup H
, this function also accepts a character
chi
= (a_j)
, expressed as usual in terms of the generators
bnr.gen
: chi(g_j) =
exp (2i
pi a_j / d_j)
, where g_j
has
order d_j = bnr.cyc[j]
. In which case, the function returns
respectively
If flag = 0
, the conductor f
of Ker
chi.
If flag = 1
, [f, Cl_f,
chi_f]
, where chi_f
is chi expressed
on the minimal ray class group, whose modulus is the conductor.
If flag = 2
, [f,
bnr(f),
chi_f]
.
The library syntax is GEN
bnrconductor0(GEN A, GEN B = NULL, GEN C = NULL, long flag)
.
Also available is GEN
bnrconductor(GEN bnr, GEN H, long flag)
(
bnr,
chi)
This function is obsolete, use bnrconductor.
The library syntax is GEN
bnrconductorofchar(GEN bnr, GEN chi)
.
(A,{B},{C},{
flag = 0})
A
, B
, C
defining a class field L
over a ground field K
(of type [
bnr]
,
[
bnr,
subgroup]
,
[
bnr,
character]
,
[
bnf,
modulus]
or
[
bnf,
modulus,
subgroup]
,
Label se:CFT), outputs data [N,r_1,D]
giving the discriminant and
signature of L
, depending on the binary digits of flag:
@3* 1: if this bit is unset, output absolute data related to L/
Q:
N
is the absolute degree [L:
Q]
, r_1
the number of real places of L
,
and D
the discriminant of L/
Q. Otherwise, output relative data for L/K
:
N
is the relative degree [L:K]
, r_1
is the number of real places of K
unramified in L
(so that the number of real places of L
is equal to r_1
times N
), and D
is the relative discriminant ideal of L/K
.
@3* 2: if this bit is set and if the modulus is not the conductor of L
,
only return 0.
The library syntax is GEN
bnrdisc0(GEN A, GEN B = NULL, GEN C = NULL, long flag)
.
bnrdisclist(
bnf,
bound,{
arch})
bnf being as output by bnfinit
(with units), computes a
list of discriminants of Abelian extensions of the number field by increasing
modulus norm up to bound bound. The ramified Archimedean places are
given by arch; all possible values are taken if arch is omitted.
The alternative syntax bnrdisclist(
bnf,
list)
is
supported, where list is as output by ideallist
or
ideallistarch
(with units), in which case arch is disregarded.
The output v
is a vector of vectors, where v[i][j]
is understood to be in
fact V[2^{15}(i-1)+j]
of a unique big vector V
. (This awkward scheme
allows for larger vectors than could be otherwise represented.)
V[k]
is itself a vector W
, whose length is the number of ideals of norm
k
. We consider first the case where arch was specified. Each
component of W
corresponds to an ideal m
of norm k
, and
gives invariants attached to the ray class field L
of bnf of
conductor [m,
arch]
. Namely, each contains a vector [m,d,r,D]
with
the following meaning: m
is the prime ideal factorization of the modulus,
d = [L:
Q]
is the absolute degree of L
, r
is the number of real places
of L
, and D
is the factorization of its absolute discriminant. We set d
= r = D = 0
if m
is not the finite part of a conductor.
If arch was omitted, all t = 2^{r_1}
possible values are taken and a
component of W
has the form [m, [[d_1,r_1,D_1],..., [d_t,r_t,D_t]]]
,
where m
is the finite part of the conductor as above, and
[d_i,r_i,D_i]
are the invariants of the ray class field of conductor
[m,v_i]
, where v_i
is the i
-th Archimedean component, ordered by
inverse lexicographic order; so v_1 = [0,...,0]
, v_2 = [1,0...,0]
,
etc. Again, we set d_i = r_i = D_i = 0
if [m,v_i]
is not a conductor.
Finally, each prime ideal pr = [p,
alpha,e,f,
beta]
in the prime
factorization m
is coded as the integer p.n^2+(f-1).n+(j-1)
,
where n
is the degree of the base field and j
is such that
pr = idealprimedec(
nf,p)[j]
.
@3m
can be decoded using bnfdecodemodule
.
Note that to compute such data for a single field, either bnrclassno
or bnrdisc
is more efficient.
The library syntax is GEN
bnrdisclist0(GEN bnf, GEN bound, GEN arch = NULL)
.
(
bnr,
mat, H)
Apply the automorphism given by its matrix mat to the congruence
subgroup H
given as a HNF matrix.
The matrix mat can be computed with bnrgaloismatrix
.
The library syntax is GEN
bnrgaloisapply(GEN bnr, GEN mat, GEN H)
.
(
bnr,
aut)
Return the matrix of the action of the automorphism aut of the base
field bnf.nf
on the generators of the ray class field bnr.gen
.
aut can be given as a polynomial, an algebraic number, or a vector of
automorphisms or a Galois group as output by galoisinit
, in which case a
vector of matrices is returned (in the later case, only for the generators
aut.gen
).
See bnrisgalois
for an example.
The library syntax is GEN
bnrgaloismatrix(GEN bnr, GEN aut)
.
When aut
is a polynomial or an algebraic number,
GEN
bnrautmatrix(GEN bnr, GEN aut)
is available.
(
bnf,f,{
flag = 0})
bnf is as
output by bnfinit
(including fundamental units), f
is a modulus,
initializes data linked to the ray class group structure corresponding to
this module, a so-called bnr
structure. One can input the attached
bid with generators for f
instead of the module itself, saving some
time. (As in idealstar
, the finite part of the conductor may be given
by a factorization into prime ideals, as produced by idealfactor
.)
The following member functions are available
on the result: .bnf
is the underlying bnf,
.mod
the modulus, .bid
the bid
structure attached to the
modulus; finally, .clgp
, .no
, .cyc
, .gen
refer to the
ray class group (as a finite abelian group), its cardinality, its elementary
divisors, its generators (only computed if flag = 1
).
The last group of functions are different from the members of the underlying
bnf, which refer to the class group; use bnr.bnf.
xxx
to access these, e.g. bnr.bnf.cyc
to get the cyclic decomposition
of the class group.
They are also different from the members of the underlying bid, which
refer to (
Z_K/f)^*
; use bnr.bid.
xxx to access these,
e.g. bnr.bid.no
to get phi(f)
.
If flag = 0
(default), the generators of the ray class group are not computed,
which saves time. Hence bnr.gen
would produce an error.
If flag = 1
, as the default, except that generators are computed.
The library syntax is GEN
bnrinit0(GEN bnf, GEN f, long flag)
.
Instead the above hardcoded numerical flags, one should rather use
GEN
Buchray(GEN bnf, GEN module, long flag)
where flag is an or-ed combination of nf_GEN
(include generators)
and nf_INIT
(if omitted, return just the cardinality of the ray class
group and its structure), possibly 0.
(A,{B},{C})
Fast variant of bnrconductor
(A,B,C)
; A
, B
, C
represent
an extension of the base field, given by class field theory
(see Label se:CFT). Outputs 1 if this modulus is the conductor, and 0
otherwise. This is slightly faster than bnrconductor
when the
character or subgroup is not primitive.
The library syntax is long
bnrisconductor0(GEN A, GEN B = NULL, GEN C = NULL)
.
(
bnr,
gal, H)
Check whether the class field attached to the subgroup H
is Galois
over the subfield of bnr.nf
fixed by the group gal, which can be
given as output by galoisinit
, or as a matrix or a vector of matrices as
output by bnrgaloismatrix
, the second option being preferable, since it
saves the recomputation of the matrices. Note: The function assumes that the
ray class field attached to bnr is Galois, which is not checked.
In the following example, we lists the congruence subgroups of subextension of
degree at most 3
of the ray class field of conductor 9
which are Galois
over the rationals.
K=bnfinit(a^4-3*a^2+253009); G=galoisinit(K); B=bnrinit(K,9,1); L1=[H|H<-subgrouplist(B,3), bnrisgalois(B,G,H)] ## M=bnrgaloismatrix(B,G) L2=[H|H<-subgrouplist(B,3), bnrisgalois(B,M,H)] ##
The second computation is much faster since bnrgaloismatrix(B,G)
is
computed only once.
The library syntax is long
bnrisgalois(GEN bnr, GEN gal, GEN H)
.
(
bnr,x,{
flag = 1})
bnr being the
number field data which is output by bnrinit
(,,1)
and x
being an
ideal in any form, outputs the components of x
on the ray class group
generators in a way similar to bnfisprincipal
. That is a 2-component
vector v
where v[1]
is the vector of components of x
on the ray class
group generators, v[2]
gives on the integral basis an element alpha such
that x =
alphaprod_ig_i^{x_i}
.
If flag = 0
, outputs only v_1
. In that case, bnr need not contain the
ray class group generators, i.e. it may be created with bnrinit
(,,0)
If x
is not coprime to the modulus of bnr the result is undefined.
The library syntax is GEN
bnrisprincipal(GEN bnr, GEN x, long flag)
.
Instead of hardcoded numerical flags, one should rather
use
GEN
isprincipalray(GEN bnr, GEN x)
for flag = 0
, and if you
want generators:
bnrisprincipal(bnr, x, nf_GEN)
(
bnr,
chi,{
flag = 0})
If chi =
chi is a
character over bnr, not necessarily primitive, let
L(s,
chi) =
sum_{id}
chi(id) N(id)^{-s}
be the attached
Artin L-function. Returns the so-called Artin root number, i.e. the
complex number W(
chi)
of modulus 1 such that
Lambda(1-s,
chi) = W(
chi)
Lambda(s,\overline{
chi})
@3where Lambda(s,
chi) = A(
chi)^{s/2}
gamma_
chi(s) L(s,
chi)
is
the enlarged L-function attached to L
.
The generators of the ray class group are needed, and you can set flag = 1
if
the character is known to be primitive. Example:
bnf = bnfinit(x^2 - x - 57); bnr = bnrinit(bnf, [7,[1,1]], 1); bnrrootnumber(bnr, [2,1])
returns the root number of the character chi of
Cl _{7 oo _1 oo _2}(
Q(
sqrt {229}))
defined by chi(g_1^ag_2^b)
=
zeta_1^{2a}
zeta_2^b
. Here g_1, g_2
are the generators of the
ray-class group given by bnr.gen
and zeta_1 = e^{2i
pi/N_1},
zeta_2 = e^{2i
pi/N_2}
where N_1, N_2
are the orders of g_1
and
g_2
respectively (N_1 = 6
and N_2 = 3
as bnr.cyc
readily tells us).
The library syntax is GEN
bnrrootnumber(GEN bnr, GEN chi, long flag, long prec)
.
bnrstark(
bnr,{
subgroup})
bnr being as output by bnrinit(,,1)
, finds a relative equation
for the class field corresponding to the modulus in bnr and the given
congruence subgroup (as usual, omit subgroup if you want the whole ray
class group).
The main variable of bnr must not be x
, and the ground field and the
class field must be totally real. When the base field is Q, the vastly
simpler galoissubcyclo
is used instead. Here is an example:
bnf = bnfinit(y^2 - 3); bnr = bnrinit(bnf, 5, 1); bnrstark(bnr)
returns the ray class field of Q(
sqrt {3})
modulo 5
. Usually, one wants
to apply to the result one of
rnfpolredabs(bnf, pol, 16) \\ compute a reduced relative polynomial rnfpolredabs(bnf, pol, 16 + 2) \\ compute a reduced absolute polynomial
The routine uses Stark units and needs to find a suitable auxiliary
conductor, which may not exist when the class field is not cyclic over the
base. In this case bnrstark
is allowed to return a vector of
polynomials defining independent relative extensions, whose compositum
is the requested class field. It was decided that it was more useful
to keep the extra information thus made available, hence the user has to take
the compositum herself.
Even if it exists, the auxiliary conductor may be so large that later
computations become unfeasible. (And of course, Stark's conjecture may simply
be wrong.) In case of difficulties, try rnfkummer
:
? bnr = bnrinit(bnfinit(y^8-12*y^6+36*y^4-36*y^2+9,1), 2, 1); ? bnrstark(bnr) *** at top-level: bnrstark(bnr) *** ^------------- *** bnrstark: need 3919350809720744 coefficients in initzeta. *** Computation impossible. ? lift( rnfkummer(bnr) ) time = 24 ms. %2 = x^2 + (1/3*y^6 - 11/3*y^4 + 8*y^2 - 5)
The library syntax is GEN
bnrstark(GEN bnr, GEN subgroup = NULL, long prec)
.
(
nf,b)
Gives as a vector the first b
coefficients of the Dedekind zeta function of the number field nf
considered as a Dirichlet series.
The library syntax is GEN
dirzetak(GEN nf, GEN b)
.
(x,t)
This function is obsolete, use nffactor
.
factorization of the univariate polynomial x
over the number field defined by the (univariate) polynomial t
. x
may
have coefficients in Q or in the number field. The algorithm reduces to
factorization over Q (Trager's trick). The direct approach of
nffactor
, which uses van Hoeij's method in a relative setting, is
in general faster.
The main variable of t
must be of lower priority than that of x
(see Label se:priority). However if non-rational number field elements
occur (as polmods or polynomials) as coefficients of x
, the variable of
these polmods must be the same as the main variable of t
. For
example
? factornf(x^2 + Mod(y, y^2+1), y^2+1); ? factornf(x^2 + y, y^2+1); \\ these two are OK ? factornf(x^2 + Mod(z,z^2+1), y^2+1) *** at top-level: factornf(x^2+Mod(z,z *** ^-------------------- *** factornf: inconsistent data in rnf function. ? factornf(x^2 + z, y^2+1) *** at top-level: factornf(x^2+z,y^2+1 *** ^-------------------- *** factornf: incorrect variable in rnf function.
The library syntax is GEN
polfnf(GEN x, GEN t)
.
(
gal,{
flag})
gal being be a Galois group as output by galoisinit
,
export the underlying permutation group as a string suitable
for (no flags or flag = 0
) GAP or (flag = 1
) Magma. The following example
compute the index of the underlying abstract group in the GAP library:
? G = galoisinit(x^6+108); ? s = galoisexport(G) %2 = "Group((1, 2, 3)(4, 5, 6), (1, 4)(2, 6)(3, 5))" ? extern("echo \"IdGroup("s");\" | gap -q") %3 = [6, 1] ? galoisidentify(G) %4 = [6, 1]
This command also accepts subgroups returned by galoissubgroups
.
To import a GAP permutation into gp (for galoissubfields
for
instance), the following GAP function may be useful:
PermToGP := function(p, n) return Permuted([1..n],p); end;
gap> p:= (1,26)(2,5)(3,17)(4,32)(6,9)(7,11)(8,24)(10,13)(12,15)(14,27) (16,22)(18,28)(19,20)(21,29)(23,31)(25,30) gap> PermToGP(p,32); [ 26, 5, 17, 32, 2, 9, 11, 24, 6, 13, 7, 15, 10, 27, 12, 22, 3, 28, 20, 19, 29, 16, 31, 8, 30, 1, 14, 18, 21, 25, 23, 4 ]
The library syntax is GEN
galoisexport(GEN gal, long flag)
.
(
gal,
perm,{
flag},{v = y})
gal being be a Galois group as output by galoisinit
and
perm an element of gal.group
, a vector of such elements
or a subgroup of gal as returned by galoissubgroups,
computes the fixed field of gal by the automorphism defined by the
permutations perm of the roots gal.roots
. P
is guaranteed to
be squarefree modulo gal.p
.
If no flags or flag = 0
, output format is the same as for nfsubfield
,
returning [P,x]
such that P
is a polynomial defining the fixed field, and
x
is a root of P
expressed as a polmod in gal.pol
.
If flag = 1
return only the polynomial P
.
If flag = 2
return [P,x,F]
where P
and x
are as above and F
is the
factorization of gal.pol
over the field defined by P
, where
variable v
(y
by default) stands for a root of P
. The priority of v
must be less than the priority of the variable of gal.pol
(see
Label se:priority). Example:
? G = galoisinit(x^4+1); ? galoisfixedfield(G,G.group[2],2) %2 = [x^2 + 2, Mod(x^3 + x, x^4 + 1), [x^2 - y*x - 1, x^2 + y*x - 1]]
computes the factorization x^4+1 = (x^2-
sqrt {-2}x-1)(x^2+
sqrt {-2}x-1)
The library syntax is GEN
galoisfixedfield(GEN gal, GEN perm, long flag, long v = -1)
where v
is a variable number.
(a,{b},{s})
Query the galpol package for a polynomial with Galois group isomorphic to
GAP4(a,b), totally real if s = 1
(default) and totally complex if s = 2
. The
output is a vector [pol
, den
] where
@3* pol
is the polynomial of degree a
@3* den
is the denominator of nfgaloisconj(pol)
.
Pass it as an optional argument to galoisinit
or nfgaloisconj
to
speed them up:
? [pol,den] = galoisgetpol(64,4,1); ? G = galoisinit(pol); time = 352ms ? galoisinit(pol, den); \\ passing 'den' speeds up the computation time = 264ms ? % == %` %4 = 1 \\ same answer
If b
and s
are omitted, return the number of isomorphism classes of
groups of order a
.
The library syntax is GEN
galoisgetpol(long a, long b, long s)
.
Also available is GEN
galoisnbpol(long a)
when b
and s
are omitted.
(
gal)
gal being be a Galois group as output by galoisinit
,
output the isomorphism class of the underlying abstract group as a
two-components vector [o,i]
, where o
is the group order, and i
is the
group index in the GAP4 Small Group library, by Hans Ulrich Besche, Bettina
Eick and Eamonn O'Brien.
This command also accepts subgroups returned by galoissubgroups
.
The current implementation is limited to degree less or equal to 127
.
Some larger ``easy'' orders are also supported.
The output is similar to the output of the function IdGroup
in GAP4.
Note that GAP4 IdGroup
handles all groups of order less than 2000
except 1024
, so you can use galoisexport
and GAP4 to identify large
Galois groups.
The library syntax is GEN
galoisidentify(GEN gal)
.
galoisinit(
pol,{
den})
Computes the Galois group
and all necessary information for computing the fixed fields of the
Galois extension K/
Q where K
is the number field defined by
pol (monic irreducible polynomial in Z[X]
or
a number field as output by nfinit
). The extension K/
Q must be
Galois with Galois group ``weakly'' super-solvable, see below;
returns 0 otherwise. Hence this permits to quickly check whether a polynomial
of order strictly less than 36
is Galois or not.
The algorithm used is an improved version of the paper ``An efficient algorithm for the computation of Galois automorphisms'', Bill Allombert, Math. Comp, vol. 73, 245, 2001, pp. 359--375.
A group G
is said to be ``weakly'' super-solvable if there exists a
normal series
{1} = H_0 \triangleleft H_1 \triangleleft...\triangleleft H_{n-1}
\triangleleft H_n
such that each H_i
is normal in G
and for i < n
, each quotient group
H_{i+1}/H_i
is cyclic, and either H_n = G
(then G
is super-solvable) or
G/H_n
is isomorphic to either A_4
or S_4
.
In practice, almost all small groups are WKSS, the exceptions having order
36(1 exception), 48(2), 56(1), 60(1), 72(5), 75(1), 80(1), 96(10) and >=
108
.
This function is a prerequisite for most of the galois
xxx
routines.
For instance:
P = x^6 + 108; G = galoisinit(P); L = galoissubgroups(G); vector(#L, i, galoisisabelian(L[i],1)) vector(#L, i, galoisidentify(L[i]))
The output is an 8-component vector gal.
gal[1]
contains the polynomial pol
(gal.pol
).
gal[2]
is a three-components vector [p,e,q]
where p
is a
prime number (gal.p
) such that pol totally split
modulo p
, e
is an integer and q = p^e
(gal.mod
) is the
modulus of the roots in gal.roots
.
gal[3]
is a vector L
containing the p
-adic roots of
pol as integers implicitly modulo gal.mod
.
(gal.roots
).
gal[4]
is the inverse of the Vandermonde matrix of the
p
-adic roots of pol, multiplied by gal[5]
.
gal[5]
is a multiple of the least common denominator of the
automorphisms expressed as polynomial in a root of pol.
gal[6]
is the Galois group G
expressed as a vector of
permutations of L
(gal.group
).
gal[7]
is a generating subset S = [s_1,...,s_g]
of G
expressed as a vector of permutations of L
(gal.gen
).
gal[8]
contains the relative orders [o_1,...,o_g]
of
the generators of S
(gal.orders
).
Let H_n
be as above, we have the following properties:
* if G/H_n ~ A_4
then [o_1,...,o_g]
ends by
[2,2,3]
.
* if G/H_n ~ S_4
then [o_1,...,o_g]
ends by
[2,2,3,2]
.
* for 1 <= i <= g
the subgroup of G
generated by
[s_1,...,s_g]
is normal, with the exception of i = g-2
in the
A_4
case and of i = g-3
in the S_A
case.
* the relative order o_i
of s_i
is its order in the
quotient group G/<s_1,...,s_{i-1}>
, with the same
exceptions.
* for any x\in G
there exists a unique family
[e_1,...,e_g]
such that (no exceptions):
-- for 1 <= i <= g
we have 0 <= e_i < o_i
-- x = g_1^{e_1}g_2^{e_2}...g_n^{e_n}
If present den
must be a suitable value for gal[5]
.
The library syntax is GEN
galoisinit(GEN pol, GEN den = NULL)
.
(
gal,{
flag = 0})
gal being as output by galoisinit
, return 0
if
gal is not an abelian group, and the HNF matrix of gal over
gal.gen
if fl = 0
, 1
if fl = 1
.
This command also accepts subgroups returned by galoissubgroups
.
The library syntax is GEN
galoisisabelian(GEN gal, long flag)
.
(
gal,
subgrp)
gal being as output by galoisinit
, and subgrp a subgroup
of gal as output by galoissubgroups
,return 1
if subgrp is a
normal subgroup of gal, else return 0.
This command also accepts subgroups returned by galoissubgroups
.
The library syntax is long
galoisisnormal(GEN gal, GEN subgrp)
.
(
gal,
perm)
gal being a
Galois group as output by galoisinit
and perm a element of
gal.group
, return the polynomial defining the Galois
automorphism, as output by nfgaloisconj
, attached to the
permutation perm of the roots gal.roots
. perm can
also be a vector or matrix, in this case, galoispermtopol
is
applied to all components recursively.
@3Note that
G = galoisinit(pol); galoispermtopol(G, G[6])~
is equivalent to nfgaloisconj(pol)
, if degree of pol is greater
or equal to 2
.
The library syntax is GEN
galoispermtopol(GEN gal, GEN perm)
.
galoissubcyclo(N,H,{
fl = 0},{v})
Computes the subextension
of Q(
zeta_n)
fixed by the subgroup H \subset (
Z/n
Z)^*
. By the
Kronecker-Weber theorem, all abelian number fields can be generated in this
way (uniquely if n
is taken to be minimal).
@3The pair (n, H)
is deduced from the parameters (N, H)
as follows
@3* N
an integer: then n = N
; H
is a generator, i.e. an
integer or an integer modulo n
; or a vector of generators.
@3* N
the output of znstar(n)
. H
as in the first case
above, or a matrix, taken to be a HNF left divisor of the SNF for (
Z/n
Z)^*
(of type N.cyc
), giving the generators of H
in terms of N.gen
.
@3* N
the output of bnrinit(bnfinit(y), m, 1)
where m
is a
module. H
as in the first case, or a matrix taken to be a HNF left
divisor of the SNF for the ray class group modulo m
(of type N.cyc
), giving the generators of H
in terms of N.gen
.
In this last case, beware that H
is understood relatively to N
; in
particular, if the infinite place does not divide the module, e.g if m
is
an integer, then it is not a subgroup of (
Z/n
Z)^*
, but of its quotient by
{+- 1}
.
If fl = 0
, compute a polynomial (in the variable v) defining
the subfield of Q(
zeta_n)
fixed by the subgroup H of (
Z/n
Z)^*
.
If fl = 1
, compute only the conductor of the abelian extension, as a module.
If fl = 2
, output [pol, N]
, where pol
is the polynomial as output when
fl = 0
and N
the conductor as output when fl = 1
.
The following function can be used to compute all subfields of
Q(
zeta_n)
(of exact degree d
, if d
is set):
polsubcyclo(n, d = -1)= { my(bnr,L,IndexBound); IndexBound = if (d < 0, n, [d]); bnr = bnrinit(bnfinit(y), [n,[1]], 1); L = subgrouplist(bnr, IndexBound, 1); vector(#L,i, galoissubcyclo(bnr,L[i])); }
Setting L = subgrouplist(bnr, IndexBound)
would produce subfields of exact
conductor n oo
.
The library syntax is GEN
galoissubcyclo(GEN N, GEN H = NULL, long fl, long v = -1)
where v
is a variable number.
(G,{
flag = 0},{v})
Outputs all the subfields of the Galois group G, as a vector.
This works by applying galoisfixedfield
to all subgroups. The meaning of
flag is the same as for galoisfixedfield
.
The library syntax is GEN
galoissubfields(GEN G, long flag, long v = -1)
where v
is a variable number.
(G)
Outputs all the subgroups of the Galois group gal
. A subgroup is a
vector [gen, orders], with the same meaning
as for gal.gen
and gal.orders
. Hence gen is a vector of
permutations generating the subgroup, and orders is the relatives
orders of the generators. The cardinality of a subgroup is the product of the
relative orders. Such subgroup can be used instead of a Galois group in the
following command: galoisisabelian
, galoissubgroups
,
galoisexport
and galoisidentify
.
To get the subfield fixed by a subgroup sub of gal, use
galoisfixedfield(gal,sub[1])
The library syntax is GEN
galoissubgroups(GEN G)
.
(
nf,x,y)
Sum of the two ideals x
and y
in the number field nf. The
result is given in HNF.
? K = nfinit(x^2 + 1); ? a = idealadd(K, 2, x + 1) \\ ideal generated by 2 and 1+I %2 = [2 1]
[0 1] ? pr = idealprimedec(K, 5)[1]; \\ a prime ideal above 5 ? idealadd(K, a, pr) \\ coprime, as expected %4 = [1 0]
[0 1]
@3 This function cannot be used to add arbitrary Z-modules, since it assumes that its arguments are ideals:
? b = Mat([1,0]~); ? idealadd(K, b, b) \\ only square t_MATs represent ideals *** idealadd: non-square t_MAT in idealtyp. ? c = [2, 0; 2, 0]; idealadd(K, c, c) \\ non-sense %6 = [2 0]
[0 2] ? d = [1, 0; 0, 2]; idealadd(K, d, d) \\ non-sense %7 = [1 0]
[0 1]
@3In the last two examples, we get wrong results since the
matrices c
and d
do not correspond to an ideal: the Z-span of their
columns (as usual interpreted as coordinates with respect to the integer basis
K.zk
) is not an O_K
-module. To add arbitrary Z-modules generated
by the columns of matrices A
and B
, use mathnf(concat(A,B))
.
The library syntax is GEN
idealadd(GEN nf, GEN x, GEN y)
.
(
nf,x,{y})
x
and y
being two co-prime
integral ideals (given in any form), this gives a two-component row vector
[a,b]
such that a\in x
, b\in y
and a+b = 1
.
The alternative syntax idealaddtoone(
nf,v)
, is supported, where
v
is a k
-component vector of ideals (given in any form) which sum to
Z_K
. This outputs a k
-component vector e
such that e[i]\in x[i]
for
1 <= i <= k
and sum_{1 <= i <= k}e[i] = 1
.
The library syntax is GEN
idealaddtoone0(GEN nf, GEN x, GEN y = NULL)
.
(
nf,x,{
flag})
If x
is a fractional ideal
(given in any form), gives an element alpha in nf such that for
all prime ideals p such that the valuation of x
at p is
non-zero, we have v_{
p}(
alpha) = v_{
p}(x)
, and
v_{
p}(
alpha) >= 0
for all other p.
The argument x
may also be given as a prime ideal factorization, as
output by idealfactor
, but allowing zero exponents.
This yields an element alpha such that for all prime ideals p
occurring in x
, v_{
p}(
alpha) = v_{
p}(x)
;
for all other prime ideals, v_{
p}(
alpha) >= 0
.
flag is deprecated (ignored), kept for backward compatibility
The library syntax is GEN
idealappr0(GEN nf, GEN x, long flag)
.
Use directly GEN
idealappr(GEN nf, GEN x)
since flag is ignored.
(
nf,x,{y})
x
being a prime ideal factorization
(i.e. a 2 by 2 matrix whose first column contains prime ideals, and the second
column integral exponents), y
a vector of elements in nf indexed by
the ideals in x
, computes an element b
such that
v_{
p}(b - y_{
p}) >= v_{
p}(x)
for all prime ideals
in x
and v_{
p}(b) >= 0
for all other p.
? K = nfinit(t^2-2); ? x = idealfactor(K, 2^2*3) %2 = [[2, [0, 1]~, 2, 1, [0, 2; 1, 0]] 4]
[ [3, [3, 0]~, 1, 2, 1] 1] ? y = [t,1]; ? idealchinese(K, x, y) %4 = [4, -3]~
The argument x
may also be of the form [x, s]
where the first component
is as above and s
is a vector of signs, with r_1
components
s_i
in {-1,0,1}
:
if sigma_i
denotes the i
-th real embedding of the number field,
the element b
returned satisfies further
s_i sign(
sigma_i(b)) >= 0
for all i
. In other words, the sign is
fixed to s_i
at the i
-th embedding whenever s_i
is non-zero.
? idealchinese(K, [x, [1,1]], y) %5 = [16, -3]~ ? idealchinese(K, [x, [-1,-1]], y) %6 = [-20, -3]~ ? idealchinese(K, [x, [1,-1]], y) %7 = [4, -3]~
If y
is omitted, return a data structure which can be used in
place of x
in later calls and allows to solve many chinese remainder
problems for a given x
more efficiently.
? C = idealchinese(K, [x, [1,1]]); ? idealchinese(K, C, y) \\ as above %9 = [16, -3]~ ? for(i=1,10^4, idealchinese(K,C,y)) \\ ... but faster ! time = 80 ms. ? for(i=1,10^4, idealchinese(K,[x,[1,1]],y)) time = 224 ms.
Finally, this structure is itself allowed in place of x
, the
new s
overriding the one already present in the structure. This allows to
initialize for different sign conditions more efficiently when the underlying
ideal factorization remains the same.
? D = idealchinese(K, [C, [1,-1]]); \\ replaces [1,1] ? idealchinese(K, D, y) %13 = [4, -3]~ ? for(i=1,10^4,idealchinese(K,[C,[1,-1]])) time = 40 ms. \\ faster than starting from scratch ? for(i=1,10^4,idealchinese(K,[x,[1,-1]])) time = 128 ms.
The library syntax is GEN
idealchinese(GEN nf, GEN x, GEN y = NULL)
.
Also available is
GEN
idealchineseinit(GEN nf, GEN x)
when y = NULL
.
(
nf,x,y)
Given two integral ideals x
and y
in the number field nf, returns a beta in the field,
such that beta.x
is an integral ideal coprime to y
.
The library syntax is GEN
idealcoprime(GEN nf, GEN x, GEN y)
.
(
nf,x,y,{
flag = 0})
Quotient x.y^{-1}
of the two ideals x
and y
in the number
field nf. The result is given in HNF.
If flag is non-zero, the quotient x.y^{-1}
is assumed to be an
integral ideal. This can be much faster when the norm of the quotient is
small even though the norms of x
and y
are large.
The library syntax is GEN
idealdiv0(GEN nf, GEN x, GEN y, long flag)
.
Also available are GEN
idealdiv(GEN nf, GEN x, GEN y)
(flag = 0
) and GEN
idealdivexact(GEN nf, GEN x, GEN y)
(flag = 1
).
(
nf,x)
Factors into prime ideal powers the
ideal x
in the number field nf. The output format is similar to the
factor
function, and the prime ideals are represented in the form
output by the idealprimedec
function.
The library syntax is GEN
idealfactor(GEN nf, GEN x)
.
(
nf,f,{e},{
flag = 0})
Gives back the ideal corresponding to a factorization. The integer 1
corresponds to the empty factorization.
If e
is present, e
and f
must be vectors of the same length (e
being
integral), and the corresponding factorization is the product of the
f[i]^{e[i]}
.
If not, and f
is vector, it is understood as in the preceding case with e
a vector of 1s: we return the product of the f[i]
. Finally, f
can be a
regular factorization, as produced by idealfactor
.
? nf = nfinit(y^2+1); idealfactor(nf, 4 + 2*y) %1 = [[2, [1, 1]~, 2, 1, [1, 1]~] 2]
[[5, [2, 1]~, 1, 1, [-2, 1]~] 1]
? idealfactorback(nf, %) %2 = [10 4]
[0 2]
? f = %1[,1]; e = %1[,2]; idealfactorback(nf, f, e) %3 = [10 4]
[0 2]
? % == idealhnf(nf, 4 + 2*y) %4 = 1
If flag
is non-zero, perform ideal reductions (idealred
) along the
way. This is most useful if the ideals involved are all extended
ideals (for instance with trivial principal part), so that the principal parts
extracted by idealred
are not lost. Here is an example:
? f = vector(#f, i, [f[i], [;]]); \\ transform to extended ideals ? idealfactorback(nf, f, e, 1) %6 = [[1, 0; 0, 1], [2, 1; [2, 1]~, 1]] ? nffactorback(nf, %[2]) %7 = [4, 2]~
The extended ideal returned in %6
is the trivial ideal 1
, extended
with a principal generator given in factored form. We use nffactorback
to recover it in standard form.
The library syntax is GEN
idealfactorback(GEN nf, GEN f, GEN e = NULL, long flag)
.
(
nf,
gal,
pr)
Let K
be the number field defined by nf
and assume K/
Q be a
Galois extension with Galois group given gal = galoisinit(nf)
,
and that pr is an unramified prime ideal p in prid
format.
This function returns a permutation of gal.group
which defines
the Frobenius element Frob _{
p}
attached to p.
If p
is the unique prime number in p, then
Frob (x) = x^p mod
p for all x\in
Z_K
.
? nf = nfinit(polcyclo(31)); ? gal = galoisinit(nf); ? pr = idealprimedec(nf,101)[1]; ? g = idealfrobenius(nf,gal,pr); ? galoispermtopol(gal,g) %5 = x^8
@3This is correct since 101 = 8 mod 31
.
The library syntax is GEN
idealfrobenius(GEN nf, GEN gal, GEN pr)
.
(
nf,u,{v})
Gives the Hermite normal form of the ideal u
Z_K+v
Z_K
, where u
and v
are elements of the number field K
defined by nf.
? nf = nfinit(y^3 - 2); ? idealhnf(nf, 2, y+1) %2 = [1 0 0]
[0 1 0]
[0 0 1] ? idealhnf(nf, y/2, [0,0,1/3]~) %3 = [1/3 0 0]
[0 1/6 0]
[0 0 1/6]
If b
is omitted, returns the HNF of the ideal defined by u
: u
may be an
algebraic number (defining a principal ideal), a maximal ideal (as given by
idealprimedec
or idealfactor
), or a matrix whose columns give
generators for the ideal. This last format is a little complicated, but
useful to reduce general modules to the canonical form once in a while:
@3* if strictly less than N = [K:
Q]
generators are given, u
is the Z_K
-module they generate,
@3* if N
or more are given, it is assumed that they form a
Z-basis of the ideal, in particular that the matrix has maximal rank N
.
This acts as mathnf
since the Z_K
-module structure is (taken for
granted hence) not taken into account in this case.
? idealhnf(nf, idealprimedec(nf,2)[1]) %4 = [2 0 0]
[0 1 0]
[0 0 1] ? idealhnf(nf, [1,2;2,3;3,4]) %5 = [1 0 0]
[0 1 0]
[0 0 1]
@3Finally, when K
is quadratic with discriminant D_K
, we
allow u =
Qfb(a,b,c)
, provided b^2 - 4ac = D_K
. As usual,
this represents the ideal a
Z + (1/2)(-b +
sqrt {D_K})
Z.
? K = nfinit(x^2 - 60); K.disc %1 = 60 ? idealhnf(K, qfbprimeform(60,2)) %2 = [2 1]
[0 1] ? idealhnf(K, Qfb(1,2,3)) *** at top-level: idealhnf(K,Qfb(1,2,3 *** ^-------------------- *** idealhnf: Qfb(1, 2, 3) has discriminant != 60 in idealhnf.
The library syntax is GEN
idealhnf0(GEN nf, GEN u, GEN v = NULL)
.
Also available is GEN
idealhnf(GEN nf, GEN a)
.
(
nf,A,B)
Intersection of the two ideals
A
and B
in the number field nf. The result is given in HNF.
? nf = nfinit(x^2+1); ? idealintersect(nf, 2, x+1) %2 = [2 0]
[0 2]
This function does not apply to general Z-modules, e.g. orders, since its
arguments are replaced by the ideals they generate. The following script
intersects Z-modules A
and B
given by matrices of compatible
dimensions with integer coefficients:
ZM_intersect(A,B) = { my(Ker = matkerint(concat(A,B))); mathnf( A * Ker[1..#A,] ) }
The library syntax is GEN
idealintersect(GEN nf, GEN A, GEN B)
.
(
nf,x)
Inverse of the ideal x
in the
number field nf, given in HNF. If x
is an extended
ideal, its principal part is suitably
updated: i.e. inverting [I,t]
, yields [I^{-1}, 1/t]
.
The library syntax is GEN
idealinv(GEN nf, GEN x)
.
(
nf,
bound,{
flag = 4})
Computes the list of all ideals of norm less or equal to bound in the number field nf. The result is a row vector with exactly bound components. Each component is itself a row vector containing the information about ideals of a given norm, in no specific order, depending on the value of flag:
The possible values of flag are:
0: give the bid attached to the ideals, without generators.
1: as 0, but include the generators in the bid.
2: in this case, nf must be a bnf with units. Each
component is of the form [
bid,U]
, where bid is as case 0
and U
is a vector of discrete logarithms of the units. More precisely, it
gives the ideallog
s with respect to bid of bnf.tufu
.
This structure is technical, and only meant to be used in conjunction with
bnrclassnolist
or bnrdisclist
.
3: as 2, but include the generators in the bid.
4: give only the HNF of the ideal.
? nf = nfinit(x^2+1); ? L = ideallist(nf, 100); ? L[1] %3 = [[1, 0; 0, 1]] \\ A single ideal of norm 1 ? #L[65] %4 = 4 \\ There are 4 ideals of norm 4 in B<Z>[i]
If one wants more information, one could do instead:
? nf = nfinit(x^2+1); ? L = ideallist(nf, 100, 0); ? l = L[25]; vector(#l, i, l[i].clgp) %3 = [[20, [20]], [16, [4, 4]], [20, [20]]] ? l[1].mod %4 = [[25, 18; 0, 1], []] ? l[2].mod %5 = [[5, 0; 0, 5], []] ? l[3].mod %6 = [[25, 7; 0, 1], []]
@3where we ask for the structures of the (
Z[i]/I)^*
for all
three ideals of norm 25
. In fact, for all moduli with finite part of norm
25
and trivial Archimedean part, as the last 3 commands show. See
ideallistarch
to treat general moduli.
The library syntax is GEN
ideallist0(GEN nf, long bound, long flag)
.
(
nf,
list,
arch)
list is a vector of vectors of bid's, as output by ideallist
with
flag 0
to 3
. Return a vector of vectors with the same number of
components as the original list. The leaves give information about
moduli whose finite part is as in original list, in the same order, and
Archimedean part is now arch (it was originally trivial). The
information contained is of the same kind as was present in the input; see
ideallist
, in particular the meaning of flag.
? bnf = bnfinit(x^2-2); ? bnf.sign %2 = [2, 0] \\ two places at infinity ? L = ideallist(bnf, 100, 0); ? l = L[98]; vector(#l, i, l[i].clgp) %4 = [[42, [42]], [36, [6, 6]], [42, [42]]] ? La = ideallistarch(bnf, L, [1,1]); \\ add them to the modulus ? l = La[98]; vector(#l, i, l[i].clgp) %6 = [[168, [42, 2, 2]], [144, [6, 6, 2, 2]], [168, [42, 2, 2]]]
Of course, the results above are obvious: adding t
places at infinity will
add t
copies of Z/2
Z to (
Z_K/f)^*
. The following application
is more typical:
? L = ideallist(bnf, 100, 2); \\ units are required now ? La = ideallistarch(bnf, L, [1,1]); ? H = bnrclassnolist(bnf, La); ? H[98]; %4 = [2, 12, 2]
The library syntax is GEN
ideallistarch(GEN nf, GEN list, GEN arch)
.
ideallog({
nf},x,
bid)
nf is a number field,
bid is as output by idealstar(nf, D,...)
and x
a
non-necessarily integral element of nf which must have valuation
equal to 0 at all prime ideals in the support of D
. This function
computes the discrete logarithm of x
on the generators given in
bid.gen
. In other words, if g_i
are these generators, of orders
d_i
respectively, the result is a column vector of integers (x_i)
such
that 0 <= x_i < d_i
and
x =
prod_i g_i^{x_i} (mod ^*D) .
Note that when the support of D
contains places at infinity, this
congruence implies also sign conditions on the attached real embeddings.
See znlog
for the limitations of the underlying discrete log algorithms.
When nf is omitted, take it to be the rational number field. In that
case, x
must be a t_INT
and bid must have been initialized by
idealstar(,N)
.
The library syntax is GEN
ideallog(GEN nf = NULL, GEN x, GEN bid)
.
Also available is
GEN
Zideallog(GEN bid, GEN x)
when nf
is NULL
.
idealmin(
nf,
ix,{
vdir})
This function is useless and kept for backward compatibility only,
use idealred
. Computes a pseudo-minimum of the ideal x
in the
direction vdir in the number field nf.
The library syntax is GEN
idealmin(GEN nf, GEN ix, GEN vdir = NULL)
.
(
nf,x,y,{
flag = 0})
Ideal multiplication of the ideals x
and y
in the number field
nf; the result is the ideal product in HNF. If either x
or y
are extended ideals, their principal part is suitably
updated: i.e. multiplying [I,t]
, [J,u]
yields [IJ, tu]
; multiplying
I
and [J, u]
yields [IJ, u]
.
? nf = nfinit(x^2 + 1); ? idealmul(nf, 2, x+1) %2 = [4 2]
[0 2] ? idealmul(nf, [2, x], x+1) \\ extended ideal * ideal %3 = [[4, 2; 0, 2], x] ? idealmul(nf, [2, x], [x+1, x]) \\ two extended ideals %4 = [[4, 2; 0, 2], [-1, 0]~]
@3
If flag is non-zero, reduce the result using idealred
.
The library syntax is GEN
idealmul0(GEN nf, GEN x, GEN y, long flag)
.
@3See also
GEN
idealmul(GEN nf, GEN x, GEN y)
(flag = 0
) and
GEN
idealmulred(GEN nf, GEN x, GEN y)
(flag != 0
).
(
nf,x)
Computes the norm of the ideal x
in the number field nf.
The library syntax is GEN
idealnorm(GEN nf, GEN x)
.
(
nf,x)
Returns [A,B]
, where A,B
are coprime integer ideals
such that x = A/B
, in the number field nf.
? nf = nfinit(x^2+1); ? idealnumden(nf, (x+1)/2) %2 = [[1, 0; 0, 1], [2, 1; 0, 1]]
The library syntax is GEN
idealnumden(GEN nf, GEN x)
.
(
nf,x,k,{
flag = 0})
Computes the k
-th power of
the ideal x
in the number field nf; k\in
Z.
If x
is an extended
ideal, its principal part is suitably
updated: i.e. raising [I,t]
to the k
-th power, yields [I^k, t^k]
.
If flag is non-zero, reduce the result using idealred
, throughout
the (binary) powering process; in particular, this is not the same
as idealpow(
nf,x,k)
followed by reduction.
The library syntax is GEN
idealpow0(GEN nf, GEN x, GEN k, long flag)
.
@3See also
GEN
idealpow(GEN nf, GEN x, GEN k)
and
GEN
idealpows(GEN nf, GEN x, long k)
(flag = 0
).
Corresponding to flag = 1
is GEN
idealpowred(GEN nf, GEN vp, GEN k)
.
(
nf,p,{f = 0})
Computes the prime ideal
decomposition of the (positive) prime number p
in the number field K
represented by nf. If a non-prime p
is given the result is undefined.
If f
is present and non-zero, restrict the result to primes of residue
degree <= f
.
The result is a vector of prid structures, each representing one of the
prime ideals above p
in the number field nf. The representation
pr = [p,a,e,f,
mb]
of a prime ideal means the following: a
and
is an algebraic integer in the maximal order Z_K
and the prime ideal is
equal to p = p
Z_K + a
Z_K
;
e
is the ramification index; f
is the residual index;
finally, mb is the multiplication table attached to the algebraic
integer b
is such that p^{-1} =
Z_K+ b/ p
Z_K
, which is used
internally to compute valuations. In other words if p
is inert,
then mb is the integer 1
, and otherwise it's a square t_MAT
whose j
-th column is b.nf.zk[j]
.
The algebraic number a
is guaranteed to have a
valuation equal to 1 at the prime ideal (this is automatic if e > 1
).
The components of pr
should be accessed by member functions: pr.p
,
pr.e
, pr.f
, and pr.gen
(returns the vector [p,a]
):
? K = nfinit(x^3-2); ? P = idealprimedec(K, 5); ? #P \\ 2 primes above 5 in Q(2^(1/3)) %3 = 2 ? [p1,p2] = P; ? [p1.e, p1.f] \\ the first is unramified of degree 1 %5 = [1, 1] ? [p2.e, p2.f] \\ the second is unramified of degree 2 %6 = [1, 2] ? p1.gen %7 = [5, [2, 1, 0]~] ? nfbasistoalg(K, %[2]) \\ a uniformizer for p1 %8 = Mod(x + 2, x^3 - 2) ? #idealprimedec(K, 5, 1) \\ restrict to f = 1 %9 = 1 \\ now only p1
The library syntax is GEN
idealprimedec_limit_f(GEN nf, GEN p, long f)
.
(
nf,
pr,k)
Given a prime ideal in idealprimedec
format,
returns the multiplicative group (1 +
pr) / (1 +
pr^k)
as an
abelian group. This function is much faster than idealstar
when the
norm of pr is large, since it avoids (useless) work in the
multiplicative group of the residue field.
? K = nfinit(y^2+1); ? P = idealprimedec(K,2)[1]; ? G = idealprincipalunits(K, P, 20); ? G.cyc %4 = [512, 256, 4] \\ Z/512 x Z/256 x Z/4 ? G.gen %5 = [[-1, -2]~, 1021, [0, -1]~] \\ minimal generators of given order
The library syntax is GEN
idealprincipalunits(GEN nf, GEN pr, long k)
.
(
nf,
gal,
pr)
Let K
be the number field defined by nf and assume that K/
Q is
Galois with Galois group G
given by gal = galoisinit(nf)
.
Let pr be the prime ideal P in prid format.
This function returns a vector g
of subgroups of gal
as follow:
@3* g[1]
is the decomposition group of P,
@3* g[2]
is G_0(
P)
, the inertia group of P,
and for i >= 2
,
@3* g[i]
is G_{i-2}(
P)
, the i-2
-th
ramification group of P.
@3The length of g
is the number of non-trivial groups in the
sequence, thus is 0
if e = 1
and f = 1
, and 1
if f > 1
and e = 1
.
The following function computes the cardinality of a subgroup of G
,
as given by the components of g
:
card(H) =my(o=H[2]); prod(i=1,#o,o[i]);
? nf=nfinit(x^6+3); gal=galoisinit(nf); pr=idealprimedec(nf,3)[1]; ? g = idealramgroups(nf, gal, pr); ? apply(card,g) %3 = [6, 6, 3, 3, 3] \\ cardinalities of the G_i
? nf=nfinit(x^6+108); gal=galoisinit(nf); pr=idealprimedec(nf,2)[1]; ? iso=idealramgroups(nf,gal,pr)[2] %5 = [[Vecsmall([2, 3, 1, 5, 6, 4])], Vecsmall([3])] ? nfdisc(galoisfixedfield(gal,iso,1)) %6 = -3
@3The field fixed by the inertia group of 2
is not ramified at
2
.
The library syntax is GEN
idealramgroups(GEN nf, GEN gal, GEN pr)
.
(
nf,I,{v = 0})
LLL reduction of
the ideal I
in the number field K
attached to nf, along the
direction v
. The v
parameter is best left omitted, but if it is present,
it must be an nf.r1 + nf.r2
-component vector of
non-negative integers. (What counts is the relative magnitude of the
entries: if all entries are equal, the effect is the same as if the vector
had been omitted.)
This function finds an a\in K^*
such that J = (a)I
is
``small'' and integral (see the end for technical details).
The result is the Hermite normal form of
the ``reduced'' ideal J
.
? K = nfinit(y^2+1); ? P = idealprimedec(K,5)[1]; ? idealred(K, P) %3 = [1 0]
[0 1]
@3More often than not, a principal ideal yields the unit
ideal as above. This is a quick and dirty way to check if ideals are principal,
but it is not a necessary condition: a non-trivial result does not prove that
the ideal is non-principal. For guaranteed results, see bnfisprincipal
,
which requires the computation of a full bnf
structure.
If the input is an extended ideal [I,s]
, the output is [J, sa]
; in
this way, one keeps track of the principal ideal part:
? idealred(K, [P, 1]) %5 = [[1, 0; 0, 1], [2, -1]~]
meaning that P
is generated by [2, -1]
. The number field element in the
extended part is an algebraic number in any form or a factorization
matrix (in terms of number field elements, not ideals!). In the latter case,
elements stay in factored form, which is a convenient way to avoid
coefficient explosion; see also idealpow
.
@3Technical note. The routine computes an LLL-reduced
basis for the lattice I^(-1)
equipped with the quadratic
form
|| x ||_v^2 =
sum_{i = 1}^{r_1+r_2} 2^{v_i}
varepsilon_i|
sigma_i(x)|^2,
where as usual the sigma_i
are the (real and) complex embeddings and
varepsilon_i = 1
, resp. 2
, for a real, resp. complex place. The element
a
is simply the first vector in the LLL basis. The only reason you may want
to try to change some directions and set some v_i != 0
is to randomize
the elements found for a fixed ideal, which is heuristically useful in index
calculus algorithms like bnfinit
and bnfisprincipal
.
@3Even more technical note. In fact, the above is a white lie.
We do not use ||.||_v
exactly but a rescaled rounded variant which
gets us faster and simpler LLLs. There's no harm since we are not using any
theoretical property of a
after all, except that it belongs to I^(-1)
and that a I
is ``expected to be small''.
The library syntax is GEN
idealred0(GEN nf, GEN I, GEN v = NULL)
.
idealstar({
nf},N,{
flag = 1})
Outputs a bid
structure,
necessary for computing in the finite abelian group G = (
Z_K/N)^*
. Here,
nf is a number field and N
is a modulus: either an ideal in any
form, or a row vector whose first component is an ideal and whose second
component is a row vector of r_1
0 or 1. Ideals can also be given
by a factorization into prime ideals, as produced by idealfactor
.
This bid is used in ideallog
to compute discrete logarithms. It
also contains useful information which can be conveniently retrieved as
bid.mod
(the modulus),
bid.clgp
(G
as a finite abelian group),
bid.no
(the cardinality of G
),
bid.cyc
(elementary divisors) and
bid.gen
(generators).
If flag = 1
(default), the result is a bid
structure without
generators: they are well defined but not explicitly computed, which saves
time.
If flag = 2
, as flag = 1
, but including generators.
If flag = 0
, only outputs (
Z_K/N)^*
as an abelian group,
i.e as a 3-component vector [h,d,g]
: h
is the order, d
is the vector of
SNF cyclic components and g
the corresponding
generators.
If nf is omitted, we take it to be the rational number fields, N
must
be an integer and we return the structure of (
Z/N
Z)^*
. In other words
idealstar(, N, flag)
is short for
idealstar(nfinit(x), N, flag)
@3but much faster. The alternative syntax znstar(N, flag)
is also available for the same effect, but due to an unfortunate historical
oversight, the default value of flag
is different in the two
functions (znstar
does not initialize by default).
The library syntax is GEN
idealstar0(GEN nf = NULL, GEN N, long flag)
.
Instead the above hardcoded numerical flags, one should rather use
GEN
Idealstar(GEN nf, GEN ideal, long flag)
, where flag
is
an or-ed combination of nf_GEN
(include generators) and nf_INIT
(return a full bid
, not a group), possibly 0
. This offers
one more combination: gen, but no init.
(
nf,x,{a})
Computes a two-element
representation of the ideal x
in the number field nf, combining a
random search and an approximation theorem; x
is an ideal
in any form (possibly an extended ideal, whose principal part is ignored)
@3* When called as idealtwoelt(nf,x)
, the result is a row vector
[a,
alpha]
with two components such that x = a
Z_K+
alphaZ_K
and a
is
chosen to be the positive generator of x
capZ, unless x
was given as a
principal ideal (in which case we may choose a = 0
). The algorithm
uses a fast lazy factorization of x
cap Z and runs in randomized
polynomial time.
@3* When called as idealtwoelt(nf,x,a)
with an explicit non-zero a
supplied as third argument, the function assumes that a \in x
and returns
alpha\in x
such that x = a
Z_K +
alphaZ_K
. Note that we must factor
a
in this case, and the algorithm is generally much slower than the
default variant.
The library syntax is GEN
idealtwoelt0(GEN nf, GEN x, GEN a = NULL)
.
Also available are
GEN
idealtwoelt(GEN nf, GEN x)
and
GEN
idealtwoelt2(GEN nf, GEN x, GEN a)
.
(
nf,x,
pr)
Gives the valuation of the ideal x
at the prime ideal pr in the
number field nf, where pr is in idealprimedec
format.
The valuation of the 0
ideal is +oo
.
The library syntax is GEN
gpidealval(GEN nf, GEN x, GEN pr)
.
Also available is
long
idealval(GEN nf, GEN x, GEN pr)
, which returns
LONG_MAX
if x = 0
and the valuation as a long
integer.
(
nf,x)
This function is deprecated, use apply
.
nf being a number field in nfinit
format, and x
a
(row or column) vector or matrix, apply nfalgtobasis
to each entry
of x
.
The library syntax is GEN
matalgtobasis(GEN nf, GEN x)
.
(
nf,x)
This function is deprecated, use apply
.
nf being a number field in nfinit
format, and x
a
(row or column) vector or matrix, apply nfbasistoalg
to each entry
of x
.
The library syntax is GEN
matbasistoalg(GEN nf, GEN x)
.
(z)
Let z = Mod(A, T)
be a polmod, and Q
be its minimal
polynomial, which must satisfy deg(Q) = deg(T)
.
Returns a ``reverse polmod'' Mod(B, Q)
, which is a root of T
.
This is quite useful when one changes the generating element in algebraic extensions:
? u = Mod(x, x^3 - x -1); v = u^5; ? w = modreverse(v) %2 = Mod(x^2 - 4*x + 1, x^3 - 5*x^2 + 4*x - 1)
which means that x^3 - 5x^2 + 4x -1
is another defining polynomial for the
cubic field
Q(u) =
Q[x]/(x^3 - x - 1) =
Q[x]/(x^3 - 5x^2 + 4x - 1) =
Q(v),
and that u \to v^2 - 4v + 1
gives an explicit isomorphism. From this, it is
easy to convert elements between the A(u)\in
Q(u)
and B(v)\in
Q(v)
representations:
? A = u^2 + 2*u + 3; subst(lift(A), 'x, w) %3 = Mod(x^2 - 3*x + 3, x^3 - 5*x^2 + 4*x - 1) ? B = v^2 + v + 1; subst(lift(B), 'x, v) %4 = Mod(26*x^2 + 31*x + 26, x^3 - x - 1)
If the minimal polynomial of z
has lower degree than expected, the routine
fails
? u = Mod(-x^3 + 9*x, x^4 - 10*x^2 + 1) ? modreverse(u) *** modreverse: domain error in modreverse: deg(minpoly(z)) < 4 *** Break loop: type 'break' to go back to GP prompt break> Vec( dbg_err() ) \\ ask for more info ["e_DOMAIN", "modreverse", "deg(minpoly(z))", "<", 4, Mod(-x^3 + 9*x, x^4 - 10*x^2 + 1)] break> minpoly(u) x^2 - 8
The library syntax is GEN
modreverse(GEN z)
.
(x,p)
Gives the vector of the slopes of the Newton
polygon of the polynomial x
with respect to the prime number p
. The n
components of the vector are in decreasing order, where n
is equal to the
degree of x
. Vertical slopes occur iff the constant coefficient of x
is
zero and are denoted by +oo
.
The library syntax is GEN
newtonpoly(GEN x, GEN p)
.
(
nf,x)
Given an algebraic number x
in the number field nf,
transforms it to a column vector on the integral basis nf.zk
.
? nf = nfinit(y^2 + 4); ? nf.zk %2 = [1, 1/2*y] ? nfalgtobasis(nf, [1,1]~) %3 = [1, 1]~ ? nfalgtobasis(nf, y) %4 = [0, 2]~ ? nfalgtobasis(nf, Mod(y, y^2+4)) %5 = [0, 2]~
This is the inverse function of nfbasistoalg
.
The library syntax is GEN
algtobasis(GEN nf, GEN x)
.
(T)
Let T(X)
be an irreducible polynomial with integral coefficients. This
function returns an integral basis of the number field defined by T
,
that is a Z-basis of its maximal order. The basis elements are given as
elements in Q[X]/(T)
:
? nfbasis(x^2 + 1) %1 = [1, x]
This function uses a modified version of the round 4 algorithm, due to David Ford, Sebastian Pauli and Xavier Roblot.
@3Local basis, orders maximal at certain primes.
Obtaining the maximal order is hard: it requires factoring the discriminant
D
of T
. Obtaining an order which is maximal at a finite explicit set of
primes is easy, but it may then be a strict suborder of the maximal order. To
specify that we are interested in a given set of places only, we can replace
the argument T
by an argument [T,
listP]
, where listP encodes
the primes we are interested in: it must be a factorization matrix, a vector
of integers or a single integer.
@3* Vector: we assume that it contains distinct prime numbers.
@3* Matrix: we assume that it is a two-column matrix of a
(partial) factorization of D
; namely the first column contains
distinct primes and the second one the valuation of D
at each of
these primes.
@3* Integer B
: this is replaced by the vector of primes up to B
. Note
that the function will use at least O(B)
time: a small value, about
10^5
, should be enough for most applications. Values larger than 2^{32}
are not supported.
In all these cases, the primes may or may not divide the discriminant D
of T
. The function then returns a Z-basis of an order whose index is
not divisible by any of these prime numbers. The result is actually a global
integral basis if all prime divisors of the field discriminant are
included! Note that nfinit
has built-in support for such
a check:
? K = nfinit([T, listP]); ? nfcertify(K) \\ we computed an actual maximal order %2 = [];
@3The first line initializes a number field structure
incorporating nfbasis([T, listP]
in place of a proven integral basis.
The second line certifies that the resulting structure is correct. This
allows to create an nf
structure attached to the number field K =
Q[X]/(T)
, when the discriminant of T
cannot be factored completely,
whereas the prime divisors of disc K
are known.
Of course, if listP contains a single prime number p
,
the function returns a local integral basis for Z_p[X]/(T)
:
? nfbasis(x^2+x-1001) %1 = [1, 1/3*x - 1/3] ? nfbasis( [x^2+x-1001, [2]] ) %2 = [1, x]
@3The Buchmann-Lenstra algorithm.
We now complicate the picture: it is in fact allowed to include
composite numbers instead of primes
in listP
(Vector or Matrix case), provided they are pairwise coprime.
The result will still be a correct integral basis if
the field discriminant factors completely over the actual primes in the list.
Adding a composite C
such that C^2
divides D
may help because
when we consider C
as a prime and run the algorithm, two good things can
happen: either we
succeed in proving that no prime dividing C
can divide the index
(without actually needing to find those primes), or the computation
exhibits a non-trivial zero divisor, thereby factoring C
and
we go on with the refined factorization. (Note that including a C
such that C^2
does not divide D
is useless.) If neither happen, then the
computed basis need not generate the maximal order. Here is an example:
? B = 10^5; ? P = factor(poldisc(T), B)[,1]; \\ primes <= B dividing D + cofactor ? basis = nfbasis([T, listP]) ? disc = nfdisc([T, listP])
@3We obtain the maximal order and its discriminant if the
field discriminant factors
completely over the primes less than B
(together with the primes
contained in the addprimes
table). This can be tested as follows:
check = factor(disc, B); lastp = check[-1..-1,1]; if (lastp > B && !setsearch(addprimes(), lastp), warning("nf may be incorrect!"))
This is a sufficient but not a necessary condition, hence the warning,
instead of an error. N.B. lastp
is the last entry
in the first column of the check
matrix, i.e. the largest prime
dividing nf.disc
if <= B
or if it belongs to the prime table.
The function nfcertify
speeds up and automates the above process:
? B = 10^5; ? nf = nfinit([T, B]); ? nfcertify(nf) %3 = [] \\ nf is unconditionally correct ? basis = nf.zk; ? disc = nf.disc;
The library syntax is nfbasis(GEN T, GEN *d, GEN listP = NULL)
, which returns the order
basis, and where *d
receives the order discriminant.
(
nf,x)
Given an algebraic number x
in the number field nf, transforms it
into t_POLMOD
form.
? nf = nfinit(y^2 + 4); ? nf.zk %2 = [1, 1/2*y] ? nfbasistoalg(nf, [1,1]~) %3 = Mod(1/2*y + 1, y^2 + 4) ? nfbasistoalg(nf, y) %4 = Mod(y, y^2 + 4) ? nfbasistoalg(nf, Mod(y, y^2+4)) %5 = Mod(y, y^2 + 4)
This is the inverse function of nfalgtobasis
.
The library syntax is GEN
basistoalg(GEN nf, GEN x)
.
(
nf)
nf being as output by
nfinit
, checks whether the integer basis is known unconditionally.
This is in particular useful when the argument to nfinit
was of the
form [T, listP]
, specifying a finite list of primes when
p
-maximality had to be proven, or a list of coprime integers to which
Buchmann-Lenstra algorithm was to be applied.
The function returns a vector of coprime composite integers. If this vector
is empty, then nf.zk
and nf.disc
are correct. Otherwise, the
result is dubious. In order to obtain a certified result, one must completely
factor each of the given integers, then addprime
each of their prime
factors, then check whether nfdisc(nf.pol)
is equal to nf.disc
.
The library syntax is GEN
nfcertify(GEN nf)
.
(
nf,P,Q,{
flag = 0})
Let nf be a number field structure attached to the field K
and let P
and Q
be squarefree polynomials in K[X]
in the same variable. Outputs
the simple factors of the étale K
-algebra A = K[X, Y] / (P(X), Q(Y))
.
The factors are given by a list of polynomials R
in K[X]
, attached to
the number field K[X]/ (R)
, and sorted by increasing degree (with respect
to lexicographic ordering for factors of equal degrees). Returns an error if
one of the polynomials is not squarefree.
Note that it is more efficient to reduce to the case where P
and Q
are
irreducible first. The routine will not perform this for you, since it may be
expensive, and the inputs are irreducible in most applications anyway. In
this case, there will be a single factor R
if and only if the number
fields defined by P
and Q
are linearly disjoint (their intersection is
K
).
The binary digits of flag mean
1: outputs a vector of 4-component vectors [R,a,b,k]
, where R
ranges through the list of all possible compositums as above, and a
(resp. b
) expresses the root of P
(resp. Q
) as an element of
K[X]/(R)
. Finally, k
is a small integer such that b + ka = X
modulo
R
.
2: assume that P
and Q
define number fields that are linearly disjoint:
both polynomials are irreducible and the corresponding number fields
have no common subfield besides K
. This allows to save a costly
factorization over K
. In this case return the single simple factor
instead of a vector with one element.
A compositum is often defined by a complicated polynomial, which it is
advisable to reduce before further work. Here is an example involving
the field K(
zeta_5, 5^{1/10})
, K =
Q(
sqrt {5})
:
? K = nfinit(y^2-5); ? L = nfcompositum(K, x^5 - y, polcyclo(5), 1); \\ list of [R,a,b,k] ? [R, a] = L[1]; \\ pick the single factor, extract R,a (ignore b,k) ? lift(R) \\ defines the compositum %4 = x^10 + (-5/2*y + 5/2)*x^9 + (-5*y + 20)*x^8 + (-20*y + 30)*x^7 + \ (-45/2*y + 145/2)*x^6 + (-71/2*y + 121/2)*x^5 + (-20*y + 60)*x^4 + \ (-25*y + 5)*x^3 + 45*x^2 + (-5*y + 15)*x + (-2*y + 6) ? a^5 - y \\ a fifth root of y %5 = 0 ? [T, X] = rnfpolredbest(K, R, 1); ? lift(T) \\ simpler defining polynomial for K[x]/(R) %7 = x^10 + (-11/2*y + 25/2) ? liftall(X) \\ root of R in K[x]/(T(x)) %8 = (3/4*y + 7/4)*x^7 + (-1/2*y - 1)*x^5 + 1/2*x^2 + (1/4*y - 1/4) ? a = subst(a.pol, 'x, X); \\ C<a> in the new coordinates ? liftall(a) %10 = (-3/4*y - 7/4)*x^7 - 1/2*x^2 ? a^5 - y %11 = 0
The main variables of P
and Q
must be the same and have higher priority
than that of nf (see varhigher
and varlower
).
The library syntax is GEN
nfcompositum(GEN nf, GEN P, GEN Q, long flag)
.
(
nf,x)
Given a pseudo-matrix x
, computes a
non-zero ideal contained in (i.e. multiple of) the determinant of x
. This
is particularly useful in conjunction with nfhnfmod
.
The library syntax is GEN
nfdetint(GEN nf, GEN x)
.
(T)
field discriminant of the number field defined by the integral,
preferably monic, irreducible polynomial T(X)
. Returns the discriminant of
the number field Q[X]/(T)
, using the Round 4
algorithm.
@3Local discriminants, valuations at certain primes.
As in nfbasis
, the argument T
can be replaced by [T,
listP]
,
where listP
is as in nfbasis
: a vector of
pairwise coprime integers (usually distinct primes), a factorization matrix,
or a single integer. In that case, the function returns the discriminant of
an order whose basis is given by nfbasis(T,listP)
, which need not be
the maximal order, and whose valuation at a prime entry in listP
is the
same as the valuation of the field discriminant.
In particular, if listP
is [p]
for a prime p
, we can
return the p
-adic discriminant of the maximal order of Z_p[X]/(T)
,
as a power of p
, as follows:
? padicdisc(T,p) = p^valuation(nfdisc(T,[p]), p); ? nfdisc(x^2 + 6) %2 = -24 ? padicdisc(x^2 + 6, 2) %3 = 8 ? padicdisc(x^2 + 6, 3) %4 = 3
The library syntax is nfdisc(GEN T)
(listP = NULL
). Also available is
GEN
nfbasis(GEN T, GEN *d, GEN listP = NULL)
, which returns the order
basis, and where *d
receives the order discriminant.
(
nf,x,y)
Given two elements x
and y
in
nf, computes their sum x+y
in the number field nf.
The library syntax is GEN
nfadd(GEN nf, GEN x, GEN y)
.
(
nf,x,y)
Given two elements x
and y
in
nf, computes their quotient x/y
in the number field nf.
The library syntax is GEN
nfdiv(GEN nf, GEN x, GEN y)
.
(
nf,x,y)
Given two elements x
and y
in
nf, computes an algebraic integer q
in the number field nf
such that the components of x-qy
are reasonably small. In fact, this is
functionally identical to round(nfdiv(
nf,x,y))
.
The library syntax is GEN
nfdiveuc(GEN nf, GEN x, GEN y)
.
(
nf,x,y,
pr)
This function is obsolete, use nfmodpr
.
Given two elements x
and y
in nf and pr a prime ideal in modpr
format (see
nfmodprinit
), computes their quotient x / y
modulo the prime ideal
pr.
The library syntax is GEN
nfdivmodpr(GEN nf, GEN x, GEN y, GEN pr)
.
This function is normally useless in library mode. Project your
inputs to the residue field using nf_to_Fq
, then work there.
(
nf,x,y)
Given two elements x
and y
in
nf, gives a two-element row vector [q,r]
such that x = qy+r
, q
is
an algebraic integer in nf, and the components of r
are
reasonably small.
The library syntax is GEN
nfdivrem(GEN nf, GEN x, GEN y)
.
(
nf,x,y)
Given two elements x
and y
in
nf, computes an element r
of nf of the form r = x-qy
with
q
and algebraic integer, and such that r
is small. This is functionally
identical to
x - nfmul(
nf,round(nfdiv(
nf,x,y)),y).
The library syntax is GEN
nfmod(GEN nf, GEN x, GEN y)
.
(
nf,x,y)
Given two elements x
and y
in
nf, computes their product x*y
in the number field nf.
The library syntax is GEN
nfmul(GEN nf, GEN x, GEN y)
.
(
nf,x,y,
pr)
This function is obsolete, use nfmodpr
.
Given two elements x
and
y
in nf and pr a prime ideal in modpr
format (see
nfmodprinit
), computes their product x*y
modulo the prime ideal
pr.
The library syntax is GEN
nfmulmodpr(GEN nf, GEN x, GEN y, GEN pr)
.
This function is normally useless in library mode. Project your
inputs to the residue field using nf_to_Fq
, then work there.
(
nf,x)
Returns the absolute norm of x
.
The library syntax is GEN
nfnorm(GEN nf, GEN x)
.
(
nf,x,k)
Given an element x
in nf, and a positive or negative integer k
,
computes x^k
in the number field nf.
The library syntax is GEN
nfpow(GEN nf, GEN x, GEN k)
.
GEN
nfinv(GEN nf, GEN x)
correspond to k = -1
, and
GEN
nfsqr(GEN nf,GEN x)
to k = 2
.
(
nf,x,k,
pr)
This function is obsolete, use nfmodpr
.
Given an element x
in nf, an integer k
and a prime ideal
pr in modpr
format
(see nfmodprinit
), computes x^k
modulo the prime ideal pr.
The library syntax is GEN
nfpowmodpr(GEN nf, GEN x, GEN k, GEN pr)
.
This function is normally useless in library mode. Project your
inputs to the residue field using nf_to_Fq
, then work there.
(
nf,a,
id)
Given an ideal id in
Hermite normal form and an element a
of the number field nf,
finds an element r
in nf such that a-r
belongs to the ideal
and r
is small.
The library syntax is GEN
nfreduce(GEN nf, GEN a, GEN id)
.
(
nf,x,
pr)
This function is obsolete, use nfmodpr
.
Given an element x
of the number field nf and a prime ideal
pr in modpr
format compute a canonical representative for the
class of x
modulo pr.
The library syntax is GEN
nfreducemodpr(GEN nf, GEN x, GEN pr)
.
This function is normally useless in library mode. Project your
inputs to the residue field using nf_to_Fq
, then work there.
(
nf,x)
Returns the absolute trace of x
.
The library syntax is GEN
nftrace(GEN nf, GEN x)
.
(
nf,x,
pr,{&y})
Given an element x
in
nf and a prime ideal pr in the format output by
idealprimedec
, computes the valuation v
at pr of the
element x
. The valuation of 0
is +oo
.
? nf = nfinit(x^2 + 1); ? P = idealprimedec(nf, 2)[1]; ? nfeltval(nf, x+1, P) %3 = 1
This particular valuation can also be obtained using
idealval(
nf,x,
pr)
, since x
is then converted to a
principal ideal.
If the y
argument is present, sets y = x
tau^v
, where tau is a
fixed ``anti-uniformizer'' for pr: its valuation at pr is -1
;
its valuation is 0
at other prime ideals dividing pr.p
and
nonnegative at all other primes. In other words y
is the part of x
coprime to pr. If x
is an algebraic integer, so is y
.
? nfeltval(nf, x+1, P, &y); y %4 = [0, 1]~
For instance if x =
prod_i x_i^{e_i}
is known to be coprime to pr,
where the x_i
are algebraic integers and e_i\in
Z then,
if v_i = nfeltval(
nf, x_i,
pr, &y_i)
, we still
have x =
prod_i y_i^{e_i}
, where the y_i
are still algebraic integers
but now all of them are coprime to pr. They can then be mapped to
the residue field of pr more efficiently than if the product had
been expanded beforehand: we can reduce mod pr after each ring
operation.
The library syntax is GEN
gpnfvalrem(GEN nf, GEN x, GEN pr, GEN *y = NULL)
.
Also available is
long
nfvalrem(GEN nf, GEN x, GEN pr, GEN *y = NULL)
, which returns
LONG_MAX
if x = 0
and the valuation as a long
integer.
(
nf,T)
Factorization of the univariate
polynomial T
over the number field nf given by nfinit
; T
has coefficients in nf (i.e. either scalar, polmod, polynomial or
column vector). The factors are sorted by increasing degree.
The main variable of nf must be of lower
priority than that of T
, see Label se:priority. However if
the polynomial defining the number field occurs explicitly in the
coefficients of T
as modulus of a t_POLMOD
or as a t_POL
coefficient, its main variable must be the same as the main variable
of T
. For example,
? nf = nfinit(y^2 + 1); ? nffactor(nf, x^2 + y); \\ OK ? nffactor(nf, x^2 + Mod(y, y^2+1)); \\ OK ? nffactor(nf, x^2 + Mod(z, z^2+1)); \\ WRONG
It is possible to input a defining polynomial for nf
instead, but this is in general less efficient since parts of an nf
structure will then be computed internally. This is useful in two
situations: when you do not need the nf
elsewhere, or when you cannot
initialize an nf
due to integer factorization difficulties when
attempting to compute the field discriminant and maximal order.
@3Caveat. nfinit([T, listP])
allows to compute in polynomial
time a conditional nf structure, which sets nf.zk
to an order
which is not guaranteed to be maximal at all primes. Always either use
nfcertify
first (which may not run in polynomial time) or make sure
to input nf.pol
instead of the conditional nf: nffactor
is
able to recover in polynomial time in this case, instead of potentially
missing a factor.
The library syntax is GEN
nffactor(GEN nf, GEN T)
.
(
nf,f,{e})
Gives back the nf element corresponding to a factorization.
The integer 1
corresponds to the empty factorization.
If e
is present, e
and f
must be vectors of the same length (e
being
integral), and the corresponding factorization is the product of the
f[i]^{e[i]}
.
If not, and f
is vector, it is understood as in the preceding case with e
a vector of 1s: we return the product of the f[i]
. Finally, f
can be a
regular factorization matrix.
? nf = nfinit(y^2+1); ? nffactorback(nf, [3, y+1, [1,2]~], [1, 2, 3]) %2 = [12, -66]~ ? 3 * (I+1)^2 * (1+2*I)^3 %3 = 12 - 66*I
The library syntax is GEN
nffactorback(GEN nf, GEN f, GEN e = NULL)
.
(
nf,Q,
pr)
This routine is obsolete, use nfmodpr
and factorff
.
Factors the univariate polynomial Q
modulo the prime ideal pr in
the number field nf. The coefficients of Q
belong to the number
field (scalar, polmod, polynomial, even column vector) and the main variable
of nf must be of lower priority than that of Q
(see
Label se:priority). The prime ideal pr is either in
idealprimedec
or (preferred) modprinit
format. The coefficients
of the polynomial factors are lifted to elements of nf:
? K = nfinit(y^2+1); ? P = idealprimedec(K, 3)[1]; ? nffactormod(K, x^2 + y*x + 18*y+1, P) %3 = [x + (2*y + 1) 1]
[x + (2*y + 2) 1] ? P = nfmodprinit(K, P); \\ convert to nfmodprinit format ? nffactormod(K, x^2 + y*x + 18*y+1) %5 = [x + (2*y + 1) 1]
[x + (2*y + 2) 1]
@3Same result, of course, here about 10% faster due to the precomputation.
The library syntax is GEN
nffactormod(GEN nf, GEN Q, GEN pr)
.
(
nf,
aut,x)
Let nf be a
number field as output by nfinit
, and let aut be a Galois
automorphism of nf expressed by its image on the field generator
(such automorphisms can be found using nfgaloisconj
). The function
computes the action of the automorphism aut on the object x
in the
number field; x
can be a number field element, or an ideal (possibly
extended). Because of possible confusion with elements and ideals, other
vector or matrix arguments are forbidden.
? nf = nfinit(x^2+1); ? L = nfgaloisconj(nf) %2 = [-x, x]~ ? aut = L[1]; /* the non-trivial automorphism */ ? nfgaloisapply(nf, aut, x) %4 = Mod(-x, x^2 + 1) ? P = idealprimedec(nf,5); /* prime ideals above 5 */ ? nfgaloisapply(nf, aut, P[2]) == P[1] %6 = 0 \\ !!!! ? idealval(nf, nfgaloisapply(nf, aut, P[2]), P[1]) %7 = 1
@3The surprising failure of the equality test (%7
) is
due to the fact that although the corresponding prime ideals are equal, their
representations are not. (A prime ideal is specified by a uniformizer, and
there is no guarantee that applying automorphisms yields the same elements
as a direct idealprimedec
call.)
The automorphism can also be given as a column vector, representing the
image of Mod(x, nf.pol)
as an algebraic number. This last
representation is more efficient and should be preferred if a given
automorphism must be used in many such calls.
? nf = nfinit(x^3 - 37*x^2 + 74*x - 37); ? aut = nfgaloisconj(nf)[2]; \\ an automorphism in basistoalg form %2 = -31/11*x^2 + 1109/11*x - 925/11 ? AUT = nfalgtobasis(nf, aut); \\ same in algtobasis form %3 = [16, -6, 5]~ ? v = [1, 2, 3]~; nfgaloisapply(nf, aut, v) == nfgaloisapply(nf, AUT, v) %4 = 1 \\ same result... ? for (i=1,10^5, nfgaloisapply(nf, aut, v)) time = 463 ms. ? for (i=1,10^5, nfgaloisapply(nf, AUT, v)) time = 343 ms. \\ but the latter is faster
The library syntax is GEN
galoisapply(GEN nf, GEN aut, GEN x)
.
(
nf,{
flag = 0},{d})
nf being a number field as output by nfinit
, computes the
conjugates of a root r
of the non-constant polynomial x =
nf[1]
expressed as polynomials in r
. This also makes sense when the number field
is not Galois since some conjugates may lie in the field.
nf can simply be a polynomial.
If no flags or flag = 0
, use a combination of flag 4
and 1
and the result
is always complete. There is no point whatsoever in using the other flags.
If flag = 1
, use nfroots
: a little slow, but guaranteed to work in
polynomial time.
If flag = 4
, use galoisinit
: very fast, but only applies to (most)
Galois fields. If the field is Galois with weakly super-solvable Galois
group (see galoisinit
), return the complete list of automorphisms, else
only the identity element. If present, d
is assumed to be a multiple of the
least common denominator of the conjugates expressed as polynomial in a root
of pol.
This routine can only compute Q-automorphisms, but it may be used to get
K
-automorphism for any base field K
as follows:
rnfgaloisconj(nfK, R) = \\ K-automorphisms of L = K[X] / (R) { my(polabs, N,al,S, ala,k, vR); R *= Mod(1, nfK.pol); \\ convert coeffs to polmod elts of K vR = variable(R); al = Mod(variable(nfK.pol),nfK.pol); [polabs,ala,k] = rnfequation(nfK, R, 1); Rt = if(k==0,R,subst(R,vR,vR-al*k)); N = nfgaloisconj(polabs) % Rt; \\ Q-automorphisms of L S = select(s->subst(Rt, vR, Mod(s,Rt)) == 0, N); if (k==0, S, apply(s->subst(s,vR,vR+k*al)-k*al,S)); } K = nfinit(y^2 + 7); rnfgaloisconj(K, x^4 - y*x^3 - 3*x^2 + y*x + 1) \\ K-automorphisms of L
The library syntax is GEN
galoisconj0(GEN nf, long flag, GEN d = NULL, long prec)
.
Use directly
GEN
galoisconj(GEN nf, GEN d)
, corresponding to flag = 0
, the others
only have historical interest.
(
nf,
Lpr,
Ld,
pl,{v = 'x})
Given nf a number field in nf or bnf format,
a t_VEC
Lpr of primes of nf and a t_VEC
Ld of
positive integers of the same length, a t_VECSMALL
pl of length
r_1
the number of real places of nf, computes a polynomial with
coefficients in nf defining a cyclic extension of nf of
minimal degree satisfying certain local conditions:
@3* at the prime Lpr[i]
, the extension has local degree a multiple of
Ld[i]
;
@3* at the i
-th real place of nf, it is complex if pl[i] = -1
(no condition if pl[i] = 0
).
The extension has degree the LCM of the local degrees. Currently, the degree
is restricted to be a prime power for the search, and to be prime for the
construction because of the rnfkummer
restrictions.
When nf is Q, prime integers are accepted instead of prid
structures. However, their primality is not checked and the behaviour is
undefined if you provide a composite number.
@3Warning. If the number field nf does not contain the n
-th
roots of unity where n
is the degree of the extension to be computed,
triggers the computation of the bnf of nf(
zeta_n)
, which may be
costly.
? nf = nfinit(y^2-5); ? pr = idealprimedec(nf,13)[1]; ? pol = nfgrunwaldwang(nf, [pr], [2], [0,-1], 'x) %3 = x^2 + Mod(3/2*y + 13/2, y^2 - 5)
The library syntax is GEN
nfgrunwaldwang(GEN nf, GEN Lpr, GEN Ld, GEN pl, long v = -1)
where v
is a variable number.
nfhilbert(
nf,a,b,{
pr})
If pr is omitted,
compute the global quadratic Hilbert symbol (a,b)
in nf, that
is 1
if x^2 - a y^2 - b z^2
has a non trivial solution (x,y,z)
in
nf, and -1
otherwise. Otherwise compute the local symbol modulo
the prime ideal pr, as output by idealprimedec
.
The library syntax is long
nfhilbert0(GEN nf, GEN a, GEN b, GEN pr = NULL)
.
Also available is long
nfhilbert(GEN bnf,GEN a,GEN b)
(global
quadratic Hilbert symbol).
(
nf,x,{
flag = 0})
Given a pseudo-matrix (A,I)
, finds a
pseudo-basis (B,J)
in Hermite normal form of the module it generates.
If flag is non-zero, also return the transformation matrix U
such that
AU = [0|B]
.
The library syntax is GEN
nfhnf0(GEN nf, GEN x, long flag)
.
Also available:
GEN
nfhnf(GEN nf, GEN x)
(flag = 0
).
GEN
rnfsimplifybasis(GEN bnf, GEN x)
simplifies the pseudo-basis
given by x = (A,I)
. The ideals in the list I
are integral, primitive and
either trivial (equal to the full ring of integer) or non-principal.
(
nf,x,
detx)
Given a pseudo-matrix (A,I)
and an ideal detx which is contained in (read integral multiple of) the
determinant of (A,I)
, finds a pseudo-basis in Hermite normal form
of the module generated by (A,I)
. This avoids coefficient explosion.
detx can be computed using the function nfdetint
.
The library syntax is GEN
nfhnfmod(GEN nf, GEN x, GEN detx)
.
(
pol,{
flag = 0})
pol being a non-constant,
preferably monic, irreducible polynomial in Z[X]
, initializes a
number field structure (nf
) attached to the field K
defined
by pol. As such, it's a technical object passed as the first argument
to most nf
xxx functions, but it contains some information which
may be directly useful. Access to this information via member
functions is preferred since the specific data organization given below
may change in the future. Currently, nf
is a row vector with 9
components:
nf[1]
contains the polynomial pol (nf.pol
).
nf[2]
contains [r1,r2]
(nf.sign
, nf.r1
,
nf.r2
), the number of real and complex places of K
.
nf[3]
contains the discriminant d(K)
(nf.disc
) of K
.
nf[4]
contains the index of nf[1]
(nf.index
),
i.e. [
Z_K :
Z[
theta]]
, where theta is any root of nf[1]
.
nf[5]
is a vector containing 7 matrices M
, G
, roundG, T
,
MD
, TI
, MDI
useful for certain computations in the number field K
.
* M
is the (r1+r2) x n
matrix whose columns represent
the numerical values of the conjugates of the elements of the integral
basis.
* G
is an n x n
matrix such that T2 = ^t G G
,
where T2
is the quadratic form T_2(x) =
sum |
sigma(x)|^2
, sigma
running over the embeddings of K
into C.
* roundG is a rescaled copy of G
, rounded to nearest
integers.
* T
is the n x n
matrix whose coefficients are
Tr(
omega_i
omega_j)
where the omega_i
are the elements of the
integral basis. Note also that det (T)
is equal to the discriminant of the
field K
. Also, when understood as an ideal, the matrix T^{-1}
generates the codifferent ideal.
* The columns of MD
(nf.diff
) express a Z-basis
of the different of K
on the integral basis.
* TI
is equal to the primitive part of T^{-1}
, which has integral
coefficients.
* Finally, MDI
is a two-element representation (for faster
ideal product) of d(K)
times the codifferent ideal
(nf.disc*
nf.codiff
, which is an integral ideal). MDI
is only used in idealinv
.
nf[6]
is the vector containing the r1+r2
roots
(nf.roots
) of nf[1]
corresponding to the r1+r2
embeddings of the number field into C (the first r1
components are real,
the next r2
have positive imaginary part).
nf[7]
is an integral basis for Z_K
(nf.zk
) expressed
on the powers of theta. Its first element is guaranteed to be 1
. This
basis is LLL-reduced with respect to T_2
(strictly speaking, it is a
permutation of such a basis, due to the condition that the first element be
1
).
nf[8]
is the n x n
integral matrix expressing the power
basis in terms of the integral basis, and finally
nf[9]
is the n x n^2
matrix giving the multiplication table
of the integral basis.
If a non monic polynomial is input, nfinit
will transform it into a
monic one, then reduce it (see flag = 3
). It is allowed, though not very
useful given the existence of nfnewprec
, to input a nf or a
bnf instead of a polynomial. It is also allowed to
input a rnf, in which case an nf
structure attached to the
absolute defining polynomial polabs
is returned (flag is then ignored).
? nf = nfinit(x^3 - 12); \\ initialize number field Q[X] / (X^3 - 12) ? nf.pol \\ defining polynomial %2 = x^3 - 12 ? nf.disc \\ field discriminant %3 = -972 ? nf.index \\ index of power basis order in maximal order %4 = 2 ? nf.zk \\ integer basis, lifted to Q[X] %5 = [1, x, 1/2*x^2] ? nf.sign \\ signature %6 = [1, 1] ? factor(abs(nf.disc )) \\ determines ramified primes %7 = [2 2]
[3 5] ? idealfactor(nf, 2) %8 = [[2, [0, 0, -1]~, 3, 1, [0, 1, 0]~] 3] \\ B<p>_2^3
@3Huge discriminants, helping nfdisc.
In case pol has a huge discriminant which is difficult to factor,
it is hard to compute from scratch the maximal order. The special input
format [
pol, B]
is also accepted where pol is a polynomial as
above and B
has one of the following forms
@3* an integer basis, as would be computed by nfbasis
: a vector of
polynomials with first element 1
. This is useful if the maximal order is
known in advance.
@3* an argument listP
which specifies a list of primes (see
nfbasis
). Instead of the maximal order, nfinit
then computes an
order which is maximal at these particular primes as well as the primes
contained in the private prime table (see addprimes
). The result is
unconditionaly correct when the discriminant nf.disc
factors
completely over this set of primes. The function nfcertify
automates
this:
? pol = polcompositum(x^5 - 101, polcyclo(7))[1]; ? nf = nfinit( [pol, 10^3] ); ? nfcertify(nf) %3 = []
@3A priori, nf.zk
defines an order which is only known
to be maximal at all primes <= 10^3
(no prime <= 10^3
divides
nf.index
). The certification step proves the correctness of the
computation.
If flag = 2
: pol is changed into another polynomial P
defining the same
number field, which is as simple as can easily be found using the
polredbest
algorithm, and all the subsequent computations are done
using this new polynomial. In particular, the first component of the result
is the modified polynomial.
If flag = 3
, apply polredbest
as in case 2, but outputs
[
nf,Mod(a,P)]
, where nf is as before and
Mod(a,P) = Mod(x,
pol)
gives the change of
variables. This is implicit when pol is not monic: first a linear change
of variables is performed, to get a monic polynomial, then polredbest
.
The library syntax is GEN
nfinit0(GEN pol, long flag, long prec)
.
Also available are
GEN
nfinit(GEN x, long prec)
(flag = 0
),
GEN
nfinitred(GEN x, long prec)
(flag = 2
),
GEN
nfinitred2(GEN x, long prec)
(flag = 3
).
Instead of the above hardcoded numerical flags in nfinit0
, one should
rather use
GEN
nfinitall(GEN x, long flag, long prec)
, where flag is an
or-ed combination of
@3* nf_RED
: find a simpler defining polynomial,
@3* nf_ORIG
: if nf_RED
set, also return the change of variable,
@3* nf_ROUND2
: Deprecated. Slow down the routine by using an
obsolete normalization algorithm (do not use this one!),
@3* nf_PARTIALFACT
: Deprecated. Lazy factorization of the
polynomial discriminant. Result is conditional unless nfcertify
can certify it.
(
nf,x)
Returns 1 if x
is an ideal in the number field nf, 0 otherwise.
The library syntax is long
isideal(GEN nf, GEN x)
.
(x,y)
Tests whether the number field K
defined
by the polynomial x
is conjugate to a subfield of the field L
defined
by y
(where x
and y
must be in Q[X]
). If they are not, the output
is the number 0. If they are, the output is a vector of polynomials, each
polynomial a
representing an embedding of K
into L
, i.e. being such
that y | x o a
.
If y
is a number field (nf), a much faster algorithm is used
(factoring x
over y
using nffactor
). Before version 2.0.14, this
wasn't guaranteed to return all the embeddings, hence was triggered by a
special flag. This is no more the case.
The library syntax is GEN
nfisincl(GEN x, GEN y)
.
(x,y)
As nfisincl
, but tests for isomorphism. If either x
or y
is a
number field, a much faster algorithm will be used.
The library syntax is GEN
nfisisom(GEN x, GEN y)
.
(
nf,
pr,a,n)
Let nf be a number field structure attached to K
,
let a \in K
and let pr be a prid attched to the
maximal ideal v
. Return 1
if a
is an n
-th power in the completed
local field K_v
, and 0
otherwise.
? K = nfinit(y^2+1); ? P = idealprimedec(K,2)[1]; \\ the ramified prime above 2 ? nfislocalpower(K,P,-1, 2) \\ -1 is a square %3 = 1 ? nfislocalpower(K,P,-1, 4) \\ ... but not a 4-th power %4 = 0 ? nfislocalpower(K,P,2, 2) \\ 2 is not a square %5 = 0
? Q = idealprimedec(K,5)[1]; \\ a prime above 5 ? nfislocalpower(K,Q, [0, 32]~, 30) \\ 32*I is locally a 30-th power %7 = 1
The library syntax is long
nfislocalpower(GEN nf, GEN pr, GEN a, GEN n)
.
(
nf,x,
pr)
This function is obsolete, use nfmodpr
.
Kernel of the matrix a
in Z_K/
pr, where pr is in
modpr format (see nfmodprinit
).
The library syntax is GEN
nfkermodpr(GEN nf, GEN x, GEN pr)
.
This function is normally useless in library mode. Project your
inputs to the residue field using nfM_to_FqM
, then work there.
(
nf,x,
pr)
Map x
to the residue field modulo pr, to a t_FFELT
.
The argument pr is either a maximal ideal in idealprimedec
format or, preferably, a modpr
structure from nfmodprinit
. The
function nfmodprlift
allows to lift back to Z_K
.
Note that the function applies to number field elements and not to
vector / matrices / polynomials of such. Use apply
to convert
recursive structures.
? K = nfinit(y^3-250); ? P = idealprimedec(K, 5)[2] ? modP = nfmodprinit(K,P); ? K.zk %4 = [1, 1/5*y, 1/25*y^2] ? apply(t->nfmodpr(K,t,modP), K.zk) %5 = [1, y, 2*y + 1]
The library syntax is GEN
nfmodpr(GEN nf, GEN x, GEN pr)
.
(
nf,
pr)
Transforms the prime ideal pr into modpr
format necessary
for all operations modulo pr in the number field nf.
The functions nfmodpr
and nfmodprlift
allow to project
to and lift from the residue field.
The library syntax is GEN
nfmodprinit(GEN nf, GEN pr)
.
(
nf,x,
pr)
Lift the t_FFELT
x
(from nfmodpr
) to the residue field
modulo pr. Vectors and matrices are also supported. For polynomials,
use apply
and the present function.
The argument pr
is either a maximal ideal in idealprimedec
format or, preferably, a modpr
structure from nfmodprinit
.
There are no compatibility checks to try and decide whether x
is attached
the same residue field as defined by pr
: the result is undefined
if not.
The function nfmodpr
allows to reduce to the residue field.
? K = nfinit(y^3-250); ? P = idealprimedec(K, 5)[2] ? modP = nfmodprinit(K,P); ? K.zk %4 = [1, 1/5*y, 1/25*y^2] ? apply(t->nfmodpr(K,t,modP), K.zk) %5 = [1, y, 2*y + 1] ? nfmodprlift(K, %, modP) %6 = [1, 1/5*y, 2/5*y + 1] ? nfeltval(K, %[3] - K.zk[3], P) %7 = 1
The library syntax is GEN
nfmodprlift(GEN nf, GEN x, GEN pr)
.
(
nf)
Transforms the number field nf into the corresponding data using current (usually larger) precision. This function works as expected if nf is in fact a bnf or a bnr (update structure to current precision) but may be quite slow: many generators of principal ideals have to be computed; note that in this latter case, the bnf must contain fundamental units.
The library syntax is GEN
nfnewprec(GEN nf, long prec)
.
See also GEN
bnfnewprec(GEN bnf, long prec)
and
GEN
bnrnewprec(GEN bnr, long prec)
.
nfroots({
nf},x)
Roots of the polynomial x
in the
number field nf given by nfinit
without multiplicity (in Q
if nf is omitted). x
has coefficients in the number field (scalar,
polmod, polynomial, column vector). The main variable of nf must be
of lower priority than that of x
(see Label se:priority). However if the
coefficients of the number field occur explicitly (as polmods) as
coefficients of x
, the variable of these polmods must be the same as
the main variable of t
(see nffactor
).
It is possible to input a defining polynomial for nf
instead, but this is in general less efficient since parts of an nf
structure will then be computed internally. This is useful in two
situations: when you do not need the nf
elsewhere, or when you cannot
initialize an nf
due to integer factorization difficulties when
attempting to compute the field discriminant and maximal order.
@3Caveat. nfinit([T, listP])
allows to compute in polynomial
time a conditional nf structure, which sets nf.zk
to an order
which is not guaranteed to be maximal at all primes. Always either use
nfcertify
first (which may not run in polynomial time) or make sure
to input nf.pol
instead of the conditional nf: nfroots
is
able to recover in polynomial time in this case, instead of potentially
missing a factor.
The library syntax is GEN
nfroots(GEN nf = NULL, GEN x)
.
See also GEN
nfrootsQ(GEN x)
,
corresponding to nf = NULL
.
(
nf)
Returns a two-component vector [w,z]
where w
is the number of roots of
unity in the number field nf, and z
is a primitive w
-th root
of unity.
? K = nfinit(polcyclo(11)); ? nfrootsof1(K) %2 = [22, [0, 0, 0, 0, 0, -1, 0, 0, 0, 0]~] ? z = nfbasistoalg(K, %[2]) \\ in algebraic form %3 = Mod(-x^5, x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) ? [lift(z^11), lift(z^2)] \\ proves that the order of z is 22 %4 = [-1, -x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1]
This function guesses the number w
as the gcd of the #k(v)^*
for
unramified v
above odd primes, then computes the roots in nf
of the w
-th cyclotomic polynomial: the algorithm is polynomial time with
respect to the field degree and the bitsize of the multiplication table in
nf (both of them polynomially bounded in terms of the size of the
discriminant). Fields of degree up to 100
or so should require less than
one minute.
The library syntax is GEN
rootsof1(GEN nf)
.
Also available is GEN
rootsof1_kannan(GEN nf)
, that computes
all algebraic integers of T_2
norm equal to the field degree
(all roots of 1
, by Kronecker's theorem). This is in general a little
faster than the default when there are roots of 1
in the field
(say twice faster), but can be much slower (say, days slower), since
the algorithm is a priori exponential in the field degree.
(
nf,x,{
flag = 0})
Given a torsion Z_K
-module x
attached to the square integral
invertible pseudo-matrix (A,I,J)
, returns an ideal list
D = [d_1,...,d_n]
which is the Smith normal form of x
. In other
words, x
is isomorphic to Z_K/d_1\oplus...\oplus
Z_K/d_n
and d_i
divides d_{i-1}
for i >= 2
. If flag is non-zero return [D,U,V]
, where
UAV
is the identity.
See Label se:ZKmodules for the definition of integral pseudo-matrix;
briefly, it is input as a 3-component row vector [A,I,J]
where
I = [b_1,...,b_n]
and J = [a_1,...,a_n]
are two ideal lists,
and A
is a square n x n
matrix with columns (A_1,...,A_n)
,
seen as elements in K^n
(with canonical basis (e_1,...,e_n)
).
This data defines the Z_K
module x
given by
(b_1e_1\oplus...\oplus b_ne_n) / (a_1A_1\oplus...\oplus a_nA_n)
,
The integrality condition is a_{i,j} \in b_i a_j^{-1}
for all i,j
. If it
is not satisfied, then the d_i
will not be integral. Note that every
finitely generated torsion module is isomorphic to a module of this form and
even with b_i = Z_K
for all i
.
The library syntax is GEN
nfsnf0(GEN nf, GEN x, long flag)
.
Also available:
GEN
nfsnf(GEN nf, GEN x)
(flag = 0
).
(
nf,a,b,P)
This function is obsolete, use nfmodpr
.
Let P
be a prime ideal in modpr format (see nfmodprinit
),
let a
be a matrix, invertible over the residue field, and let b
be
a column vector or matrix. This function returns a solution of a.x =
b
; the coefficients of x
are lifted to nf elements.
? K = nfinit(y^2+1); ? P = idealprimedec(K, 3)[1]; ? P = nfmodprinit(K, P); ? a = [y+1, y; y, 0]; b = [1, y]~ ? nfsolvemodpr(K, a,b, P) %5 = [1, 2]~
The library syntax is GEN
nfsolvemodpr(GEN nf, GEN a, GEN b, GEN P)
.
This function is normally useless in library mode. Project your
inputs to the residue field using nfM_to_FqM
, then work there.
(
nf,{d})
Defining polynomial over Q for the splitting field of nf;
if d
is given, it must be a multiple of the splitting field degree.
Instead of nf
, it is possible to input a defining (irreducible)
polynomial T
for nf
, but in general this is less efficient.
? K = nfinit(x^3-2); ? nfsplitting(K) %2 = x^6 + 108 ? nfsplitting(x^8-2) %3 = x^16 + 272*x^8 + 64
Specifying the degree of the splitting field can make the computation faster.
? nfsplitting(x^17-123); time = 3,607 ms. ? poldegree(%) %2 = 272 ? nfsplitting(x^17-123,272); time = 150 ms. ? nfsplitting(x^17-123,273); *** nfsplitting: Warning: ignoring incorrect degree bound 273 time = 3,611 ms.
The complexity of the algorithm is polynomial in the degree d
of the
splitting field and the bitsize of T
; if d
is large the result will
likely be unusable, e.g. nfinit
will not be an option:
? nfsplitting(x^6-x-1) [... degree 720 polynomial deleted ...] time = 11,020 ms.
The library syntax is GEN
nfsplitting(GEN nf, GEN d = NULL)
.
(
pol,{d = 0})
Finds all subfields of degree
d
of the number field defined by the (monic, integral) polynomial
pol (all subfields if d
is null or omitted). The result is a vector
of subfields, each being given by [g,h]
, where g
is an absolute equation
and h
expresses one of the roots of g
in terms of the root x
of the
polynomial defining nf. This routine uses J. Klüners's algorithm
in the general case, and B. Allombert's galoissubfields
when nf
is Galois (with weakly supersolvable Galois group).
The library syntax is GEN
nfsubfields(GEN pol, long d)
.
(P,Q,{
flag = 0})
P
and Q
being squarefree polynomials in Z[X]
in the same variable, outputs
the simple factors of the étale Q-algebra A =
Q(X, Y) / (P(X), Q(Y))
.
The factors are given by a list of polynomials R
in Z[X]
, attached to
the number field Q(X)/ (R)
, and sorted by increasing degree (with respect
to lexicographic ordering for factors of equal degrees). Returns an error if
one of the polynomials is not squarefree.
Note that it is more efficient to reduce to the case where P
and Q
are
irreducible first. The routine will not perform this for you, since it may be
expensive, and the inputs are irreducible in most applications anyway. In
this case, there will be a single factor R
if and only if the number
fields defined by P
and Q
are linearly disjoint (their intersection is
Q).
Assuming P
is irreducible (of smaller degree than Q
for efficiency), it
is in general much faster to proceed as follows
nf = nfinit(P); L = nffactor(nf, Q)[,1]; vector(#L, i, rnfequation(nf, L[i]))
to obtain the same result. If you are only interested in the degrees of the
simple factors, the rnfequation
instruction can be replaced by a
trivial poldegree(P) * poldegree(L[i])
.
The binary digits of flag mean
1: outputs a vector of 4-component vectors [R,a,b,k]
, where R
ranges through the list of all possible compositums as above, and a
(resp. b
) expresses the root of P
(resp. Q
) as an element of
Q(X)/(R)
. Finally, k
is a small integer such that b + ka = X
modulo
R
.
2: assume that P
and Q
define number fields which are linearly disjoint:
both polynomials are irreducible and the corresponding number fields
have no common subfield besides Q. This allows to save a costly
factorization over Q. In this case return the single simple factor
instead of a vector with one element.
A compositum is often defined by a complicated polynomial, which it is
advisable to reduce before further work. Here is an example involving
the field Q(
zeta_5, 5^{1/5})
:
? L = polcompositum(x^5 - 5, polcyclo(5), 1); \\ list of [R,a,b,k] ? [R, a] = L[1]; \\ pick the single factor, extract R,a (ignore b,k) ? R \\ defines the compositum %3 = x^20 + 5*x^19 + 15*x^18 + 35*x^17 + 70*x^16 + 141*x^15 + 260*x^14\ + 355*x^13 + 95*x^12 - 1460*x^11 - 3279*x^10 - 3660*x^9 - 2005*x^8 \ + 705*x^7 + 9210*x^6 + 13506*x^5 + 7145*x^4 - 2740*x^3 + 1040*x^2 \ - 320*x + 256 ? a^5 - 5 \\ a fifth root of 5 %4 = 0 ? [T, X] = polredbest(R, 1); ? T \\ simpler defining polynomial for B<Q>[x]/(R) %6 = x^20 + 25*x^10 + 5 ? X \\ root of R in B<Q>[y]/(T(y)) %7 = Mod(-1/11*x^15 - 1/11*x^14 + 1/22*x^10 - 47/22*x^5 - 29/11*x^4 + 7/22,\ x^20 + 25*x^10 + 5) ? a = subst(a.pol, 'x, X) \\ C<a> in the new coordinates %8 = Mod(1/11*x^14 + 29/11*x^4, x^20 + 25*x^10 + 5) ? a^5 - 5 %9 = 0
@3In the above example, x^5-5
and the 5
-th cyclotomic
polynomial are irreducible over Q; they have coprime degrees so
define linearly disjoint extensions and we could have started by
? [R,a] = polcompositum(x^5 - 5, polcyclo(5), 3); \\ [R,a,b,k]
The library syntax is GEN
polcompositum0(GEN P, GEN Q, long flag)
.
Also available are
GEN
compositum(GEN P, GEN Q)
(flag = 0
) and
GEN
compositum2(GEN P, GEN Q)
(flag = 1
).
(T)
Galois group of the non-constant
polynomial T\in
Q[X]
. In the present version 2.9.1, T
must be irreducible
and the degree d
of T
must be less than or equal to 7. If the
galdata
package has been installed, degrees 8, 9, 10 and 11 are also
implemented. By definition, if K =
Q[x]/(T)
, this computes the action of
the Galois group of the Galois closure of K
on the d
distinct roots of
T
, up to conjugacy (corresponding to different root orderings).
The output is a 4-component vector [n,s,k,name]
with the
following meaning: n
is the cardinality of the group, s
is its signature
(s = 1
if the group is a subgroup of the alternating group A_d
, s = -1
otherwise) and name is a character string containing name of the transitive
group according to the GAP 4 transitive groups library by Alexander Hulpke.
k
is more arbitrary and the choice made up to version 2.2.3 of PARI is rather
unfortunate: for d > 7
, k
is the numbering of the group among all
transitive subgroups of S_d
, as given in ``The transitive groups of degree up
to eleven'', G. Butler and J. McKay, Communications in Algebra, vol. 11,
1983,
pp. 863--911 (group k
is denoted T_k
there). And for d <= 7
, it was ad
hoc, so as to ensure that a given triple would denote a unique group.
Specifically, for polynomials of degree d <= 7
, the groups are coded as
follows, using standard notations
In degree 1: S_1 = [1,1,1]
.
In degree 2: S_2 = [2,-1,1]
.
In degree 3: A_3 = C_3 = [3,1,1]
, S_3 = [6,-1,1]
.
In degree 4: C_4 = [4,-1,1]
, V_4 = [4,1,1]
, D_4 = [8,-1,1]
, A_4 = [12,1,1]
,
S_4 = [24,-1,1]
.
In degree 5: C_5 = [5,1,1]
, D_5 = [10,1,1]
, M_{20} = [20,-1,1]
,
A_5 = [60,1,1]
, S_5 = [120,-1,1]
.
In degree 6: C_6 = [6,-1,1]
, S_3 = [6,-1,2]
, D_6 = [12,-1,1]
, A_4 = [12,1,1]
,
G_{18} = [18,-1,1]
, S_4^ -= [24,-1,1]
, A_4 x C_2 = [24,-1,2]
,
S_4^ += [24,1,1]
, G_{36}^ -= [36,-1,1]
, G_{36}^ += [36,1,1]
,
S_4 x C_2 = [48,-1,1]
, A_5 = PSL_2(5) = [60,1,1]
, G_{72} = [72,-1,1]
,
S_5 = PGL_2(5) = [120,-1,1]
, A_6 = [360,1,1]
, S_6 = [720,-1,1]
.
In degree 7: C_7 = [7,1,1]
, D_7 = [14,-1,1]
, M_{21} = [21,1,1]
,
M_{42} = [42,-1,1]
, PSL_2(7) = PSL_3(2) = [168,1,1]
, A_7 = [2520,1,1]
,
S_7 = [5040,-1,1]
.
This is deprecated and obsolete, but for reasons of backward compatibility,
we cannot change this behavior yet. So you can use the default
new_galois_format
to switch to a consistent naming scheme, namely k
is
always the standard numbering of the group among all transitive subgroups of
S_n
. If this default is in effect, the above groups will be coded as:
In degree 1: S_1 = [1,1,1]
.
In degree 2: S_2 = [2,-1,1]
.
In degree 3: A_3 = C_3 = [3,1,1]
, S_3 = [6,-1,2]
.
In degree 4: C_4 = [4,-1,1]
, V_4 = [4,1,2]
, D_4 = [8,-1,3]
, A_4 = [12,1,4]
,
S_4 = [24,-1,5]
.
In degree 5: C_5 = [5,1,1]
, D_5 = [10,1,2]
, M_{20} = [20,-1,3]
,
A_5 = [60,1,4]
, S_5 = [120,-1,5]
.
In degree 6: C_6 = [6,-1,1]
, S_3 = [6,-1,2]
, D_6 = [12,-1,3]
, A_4 = [12,1,4]
,
G_{18} = [18,-1,5]
, A_4 x C_2 = [24,-1,6]
, S_4^ += [24,1,7]
,
S_4^ -= [24,-1,8]
, G_{36}^ -= [36,-1,9]
, G_{36}^ += [36,1,10]
,
S_4 x C_2 = [48,-1,11]
, A_5 = PSL_2(5) = [60,1,12]
, G_{72} = [72,-1,13]
,
S_5 = PGL_2(5) = [120,-1,14]
, A_6 = [360,1,15]
, S_6 = [720,-1,16]
.
In degree 7: C_7 = [7,1,1]
, D_7 = [14,-1,2]
, M_{21} = [21,1,3]
,
M_{42} = [42,-1,4]
, PSL_2(7) = PSL_3(2) = [168,1,5]
, A_7 = [2520,1,6]
,
S_7 = [5040,-1,7]
.
@3Warning. The method used is that of resolvent polynomials and is sensitive to the current precision. The precision is updated internally but, in very rare cases, a wrong result may be returned if the initial precision was not sufficient.
The library syntax is GEN
polgalois(GEN T, long prec)
.
To enable the new format in library mode,
set the global variable new_galois_format
to 1
.
(T,{
flag = 0})
This function is deprecated, use polredbest
instead.
Finds polynomials with reasonably small coefficients defining subfields of
the number field defined by T
. One of the polynomials always defines Q
(hence is equal to x-1
), and another always defines the same number field
as T
if T
is irreducible.
All T
accepted by nfinit
are also allowed here;
in particular, the format [T, listP]
is recommended, e.g. with
listP = 10^5
or a vector containing all ramified primes. Otherwise,
the maximal order of Q[x]/(T)
must be computed.
The following binary digits of flag are significant:
1: Possibly use a suborder of the maximal order. The
primes dividing the index of the order chosen are larger than
primelimit
or divide integers stored in the addprimes
table.
This flag is deprecated, the [T, listP]
format is more
flexible.
2: gives also elements. The result is a two-column matrix, the first column giving primitive elements defining these subfields, the second giving the corresponding minimal polynomials.
? M = polred(x^4 + 8, 2) %1 = [1 x - 1]
[1/2*x^2 x^2 + 2]
[1/4*x^3 x^4 + 2]
[x x^4 + 8] ? minpoly(Mod(M[2,1], x^4+8)) %2 = x^2 + 2
The library syntax is polred(GEN T)
(flag = 0
). Also available is
GEN
polred2(GEN T)
(flag = 2
). The function polred0
is
deprecated, provided for backward compatibility.
(T,{
flag = 0})
Returns a canonical defining polynomial P
for the number field
Q[X]/(T)
defined by T
, such that the sum of the squares of the modulus
of the roots (i.e. the T_2
-norm) is minimal. Different T
defining
isomorphic number fields will yield the same P
. All T
accepted by
nfinit
are also allowed here, e.g. non-monic polynomials, or pairs
[T, listP]
specifying that a non-maximal order may be used. For
convenience, any number field structure (nf, bnf,...) can also
be used instead of T
.
? polredabs(x^2 + 16) %1 = x^2 + 1 ? K = bnfinit(x^2 + 16); polredabs(K) %2 = x^2 + 1
@3Warning 1. Using a t_POL
T
requires computing
and fully factoring the discriminant d_K
of the maximal order which may be
very hard. You can use the format [T, listP]
, where listP
encodes a list of known coprime divisors of disc (T)
(see ??nfbasis
),
to help the routine, thereby replacing this part of the algorithm by a
polynomial time computation But this may only compute a suborder of the
maximal order, when the divisors are not squarefree or do not include all
primes dividing d_K
. The routine attempts to certify the result
independently of this order computation as per nfcertify
: we try to
prove that the computed order is maximal. If the certification fails,
the routine then fully factors the integers returned by nfcertify
.
You can use polredbest
or polredabs(,16)
to avoid this
factorization step; in both cases, the result is no longer canonical.
@3Warning 2. Apart from the factorization of the discriminant of
T
, this routine runs in polynomial time for a fixed degree.
But the complexity is exponential in the degree: this routine
may be exceedingly slow when the number field has many subfields, hence a
lot of elements of small T_2
-norm. If you do not need a canonical
polynomial, the function polredbest
is in general much faster (it runs
in polynomial time), and tends to return polynomials with smaller
discriminants.
The binary digits of flag mean
1: outputs a two-component row vector [P,a]
, where P
is the default
output and Mod(a, P)
is a root of the original T
.
4: gives all polynomials of minimal T_2
norm; of the two polynomials
P(x)
and +- P(-x)
, only one is given.
16: Possibly use a suborder of the maximal order, without attempting to
certify the result as in Warning 1: we always return a polynomial and never
0
. The result is a priori not canonical.
? T = x^16 - 136*x^14 + 6476*x^12 - 141912*x^10 + 1513334*x^8 \ - 7453176*x^6 + 13950764*x^4 - 5596840*x^2 + 46225 ? T1 = polredabs(T); T2 = polredbest(T); ? [ norml2(polroots(T1)), norml2(polroots(T2)) ] %3 = [88.0000000, 120.000000] ? [ sizedigit(poldisc(T1)), sizedigit(poldisc(T2)) ] %4 = [75, 67]
The library syntax is GEN
polredabs0(GEN T, long flag)
.
Instead of the above hardcoded numerical flags, one should use an
or-ed combination of
@3* nf_PARTIALFACT
: possibly use a suborder of the maximal order,
without attempting to certify the result.
@3* nf_ORIG
: return [P, a]
, where Mod(a, P)
is a root of T
.
@3* nf_RAW
: return [P, b]
, where Mod(b, T)
is a root of P
.
The algebraic integer b
is the raw result produced by the small vectors
enumeration in the maximal order; P
was computed as the characteristic
polynomial of Mod(b, T)
. Mod(a, P)
as in nf_ORIG
is obtained with modreverse
.
@3* nf_ADDZK
: if r
is the result produced with some of the above
flags (of the form P
or [P,c]
), return [r,zk]
, where zk
is a
Z-basis for the maximal order of Q[X]/(P)
.
@3* nf_ALL
: return a vector of results of the above form, for all
polynomials of minimal T_2
-norm.
(T,{
flag = 0})
Finds a polynomial with reasonably
small coefficients defining the same number field as T
.
All T
accepted by nfinit
are also allowed here (e.g. non-monic
polynomials, nf
, bnf
, [T,Z_K_basis]
). Contrary to
polredabs
, this routine runs in polynomial time, but it offers no
guarantee as to the minimality of its result.
This routine computes an LLL-reduced basis for the ring of integers of
Q[X]/(T)
, then examines small linear combinations of the basis vectors,
computing their characteristic polynomials. It returns the separable
P
polynomial of smallest discriminant (the one with lexicographically
smallest abs(Vec(P))
in case of ties). This is a good candidate
for subsequent number field computations, since it guarantees that
the denominators of algebraic integers, when expressed in the power basis,
are reasonably small. With no claim of minimality, though.
It can happen that iterating this functions yields better and better polynomials, until it stabilizes:
? \p5 ? P = X^12+8*X^8-50*X^6+16*X^4-3069*X^2+625; ? poldisc(P)*1. %2 = 1.2622 E55 ? P = polredbest(P); ? poldisc(P)*1. %4 = 2.9012 E51 ? P = polredbest(P); ? poldisc(P)*1. %6 = 8.8704 E44
@3In this example, the initial polynomial P
is the one
returned by polredabs
, and the last one is stable.
If flag = 1
: outputs a two-component row vector [P,a]
, where P
is the
default output and Mod(a, P)
is a root of the original T
.
? [P,a] = polredbest(x^4 + 8, 1) %1 = [x^4 + 2, Mod(x^3, x^4 + 2)] ? charpoly(a) %2 = x^4 + 8
@3In particular, the map Q[x]/(T) \to
Q[x]/(P)
,
x:--->Mod(a,P)
defines an isomorphism of number fields, which can
be computed as
subst(lift(Q), 'x, a)
@3if Q
is a t_POLMOD
modulo T
; b = modreverse(a)
returns a t_POLMOD
giving the inverse of the above map (which should be
useless since Q[x]/(P)
is a priori a better representation for the number
field and its elements).
The library syntax is GEN
polredbest(GEN T, long flag)
.
(x)
This function is obsolete, use polredbest.
The library syntax is GEN
polredord(GEN x)
.
(x)
Applies a random Tschirnhausen
transformation to the polynomial x
, which is assumed to be non-constant
and separable, so as to obtain a new equation for the étale algebra
defined by x
. This is for instance useful when computing resolvents,
hence is used by the polgalois
function.
The library syntax is GEN
tschirnhaus(GEN x)
.
(
rnf,x)
Expresses x
on the relative
integral basis. Here, rnf is a relative number field extension L/K
as output by rnfinit
, and x
an element of L
in absolute form, i.e.
expressed as a polynomial or polmod with polmod coefficients, not on
the relative integral basis.
The library syntax is GEN
rnfalgtobasis(GEN rnf, GEN x)
.
(
bnf,M)
Let K
the field represented by
bnf, as output by bnfinit
. M
is a projective Z_K
-module
of rank n
(M\otimes K
is an n
-dimensional K
-vector space), given by a
pseudo-basis of size n
. The routine returns either a true Z_K
-basis of
M
(of size n
) if it exists, or an n+1
-element generating set of M
if
not.
It is allowed to use an irreducible polynomial P
in K[X]
instead of M
,
in which case, M
is defined as the ring of integers of K[X]/(P)
, viewed
as a Z_K
-module.
The library syntax is GEN
rnfbasis(GEN bnf, GEN M)
.
(
rnf,x)
Computes the representation of x
as a polmod with polmods coefficients. Here, rnf is a relative number
field extension L/K
as output by rnfinit
, and x
an element of
L
expressed on the relative integral basis.
The library syntax is GEN
rnfbasistoalg(GEN rnf, GEN x)
.
rnfcharpoly(
nf,T,a,{
var = 'x})
Characteristic polynomial of
a
over nf, where a
belongs to the algebra defined by T
over
nf, i.e. nf[X]/(T)
. Returns a polynomial in variable v
(x
by default).
? nf = nfinit(y^2+1); ? rnfcharpoly(nf, x^2+y*x+1, x+y) %2 = x^2 + Mod(-y, y^2 + 1)*x + 1
The library syntax is GEN
rnfcharpoly(GEN nf, GEN T, GEN a, long var = -1)
where var
is a variable number.
(
bnf,
pol)
Given bnf
as output by bnfinit
, and pol a relative polynomial defining an
Abelian extension, computes the class field theory conductor of this
Abelian extension. The result is a 3-component vector
[
conductor,
bnr,
subgroup]
, where conductor is
the conductor of the extension given as a 2-component row vector
[f_0,f_ oo ]
, bnr is the attached bnr
structure
and subgroup is a matrix in HNF defining the subgroup of the ray class
group on bnr.gen
.
The library syntax is GEN
rnfconductor(GEN bnf, GEN pol)
.
rnfdedekind(
nf,
pol,{
pr},{
flag = 0})
Given a number field K
coded by nf and a monic
polynomial P\in
Z_K[X]
, irreducible over K
and thus defining a relative
extension L
of K
, applies Dedekind's criterion to the order
Z_K[X]/(P)
, at the prime ideal pr. It is possible to set pr
to a vector of prime ideals (test maximality at all primes in the vector),
or to omit altogether, in which case maximality at all primes is tested;
in this situation flag is automatically set to 1
.
The default historic behavior (flag is 0 or omitted and pr is a
single prime ideal) is not so useful since
rnfpseudobasis
gives more information and is generally not that
much slower. It returns a 3-component vector [
max,
basis, v]
:
@3* basis is a pseudo-basis of an enlarged order O
produced by
Dedekind's criterion, containing the original order Z_K[X]/(P)
with index a power of pr. Possibly equal to the original order.
@3* max is a flag equal to 1 if the enlarged order O
could be proven to be pr-maximal and to 0 otherwise; it may still be
maximal in the latter case if pr is ramified in L
,
@3* v
is the valuation at pr of the order discriminant.
If flag is non-zero, on the other hand, we just return 1
if the order
Z_K[X]/(P)
is pr-maximal (resp. maximal at all relevant primes, as
described above), and 0
if not. This is much faster than the default,
since the enlarged order is not computed.
? nf = nfinit(y^2-3); P = x^3 - 2*y; ? pr3 = idealprimedec(nf,3)[1]; ? rnfdedekind(nf, P, pr3) %3 = [1, [[1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 1, 1]], 8] ? rnfdedekind(nf, P, pr3, 1) %4 = 1
@3In this example, pr3
is the ramified ideal above 3
,
and the order generated by the cube roots of y
is already
pr3
-maximal. The order-discriminant has valuation 8
. On the other
hand, the order is not maximal at the prime above 2:
? pr2 = idealprimedec(nf,2)[1]; ? rnfdedekind(nf, P, pr2, 1) %6 = 0 ? rnfdedekind(nf, P, pr2) %7 = [0, [[2, 0, 0; 0, 1, 0; 0, 0, 1], [[1, 0; 0, 1], [1, 0; 0, 1], [1, 1/2; 0, 1/2]]], 2]
The enlarged order is not proven to be pr2
-maximal yet. In fact, it
is; it is in fact the maximal order:
? B = rnfpseudobasis(nf, P) %8 = [[1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 1, [1, 1/2; 0, 1/2]], [162, 0; 0, 162], -1] ? idealval(nf,B[3], pr2) %9 = 2
It is possible to use this routine with non-monic
P =
sum_{i <= n} a_i X^i \in
Z_K[X]
if flag = 1
;
in this case, we test maximality of Dedekind's order generated by
1, a_n
alpha, a_n
alpha^2 + a_{n-1}
alpha,...,
a_n
alpha^{n-1} + a_{n-1}
alpha^{n-2} +...+ a_1
alpha.
The routine will fail if P
is 0
on the projective line over the residue
field Z_K/pr
(FIXME).
The library syntax is GEN
rnfdedekind(GEN nf, GEN pol, GEN pr = NULL, long flag)
.
(
nf,M)
Given a pseudo-matrix M
over the maximal
order of nf, computes its determinant.
The library syntax is GEN
rnfdet(GEN nf, GEN M)
.
(
nf,
pol)
Given a number field nf as
output by nfinit
and a polynomial pol with coefficients in
nf defining a relative extension L
of nf, computes the
relative discriminant of L
. This is a two-element row vector [D,d]
, where
D
is the relative ideal discriminant and d
is the relative discriminant
considered as an element of nf^*/{
nf^*}^2
. The main variable of
nf must be of lower priority than that of pol, see
Label se:priority.
The library syntax is GEN
rnfdiscf(GEN nf, GEN pol)
.
(
rnf,x)
Let rnf be a relative
number field extension L/K
as output by rnfinit
and let x
be an
element of L
expressed as a polynomial modulo the absolute equation
rnf.pol
, or in terms of the absolute Z-basis for Z_L
if rnf contains one (as in rnfinit(nf,pol,1)
, or after
a call to nfinit(rnf)
).
Computes x
as an element of the relative extension
L/K
as a polmod with polmod coefficients.
? K = nfinit(y^2+1); L = rnfinit(K, x^2-y); ? L.polabs %2 = x^4 + 1 ? rnfeltabstorel(L, Mod(x, L.polabs)) %3 = Mod(x, x^2 + Mod(-y, y^2 + 1)) ? rnfeltabstorel(L, 1/3) %4 = 1/3 ? rnfeltabstorel(L, Mod(x, x^2-y)) %5 = Mod(x, x^2 + Mod(-y, y^2 + 1))
? rnfeltabstorel(L, [0,0,0,1]~) \\ Z_L not initialized yet *** at top-level: rnfeltabstorel(L,[0, *** ^-------------------- *** rnfeltabstorel: incorrect type in rnfeltabstorel, apply nfinit(rnf). ? nfinit(L); \\ initialize now ? rnfeltabstorel(L, [0,0,0,1]~) %6 = Mod(Mod(y, y^2 + 1)*x, x^2 + Mod(-y, y^2 + 1))
The library syntax is GEN
rnfeltabstorel(GEN rnf, GEN x)
.
(
rnf,x,{
flag = 0})
rnf being a relative number
field extension L/K
as output by rnfinit
and x
being an element of
L
expressed as a polynomial or polmod with polmod coefficients (or as a
t_COL
on nfinit(rnf).zk
), computes
x
as an element of K
as a t_POLMOD
if flag = 0
and as a t_COL
otherwise. If x
is not in K
, a domain error occurs.
? K = nfinit(y^2+1); L = rnfinit(K, x^2-y); ? L.pol %2 = x^4 + 1 ? rnfeltdown(L, Mod(x^2, L.pol)) %3 = Mod(y, y^2 + 1) ? rnfeltdown(L, Mod(x^2, L.pol), 1) %4 = [0, 1]~ ? rnfeltdown(L, Mod(y, x^2-y)) %5 = Mod(y, y^2 + 1) ? rnfeltdown(L, Mod(y,K.pol)) %6 = Mod(y, y^2 + 1) ? rnfeltdown(L, Mod(x, L.pol)) *** at top-level: rnfeltdown(L,Mod(x,x *** ^-------------------- *** rnfeltdown: domain error in rnfeltdown: element not in the base field ? rnfeltdown(L, Mod(y, x^2-y), 1) \\ as a t_COL %7 = [0, 1]~ ? rnfeltdown(L, [0,1,0,0]~) \\ not allowed without absolute nf struct *** rnfeltdown: incorrect type in rnfeltdown (t_COL). ? nfinit(L); \\ add absolute nf structure to L ? rnfeltdown(L, [0,1,0,0]~) \\ now OK %8 = Mod(y, y^2 + 1)
@3If we had started with
L = rnfinit(K, x^2-y, 1)
, then the final would have worked directly.
The library syntax is GEN
rnfeltdown0(GEN rnf, GEN x, long flag)
.
Also available is
GEN
rnfeltdown(GEN rnf, GEN x)
(flag = 0
).
(
rnf,x)
rnf being a relative number field extension L/K
as output by
rnfinit
and x
being an element of L
, returns the relative norm
N_{L/K}(x)
as an element of K
.
? K = nfinit(y^2+1); L = rnfinit(K, x^2-y); ? rnfeltnorm(L, Mod(x, L.pol)) %2 = Mod(x, x^2 + Mod(-y, y^2 + 1)) ? rnfeltnorm(L, 2) %3 = 4 ? rnfeltnorm(L, Mod(x, x^2-y))
The library syntax is GEN
rnfeltnorm(GEN rnf, GEN x)
.
(
rnf,x)
rnf being a relative
number field extension L/K
as output by rnfinit
and x
being an
element of L
expressed as a polynomial or polmod with polmod
coefficients, computes x
as an element of the absolute extension L/
Q as
a polynomial modulo the absolute equation rnf.pol
.
? K = nfinit(y^2+1); L = rnfinit(K, x^2-y); ? L.pol %2 = x^4 + 1 ? rnfeltreltoabs(L, Mod(x, L.pol)) %3 = Mod(x, x^4 + 1) ? rnfeltreltoabs(L, Mod(y, x^2-y)) %4 = Mod(x^2, x^4 + 1) ? rnfeltreltoabs(L, Mod(y,K.pol)) %5 = Mod(x^2, x^4 + 1)
The library syntax is GEN
rnfeltreltoabs(GEN rnf, GEN x)
.
(
rnf,x)
rnf being a relative number field extension L/K
as output by
rnfinit
and x
being an element of L
, returns the relative trace
Tr_{L/K}(x)
as an element of K
.
? K = nfinit(y^2+1); L = rnfinit(K, x^2-y); ? rnfelttrace(L, Mod(x, L.pol)) %2 = 0 ? rnfelttrace(L, 2) %3 = 4 ? rnfelttrace(L, Mod(x, x^2-y))
The library syntax is GEN
rnfelttrace(GEN rnf, GEN x)
.
(
rnf,x,{
flag = 0})
rnf being a relative number field extension L/K
as output by
rnfinit
and x
being an element of K
, computes x
as an element of
the absolute extension L/
Q. As a t_POLMOD
modulo rnf.pol
if flag = 0
and as a t_COL
on the absolute field integer basis if
flag = 1
.
? K = nfinit(y^2+1); L = rnfinit(K, x^2-y); ? L.pol %2 = x^4 + 1 ? rnfeltup(L, Mod(y, K.pol)) %3 = Mod(x^2, x^4 + 1) ? rnfeltup(L, y) %4 = Mod(x^2, x^4 + 1) ? rnfeltup(L, [1,2]~) \\ in terms of K.zk %5 = Mod(2*x^2 + 1, x^4 + 1) ? rnfeltup(L, y, 1) \\ in terms of nfinit(L).zk %6 = [0, 1, 0, 0]~ ? rnfeltup(L, [1,2]~, 1) %7 = [1, 2, 0, 0]~
The library syntax is GEN
rnfeltup0(GEN rnf, GEN x, long flag)
.
(
nf,
pol,{
flag = 0})
Given a number field
nf as output by nfinit
(or simply a polynomial) and a
polynomial pol with coefficients in nf defining a relative
extension L
of nf, computes an absolute equation of L
over
Q.
The main variable of nf must be of lower priority than that
of pol (see Label se:priority). Note that for efficiency, this does
not check whether the relative equation is irreducible over nf, but
only if it is squarefree. If it is reducible but squarefree, the result will
be the absolute equation of the étale algebra defined by pol. If
pol is not squarefree, raise an e_DOMAIN
exception.
? rnfequation(y^2+1, x^2 - y) %1 = x^4 + 1 ? T = y^3-2; rnfequation(nfinit(T), (x^3-2)/(x-Mod(y,T))) %2 = x^6 + 108 \\ Galois closure of Q(2^(1/3))
If flag is non-zero, outputs a 3-component row vector [z,a,k]
, where
@3* z
is the absolute equation of L
over Q, as in the default
behavior,
@3* a
expresses as a t_POLMOD
modulo z
a root alpha of the
polynomial defining the base field nf,
@3* k
is a small integer such that theta =
beta+k
alpha
is a root of z
, where beta is a root of pol.
? T = y^3-2; pol = x^2 +x*y + y^2; ? [z,a,k] = rnfequation(T, pol, 1); ? z %3 = x^6 + 108 ? subst(T, y, a) %4 = 0 ? alpha= Mod(y, T); ? beta = Mod(x*Mod(1,T), pol); ? subst(z, x, beta + k*alpha) %7 = 0
The library syntax is GEN
rnfequation0(GEN nf, GEN pol, long flag)
.
Also available are
GEN
rnfequation(GEN nf, GEN pol)
(flag = 0
) and
GEN
rnfequation2(GEN nf, GEN pol)
(flag = 1
).
(
bnf,x)
Given bnf as output by
bnfinit
, and either a polynomial x
with coefficients in bnf
defining a relative extension L
of bnf, or a pseudo-basis x
of
such an extension, gives either a true bnf-basis of L
in upper
triangular Hermite normal form, if it exists, and returns 0
otherwise.
The library syntax is GEN
rnfhnfbasis(GEN bnf, GEN x)
.
(
rnf,x)
Let rnf be a relative
number field extension L/K
as output by rnfinit
and x
be an ideal of
the absolute extension L/
Q given by a Z-basis of elements of L
.
Returns the relative pseudo-matrix in HNF giving the ideal x
considered as
an ideal of the relative extension L/K
, i.e. as a Z_K
-module.
The reason why the input does not use the customary HNF in terms of a fixed
Z-basis for Z_L
is precisely that no such basis has been explicitly
specified. On the other hand, if you already computed an (absolute) nf
structure Labs
attached to L
, and m
is in HNF, defining
an (absolute) ideal with respect to the Z-basis Labs.zk
, then
Labs.zk * m
is a suitable Z-basis for the ideal, and
rnfidealabstorel(rnf, Labs.zk * m)
@3converts m
to a relative ideal.
? K = nfinit(y^2+1); L = rnfinit(K, x^2-y); Labs = nfinit(L); ? m = idealhnf(Labs, 17, x^3+2); ? B = rnfidealabstorel(L, Labs.zk * m) %3 = [[1, 8; 0, 1], [[17, 4; 0, 1], 1]] \\ pseudo-basis for m as Z_K-module ? A = rnfidealreltoabs(L, B) %4 = [17, x^2 + 4, x + 8, x^3 + 8*x^2] \\ Z-basis for m in Q[x]/(L.pol) ? mathnf(matalgtobasis(Labs, A)) %5 = [17 8 4 2]
[ 0 1 0 0]
[ 0 0 1 0]
[ 0 0 0 1] ? % == m %6 = 1
The library syntax is GEN
rnfidealabstorel(GEN rnf, GEN x)
.
(
rnf,x)
Let rnf be a relative number
field extension L/K
as output by rnfinit
, and x
an ideal of
L
, given either in relative form or by a Z-basis of elements of L
(see Label se:rnfidealabstorel). This function returns the ideal of K
below x
, i.e. the intersection of x
with K
.
The library syntax is GEN
rnfidealdown(GEN rnf, GEN x)
.
(
rnf,x)
Factors into prime ideal powers the
ideal x
in the attached absolute number field L = nfinit(
rnf)
.
The output format is similar to the factor
function, and the prime
ideals are represented in the form output by the idealprimedec
function for L
.
? rnf = rnfinit(nfinit(y^2+1), x^2-y+1); ? rnfidealfactor(rnf, y+1) \\ P_2^2 %2 = [[2, [0,0,1,0]~, 4, 1, [0,0,0,2;0,0,-2,0;-1,-1,0,0;1,-1,0,0]] 2]
? rnfidealfactor(rnf, x) \\ P_2 %3 = [[2, [0,0,1,0]~, 4, 1, [0,0,0,2;0,0,-2,0;-1,-1,0,0;1,-1,0,0]] 1]
? L = nfinit(rnf); ? id = idealhnf(L, idealhnf(L, 25, (x+1)^2)); ? idealfactor(L, id) == rnfidealfactor(rnf, id) %6 = 1
@3Note that ideals of the base field K
must be explicitly
lifted to L
via rnfidealup
before they can be factored.
The library syntax is GEN
rnfidealfactor(GEN rnf, GEN x)
.
(
rnf,x)
rnf being a relative number
field extension L/K
as output by rnfinit
and x
being a relative
ideal (which can be, as in the absolute case, of many different types,
including of course elements), computes the HNF pseudo-matrix attached to
x
, viewed as a Z_K
-module.
The library syntax is GEN
rnfidealhnf(GEN rnf, GEN x)
.
(
rnf,x,y)
rnf being a relative number
field extension L/K
as output by rnfinit
and x
and y
being ideals
of the relative extension L/K
given by pseudo-matrices, outputs the ideal
product, again as a relative ideal.
The library syntax is GEN
rnfidealmul(GEN rnf, GEN x, GEN y)
.
(
rnf,x)
Let rnf be a relative
number field extension L/K
as output by rnfinit
and let x
be a
relative ideal (which can be, as in the absolute case, of many different
types, including of course elements). This function computes the norm of the
x
considered as an ideal of the absolute extension L/
Q. This is
identical to
idealnorm(rnf, rnfidealnormrel(rnf,x))
@3but faster.
The library syntax is GEN
rnfidealnormabs(GEN rnf, GEN x)
.
(
rnf,x)
Let rnf be a relative
number field extension L/K
as output by rnfinit
and let x
be a
relative ideal (which can be, as in the absolute case, of many different
types, including of course elements). This function computes the relative
norm of x
as an ideal of K
in HNF.
The library syntax is GEN
rnfidealnormrel(GEN rnf, GEN x)
.
(
rnf,
pr)
Let rnf be a relative number
field extension L/K
as output by rnfinit
, and pr
a maximal
ideal of K
(prid
), this function completes the rnf
with a nf structure attached to L
(see Label se:rnfinit)
and returns the prime ideal decomposition of pr
in L/K
.
? K = nfinit(y^2+1); rnf = rnfinit(K, x^3+y+1); ? P = idealprimedec(K, 2)[1]; ? S = rnfidealprimedec(rnf, P); ? #S %4 = 1
The argument pr
is also allowed to be a prime number p
, in which
case we return a pair of vectors [SK,SL]
, where SK
contains
the primes of K
above p
and SL
[i]
is the vector of primes of L
above SK
[i]
.
? [SK,SL] = rnfidealprimedec(rnf, 5); ? [#SK, vector(#SL,i,#SL[i])] %6 = [2, [2, 2]]
The library syntax is GEN
rnfidealprimedec(GEN rnf, GEN pr)
.
(
rnf,x,{
flag = 0})
Let rnf be a relative
number field extension L/K
as output by rnfinit
and let x
be a
relative ideal, given as a Z_K
-module by a pseudo matrix [A,I]
.
This function returns the ideal x
as an absolute ideal of L/
Q.
If flag = 0
, the result is given by a vector of t_POLMOD
s modulo
rnf.pol
forming a Z-basis; if flag = 1
, it is given in HNF in terms
of the fixed Z-basis for Z_L
, see Label se:rnfinit.
? K = nfinit(y^2+1); rnf = rnfinit(K, x^2-y); ? P = idealprimedec(K,2)[1]; ? P = rnfidealup(rnf, P) %3 = [2, x^2 + 1, 2*x, x^3 + x] ? Prel = rnfidealhnf(rnf, P) %4 = [[1, 0; 0, 1], [[2, 1; 0, 1], [2, 1; 0, 1]]] ? rnfidealreltoabs(rnf,Prel) %5 = [2, x^2 + 1, 2*x, x^3 + x] ? rnfidealreltoabs(rnf,Prel,1) %6 = [2 1 0 0]
[0 1 0 0]
[0 0 2 1]
[0 0 0 1]
The reason why we do not return by default (flag = 0
) the customary HNF in
terms of a fixed Z-basis for Z_L
is precisely because
a rnf does not contain such a basis by default. Completing the
structure so that it contains a nf structure for L
is polynomial
time but costly when the absolute degree is large, thus it is not done by
default. Note that setting flag = 1
will complete the rnf.
The library syntax is GEN
rnfidealreltoabs0(GEN rnf, GEN x, long flag)
.
Also available is
GEN
rnfidealreltoabs(GEN rnf, GEN x)
(flag = 0
).
(
rnf,x)
rnf being a relative
number field extension L/K
as output by rnfinit
and x
being an
ideal of the relative extension L/K
given by a pseudo-matrix, gives a
vector of two generators of x
over Z_L
expressed as polmods with polmod
coefficients.
The library syntax is GEN
rnfidealtwoelement(GEN rnf, GEN x)
.
(
rnf,x,{
flag = 0})
Let rnf be a relative number
field extension L/K
as output by rnfinit
and let x
be an ideal of
K
. This function returns the ideal x
Z_L
as an absolute ideal of L/
Q,
in the form of a Z-basis. If flag = 0
, the result is given by a vector of
polynomials (modulo rnf.pol
); if flag = 1
, it is given in HNF in terms
of the fixed Z-basis for Z_L
, see Label se:rnfinit.
? K = nfinit(y^2+1); rnf = rnfinit(K, x^2-y); ? P = idealprimedec(K,2)[1]; ? rnfidealup(rnf, P) %3 = [2, x^2 + 1, 2*x, x^3 + x] ? rnfidealup(rnf, P,1) %4 = [2 1 0 0]
[0 1 0 0]
[0 0 2 1]
[0 0 0 1]
The reason why we do not return by default (flag = 0
) the customary HNF in
terms of a fixed Z-basis for Z_L
is precisely because
a rnf does not contain such a basis by default. Completing the
structure so that it contains a nf structure for L
is polynomial
time but costly when the absolute degree is large, thus it is not done by
default. Note that setting flag = 1
will complete the rnf.
The library syntax is GEN
rnfidealup0(GEN rnf, GEN x, long flag)
.
Also available is
GEN
rnfidealup(GEN rnf, GEN x)
(flag = 0
).
(
nf,
pol,{
flag = 0})
nf being a number field in nfinit
format considered as base field, and pol a polynomial defining a relative
extension over nf, this computes data to work in the
relative extension. The main variable of pol must be of higher priority
(see Label se:priority) than that of nf, and the coefficients of
pol must be in nf.
The result is a row vector, whose components are technical. In the following
description, we let K
be the base field defined by nf and L/K
the extension attached to the rnf. Furthermore, we let
m = [K:
Q]
the degree of the base field, n = [L:K]
the relative degree,
r_1
and r_2
the number of real and complex places of K
. Access to this
information via member functions is preferred since the specific
data organization specified below will change in the future.
If flag = 1
, add an nf structure attached to L
to rnf.
This is likely to be very expensive if the absolute degree mn
is large,
but fixes an integer basis for Z_L
as a Z-module and allows to input
and output elements of L
in absolute form: as t_COL
for elements,
as t_MAT
in HNF for ideals, as prid
for prime ideals. Without such
a call, elements of L
are represented as t_POLMOD
, etc.
Note that a subsequent nfinit
(
rnf)
will also explicitly
add such a component, and so will the following functions rnfidealmul
,
rnfidealtwoelt
, rnfidealprimedec
, rnfidealup
(with flag 1)
and rnfidealreltoabs
(with flag 1). The absolute nf structure
attached to L
can be recovered using nfinit(rnf)
.
rnf[1]
(rnf.pol
) contains the relative polynomial pol.
rnf[2]
contains the integer basis [A,d]
of K
, as
(integral) elements of L/
Q. More precisely, A
is a vector of
polynomial with integer coefficients, d
is a denominator, and the integer
basis is given by A/d
.
rnf[3]
(rnf.disc
) is a two-component row vector
[
d(L/K),s]
where d(L/K)
is the relative ideal discriminant
of L/K
and s
is the discriminant of L/K
viewed as an element of
K^*/(K^*)^2
, in other words it is the output of rnfdisc
.
rnf[4]
(rnf.index
) is the ideal index f, i.e. such
that d(pol)
Z_K =
f^2
d(L/K)
.
rnf[5]
is currently unused.
rnf[6]
is currently unused.
rnf[7]
(rnf.zk
) is the pseudo-basis (A,I)
for the maximal
order Z_L
as a Z_K
-module: A
is the relative integral pseudo basis
expressed as polynomials (in the variable of pol
) with polmod coefficients
in nf, and the second component I
is the ideal list of the
pseudobasis in HNF.
rnf[8]
is the inverse matrix of the integral basis matrix, with
coefficients polmods in nf.
rnf[9]
is currently unused.
rnf[10]
(rnf.nf
) is nf.
rnf[11]
is an extension of rnfequation(K, pol, 1)
. Namely, a
vector [P, a, k, K.pol, pol]
describing the absolute
extension
L/
Q: P
is an absolute equation, more conveniently obtained
as rnf.polabs
; a
expresses the generator alpha = y mod K.pol
of the number field K
as an element of L
, i.e. a polynomial modulo the
absolute equation P
;
k
is a small integer such that, if beta is an abstract root of pol
and alpha the generator of K
given above, then P(
beta + k
alpha) = 0
.
@3Caveat. Be careful if k != 0
when dealing simultaneously with
absolute and relative quantities since L =
Q(
beta + k
alpha) =
K(
alpha)
, and the generator chosen for the absolute extension is not the
same as for the relative one. If this happens, one can of course go on
working, but we advise to change the relative polynomial so that its root
becomes beta + k
alpha. Typical GP instructions would be
[P,a,k] = rnfequation(K, pol, 1); if (k, pol = subst(pol, x, x - k*Mod(y, K.pol))); L = rnfinit(K, pol);
rnf[12]
is by default unused and set equal to 0. This field is used
to store further information about the field as it becomes available (which
is rarely needed, hence would be too expensive to compute during the initial
rnfinit
call).
The library syntax is GEN
rnfinit0(GEN nf, GEN pol, long flag)
.
Also available is
GEN
rnfinit(GEN nf,GEN pol)
(flag = 0
).
(
nf,T)
T
being a relative polynomial with coefficients
in nf, return 1 if it defines an abelian extension, and 0 otherwise.
? K = nfinit(y^2 + 23); ? rnfisabelian(K, x^3 - 3*x - y) %2 = 1
The library syntax is long
rnfisabelian(GEN nf, GEN T)
.
(
bnf,x)
Given bnf as output by
bnfinit
, and either a polynomial x
with coefficients in bnf
defining a relative extension L
of bnf, or a pseudo-basis x
of
such an extension, returns true (1) if L/
bnf is free, false (0) if
not.
The library syntax is long
rnfisfree(GEN bnf, GEN x)
.
(
rnf)
Let rnf a a relative number field extension L/K
as output
by rnfinit
whole degree [L:K]
is a power of a prime ell.
Return 1
if the ell-extension is locally cyclotomic (locally contained in
the cyclotomic Z_
ell-extension of K_v
at all places v |
ell), and
0
if not.
? K = nfinit(y^2 + y + 1); ? L = rnfinit(K, x^3 - y); /* = K(zeta_9), globally cyclotomic */ ? rnfislocalcyclo(L) %3 = 1 \\ we expect 3-adic continuity by Krasner's lemma ? vector(5, i, rnfislocalcyclo(rnfinit(K, x^3 - y + 3^i))) %5 = [0, 1, 1, 1, 1]
The library syntax is long
rnfislocalcyclo(GEN rnf)
.
(T,a,{
flag = 0})
Similar to
bnfisnorm
but in the relative case. T
is as output by
rnfisnorminit
applied to the extension L/K
. This tries to decide
whether the element a
in K
is the norm of some x
in the extension
L/K
.
The output is a vector [x,q]
, where a =
Norm (x)*q
. The
algorithm looks for a solution x
which is an S
-integer, with S
a list
of places of K
containing at least the ramified primes, the generators of
the class group of L
, as well as those primes dividing a
. If L/K
is
Galois, then this is enough; otherwise, flag is used to add more primes to
S
: all the places above the primes p <=
flag (resp. p|
flag) if flag > 0
(resp. flag < 0
).
The answer is guaranteed (i.e. a
is a norm iff q = 1
) if the field is
Galois, or, under GRH, if S
contains all primes less than
12
log ^2|
disc (M)|
, where M
is the normal
closure of L/K
.
If rnfisnorminit
has determined (or was told) that L/K
is
Galois, and flag != 0
, a Warning is issued (so that you can set
flag = 1
to check whether L/K
is known to be Galois, according to T
).
Example:
bnf = bnfinit(y^3 + y^2 - 2*y - 1); p = x^2 + Mod(y^2 + 2*y + 1, bnf.pol); T = rnfisnorminit(bnf, p); rnfisnorm(T, 17)
checks whether 17
is a norm in the Galois extension Q(
beta) /
Q(
alpha)
, where alpha^3 +
alpha^2 - 2
alpha - 1 = 0
and beta^2 +
alpha^2 + 2
alpha + 1 = 0
(it is).
The library syntax is GEN
rnfisnorm(GEN T, GEN a, long flag)
.
(
pol,
polrel,{
flag = 2})
Let K
be defined by a root of pol, and L/K
the extension defined
by the polynomial polrel. As usual, pol can in fact be an nf,
or bnf, etc; if pol has degree 1
(the base field is Q),
polrel is also allowed to be an nf, etc. Computes technical data needed
by rnfisnorm
to solve norm equations Nx = a
, for x
in L
, and a
in K
.
If flag = 0
, do not care whether L/K
is Galois or not.
If flag = 1
, L/K
is assumed to be Galois (unchecked), which speeds up
rnfisnorm
.
If flag = 2
, let the routine determine whether L/K
is Galois.
The library syntax is GEN
rnfisnorminit(GEN pol, GEN polrel, long flag)
.
rnfkummer(
bnr,{
subgp},{d = 0})
bnr
being as output by bnrinit
, finds a relative equation for the
class field corresponding to the module in bnr and the given
congruence subgroup (the full ray class field if subgp is omitted).
If d
is positive, outputs the list of all relative equations of
degree d
contained in the ray class field defined by bnr, with
the same conductor as (
bnr,
subgp)
.
@3Warning. This routine only works for subgroups of prime index. It
uses Kummer theory, adjoining necessary roots of unity (it needs to compute a
tough bnfinit
here), and finds a generator via Hecke's characterization
of ramification in Kummer extensions of prime degree. If your extension does
not have prime degree, for the time being, you have to split it by hand as a
tower / compositum of such extensions.
The library syntax is GEN
rnfkummer(GEN bnr, GEN subgp = NULL, long d, long prec)
.
(
nf,
pol,
order)
Given a polynomial
pol with coefficients in nf defining a relative extension L
and
a suborder order of L
(of maximal rank), as output by
rnfpseudobasis
(
nf,
pol)
or similar, gives
[[
neworder],U]
, where neworder is a reduced order and U
is
the unimodular transformation matrix.
The library syntax is GEN
rnflllgram(GEN nf, GEN pol, GEN order, long prec)
.
(
bnr,
pol)
bnr being a big ray
class field as output by bnrinit
and pol a relative polynomial
defining an Abelian extension, computes the norm group (alias Artin
or Takagi group) corresponding to the Abelian extension of
bnf =
bnr.bnf
defined by pol, where the module corresponding to bnr is assumed
to be a multiple of the conductor (i.e. pol defines a subextension of
bnr). The result is the HNF defining the norm group on the given generators
of bnr.gen
. Note that neither the fact that pol defines an
Abelian extension nor the fact that the module is a multiple of the conductor
is checked. The result is undefined if the assumption is not correct,
but the function will return the empty matrix [;]
if it detects a
problem; it may also not detect the problem and return a wrong result.
The library syntax is GEN
rnfnormgroup(GEN bnr, GEN pol)
.
(
nf,
pol)
This function is obsolete: use rnfpolredbest
instead.
Relative version of polred
. Given a monic polynomial pol with
coefficients in nf, finds a list of relative polynomials defining some
subfields, hopefully simpler and containing the original field. In the present
version 2.9.1, this is slower and less efficient than rnfpolredbest
.
@3Remark. this function is based on an incomplete reduction
theory of lattices over number fields, implemented by rnflllgram
, which
deserves to be improved.
The library syntax is GEN
rnfpolred(GEN nf, GEN pol, long prec)
.
(
nf,
pol,{
flag = 0})
This function is obsolete: use rnfpolredbest
instead.
Relative version of polredabs
. Given a monic polynomial pol
with coefficients in nf, finds a simpler relative polynomial defining
the same field. The binary digits of flag mean
The binary digits of flag correspond to 1
: add information to convert
elements to the new representation, 2
: absolute polynomial, instead of
relative, 16
: possibly use a suborder of the maximal order. More precisely:
0: default, return P
1: returns [P,a]
where P
is the default output and a
,
a t_POLMOD
modulo P
, is a root of pol.
2: returns Pabs, an absolute, instead of a relative, polynomial. Same as but faster than
rnfequation(nf, rnfpolredabs(nf,pol))
3: returns [
Pabs,a,b]
, where Pabs is an absolute polynomial
as above, a
, b
are t_POLMOD
modulo Pabs, roots of nf.pol
and pol respectively.
16: possibly use a suborder of the maximal order. This is slower than the default when the relative discriminant is smooth, and much faster otherwise. See Label se:polredabs.
@3Warning. In the present implementation, rnfpolredabs
produces smaller polynomials than rnfpolred
and is usually
faster, but its complexity is still exponential in the absolute degree.
The function rnfpolredbest
runs in polynomial time, and tends to
return polynomials with smaller discriminants.
The library syntax is GEN
rnfpolredabs(GEN nf, GEN pol, long flag)
.
(
nf,
pol,{
flag = 0})
Relative version of polredbest
. Given a monic polynomial pol
with coefficients in nf, finds a simpler relative polynomial P
defining the same field. As opposed to rnfpolredabs
this function does
not return a smallest (canonical) polynomial with respect to some
measure, but it does run in polynomial time.
The binary digits of flag correspond to 1
: add information to convert
elements to the new representation, 2
: absolute polynomial, instead of
relative. More precisely:
0: default, return P
1: returns [P,a]
where P
is the default output and a
,
a t_POLMOD
modulo P
, is a root of pol.
2: returns Pabs, an absolute, instead of a relative, polynomial. Same as but faster than
rnfequation(nf, rnfpolredbest(nf,pol))
3: returns [
Pabs,a,b]
, where Pabs is an absolute polynomial
as above, a
, b
are t_POLMOD
modulo Pabs, roots of nf.pol
and pol respectively.
? K = nfinit(y^3-2); pol = x^2 +x*y + y^2; ? [P, a] = rnfpolredbest(K,pol,1); ? P %3 = x^2 - x + Mod(y - 1, y^3 - 2) ? a %4 = Mod(Mod(2*y^2+3*y+4,y^3-2)*x + Mod(-y^2-2*y-2,y^3-2), x^2 - x + Mod(y-1,y^3-2)) ? subst(K.pol,y,a) %5 = 0 ? [Pabs, a, b] = rnfpolredbest(K,pol,3); ? Pabs %7 = x^6 - 3*x^5 + 5*x^3 - 3*x + 1 ? a %8 = Mod(-x^2+x+1, x^6-3*x^5+5*x^3-3*x+1) ? b %9 = Mod(2*x^5-5*x^4-3*x^3+10*x^2+5*x-5, x^6-3*x^5+5*x^3-3*x+1) ? subst(K.pol,y,a) %10 = 0 ? substvec(pol,[x,y],[a,b]) %11 = 0
The library syntax is GEN
rnfpolredbest(GEN nf, GEN pol, long flag)
.
(
nf,
pol)
Given a number field
nf as output by nfinit
and a polynomial pol with
coefficients in nf defining a relative extension L
of nf,
computes a pseudo-basis (A,I)
for the maximal order Z_L
viewed as a
Z_K
-module, and the relative discriminant of L
. This is output as a
four-element row vector [A,I,D,d]
, where D
is the relative ideal
discriminant and d
is the relative discriminant considered as an element of
nf^*/{
nf^*}^2
.
The library syntax is GEN
rnfpseudobasis(GEN nf, GEN pol)
.
(
nf,x)
Given a number field nf as
output by nfinit
and either a polynomial x
with coefficients in
nf defining a relative extension L
of nf, or a pseudo-basis
x
of such an extension as output for example by rnfpseudobasis
,
computes another pseudo-basis (A,I)
(not in HNF in general) such that all
the ideals of I
except perhaps the last one are equal to the ring of
integers of nf, and outputs the four-component row vector [A,I,D,d]
as in rnfpseudobasis
. The name of this function comes from the fact
that the ideal class of the last ideal of I
, which is well defined, is the
Steinitz class of the Z_K
-module Z_L
(its image in SK_0(
Z_K)
).
The library syntax is GEN
rnfsteinitz(GEN nf, GEN x)
.
subgrouplist(
bnr,{
bound},{
flag = 0})
bnr being as output by bnrinit
or a list of cyclic components
of a finite Abelian group G
, outputs the list of subgroups of G
. Subgroups
are given as HNF left divisors of the SNF matrix corresponding to G
.
If flag = 0
(default) and bnr is as output by bnrinit
, gives
only the subgroups whose modulus is the conductor. Otherwise, the modulus is
not taken into account.
If bound is present, and is a positive integer, restrict the output to
subgroups of index less than bound. If bound is a vector
containing a single positive integer B
, then only subgroups of index
exactly equal to B
are computed. For instance
? subgrouplist([6,2]) %1 = [[6, 0; 0, 2], [2, 0; 0, 2], [6, 3; 0, 1], [2, 1; 0, 1], [3, 0; 0, 2], [1, 0; 0, 2], [6, 0; 0, 1], [2, 0; 0, 1], [3, 0; 0, 1], [1, 0; 0, 1]] ? subgrouplist([6,2],3) \\ index less than 3 %2 = [[2, 1; 0, 1], [1, 0; 0, 2], [2, 0; 0, 1], [3, 0; 0, 1], [1, 0; 0, 1]] ? subgrouplist([6,2],[3]) \\ index 3 %3 = [[3, 0; 0, 1]] ? bnr = bnrinit(bnfinit(x), [120,[1]], 1); ? L = subgrouplist(bnr, [8]);
In the last example, L
corresponds to the 24 subfields of
Q(
zeta_{120})
, of degree 8
and conductor 120 oo
(by setting flag,
we see there are a total of 43
subgroups of degree 8
).
? vector(#L, i, galoissubcyclo(bnr, L[i]))
will produce their equations. (For a general base field, you would
have to rely on bnrstark
, or rnfkummer
.)
The library syntax is GEN
subgrouplist0(GEN bnr, GEN bound = NULL, long flag)
.
This section collects functions related to associative algebras and central
simple algebras over number fields. Let A
be a finite-dimensional unitary
associative algebra over a field K
. We say that A
is central if
the center of A
is K
, and that A
is simple if it has no
nontrivial two-sided ideals.
We provide functions to manipulate associative algebras of finite
dimension over Q or F_p
. We represent them by the left multiplication
table on a basis over the prime subfield. The function algtableinit
creates the object representing an associative algebra. We also provide
functions to manipulate central simple algebras over number fields. We
represent them either by the left multiplication table on a basis over the
center, or by a cyclic algebra (see below). The function alginit
creates
the object representing a central simple algebra.
The set of elements of an algebra A
that annihilate every simple left
A
-module is a two-sided ideal, called the Jacobson radical of A
.
An algebra is semisimple if its Jacobson radical is trivial. A
semisimple algebra is isomorphic to a direct sum of simple algebras. The
dimension of a central simple algebra A
over K
is always a square d^2
,
and the integer d
is called the degree of the algebra A
over K
.
A central simple algebra A
over a field K
is always isomorphic to M_d(D)
for some integer d
and some central division algebra D
of degree e
: the
integer e
is called the index of A
.
Let L/K
be a cyclic extension of degree d
, let sigma be a
generator of Gal(L/K)
and let b\in K^*
. Then the
cyclic algebra (L/K,
sigma,b)
is the algebra
\bigoplus_{i = 0}^{d-1}x^iL
with x^d = b
and ell x = x
sigma(
ell)
for
all ell\in L
. The algebra (L/K,
sigma,b)
is a central simple K
-algebra
of degree d
, and it is an L
-vector space. Left multiplication is
L
-linear and induces a K
-algebra homomorphism (L/K,
sigma,b)\to M_d(L)
.
Let K
be a nonarchimedean local field with uniformizer pi, and let
L/K
be the unique unramified extension of degree d
. Then every central
simple algebra A
of degree d
over K
is isomorphic to
(L/K,
Frob ,
pi^h)
for some integer h
. The element h/d\in
(1/d)
Z/
Z\subset
Q/
Z is called the Hasse invariant of A
.
Let A
be an algebra of finite dimension over Q. An order
in A
is a finitely generated Z-submodule O such
that QO = A
, that is also a subring with unit. We define natural
orders in central simple algebras defined by a cyclic algebra or by a
multiplication table over the center. Let A = (L/K,
sigma,b) =
\bigoplus_{i = 0}^{d-1}x^iL
be a cyclic algebra over a number field K
of
degree n
with ring of integers Z_K
. Let Z_L
be the ring of integers
of L
, and assume that b
is integral. Then the submodule O =
\bigoplus_{i = 0}^{d-1}x^i
Z_L
is an order in A
, called the
natural order. Let omega_0,...,
omega_{nd-1}
be a Z-basis
of Z_L
. The natural basis of O is b_0,...,b_{nd^2-1}
where b_i = x^{i/(nd)}
omega_{(i mod nd)}
. Now let A
be a central simple
algebra of degree d
over a number field K
of degree n
with ring of
integers Z_K
. Let e_0,...,e_{d^2-1}
be a basis of A
over K
and
assume that the left multiplication table of A
on (e_i)
is integral. Then
the submodule O = \bigoplus_{i = 0}^{d^2-1}
Z_K e_i
is an order
in A
, called the natural order. Let omega_0,...,
omega_{n-1}
be
a Z-basis of Z_K
. The natural basis of O
is b_0,...,b_{nd^2-1}
where b_i =
omega_{(i mod n)}e_{i/n}
.
As with number fields, we represent elements of central simple algebras
in two ways, called the algebraic representation and the basis
representation, and you can convert betweeen the two with the functions
algalgtobasis
and algbasistoalg
. In every central simple algebra
object, we store a Z-basis of an order O_0
, and the basis
representation is simply a t_COL
with coefficients in Q expressing the
element in that basis. If no maximal order was computed, then O_0
is
the natural order. If a maximal order was computed, then O_0
is a
maximal order containing the natural order. For a cyclic algebra A =
(L/K,
sigma,b)
, the algebraic representation is a t_COL
with coefficients
in L
representing the element in the decomposition A =
\bigoplus_{i = 0}^{d-1}x^iL
. For a central simple algebra defined by a
multiplication table over its center K
on a basis (e_i)
, the algebraic
representation is a t_COL
with coefficients in K
representing the element
on the basis (e_i)
.
@3Warning. The coefficients in the decomposition A =
\bigoplus_{i = 0}^{d-1}x^iL
are not the same as those in the decomposition A
= \bigoplus_{i = 0}^{d-1}Lx^i
! The i
-th coefficients are related by
conjugating by x^i
, which on L
amounts to acting by sigma^i
.
@3Warning. For a central simple algebra over Q defined by a
multiplication table, we cannot distinguish between the basis and the algebraic
representations from the size of the vectors. The behaviour is then to always
interpret the column vector as a basis representation if the coefficients are
t_INT
or t_FRAC
, and as an algebraic representation if the coefficients
are t_POL
or t_POLMOD
.
(
al)
Given an algebra al output by alginit
or by
algtableinit
, returns the dimension of al over its prime subfield
(Q or F_p
).
? nf = nfinit(y^3-y+1); ? A = alginit(nf, [-1,-1]); ? algabsdim(A) %3 = 12
The library syntax is long
algabsdim(GEN al)
.
(
al,x,y)
Given two elements x
and y
in al, computes their sum x+y
in
the algebra al.
? A = alginit(nfinit(y),[-1,1]); ? algadd(A,[1,0]~,[1,2]~) %2 = [2, 2]~
Also accepts matrices with coefficients in al.
The library syntax is GEN
algadd(GEN al, GEN x, GEN y)
.
(
al,x)
Given an element x in the central simple algebra al output
by alginit
, transforms it to a column vector on the integral basis of
al. This is the inverse function of algbasistoalg
.
? A = alginit(nfinit(y^2-5),[2,y]); ? algalgtobasis(A,[y,1]~) %2 = [0, 2, 0, -1, 2, 0, 0, 0]~ ? algbasistoalg(A,algalgtobasis(A,[y,1]~)) %3 = [Mod(Mod(y, y^2 - 5), x^2 - 2), 1]~
The library syntax is GEN
algalgtobasis(GEN al, GEN x)
.
(
al)
Given a cyclic algebra al = (L/K,
sigma,b)
output by
alginit
, returns the automorphism sigma.
? nf = nfinit(y); ? p = idealprimedec(nf,7)[1]; ? p2 = idealprimedec(nf,11)[1]; ? A = alginit(nf,[3,[[p,p2],[1/3,2/3]],[0]]); ? algaut(A) %5 = -1/3*x^2 + 1/3*x + 26/3
The library syntax is GEN
algaut(GEN al)
.
(
al)
Given a cyclic algebra al = (L/K,
sigma,b)
output by
alginit
, returns the element b\in K
.
nf = nfinit(y); ? p = idealprimedec(nf,7)[1]; ? p2 = idealprimedec(nf,11)[1]; ? A = alginit(nf,[3,[[p,p2],[1/3,2/3]],[0]]); ? algb(A) %5 = Mod(-77, y)
The library syntax is GEN
algb(GEN al)
.
(
al)
Given an central simple algebra al output by alginit
, returns
a Z-basis of the order O_0
stored in al with respect to the
natural order in al. It is a maximal order if one has been computed.
A = alginit(nfinit(y), [-1,-1]); ? algbasis(A) %2 = [1 0 0 1/2]
[0 1 0 1/2]
[0 0 1 1/2]
[0 0 0 1/2]
The library syntax is GEN
algbasis(GEN al)
.
(
al,x)
Given an element x in the central simple algebra al output
by alginit
, transforms it to its algebraic representation in al.
This is the inverse function of algalgtobasis
.
? A = alginit(nfinit(y^2-5),[2,y]); ? z = algbasistoalg(A,[0,1,0,0,2,-3,0,0]~); ? liftall(z) %3 = [(-1/2*y - 2)*x + (-1/4*y + 5/4), -3/4*y + 7/4]~ ? algalgtobasis(A,z) %4 = [0, 1, 0, 0, 2, -3, 0, 0]~
The library syntax is GEN
algbasistoalg(GEN al, GEN x)
.
(
al)
If al is a table algebra output by algtableinit
, returns a
basis of the center of the algebra al over its prime field (Q or
F_p
). If al is a central simple algebra output by alginit
,
returns the center of al, which is stored in al.
A simple example: the 2 x 2
upper triangular matrices over Q,
generated by I_2
, a = [0,1;0,0]
and b = [0,0;0,1]
,
such that a^2 = 0
, ab = a
, ba = 0
, b^2 = b
: the diagonal matrices
form the center.
? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]]; ? A = algtableinit(mt); ? algcenter(A) \\ = (I_2) %3 = [1]
[0]
[0]
An example in the central simple case:
? nf = nfinit(y^3-y+1); ? A = alginit(nf, [-1,-1]); ? algcenter(A).pol %3 = y^3 - y + 1
The library syntax is GEN
algcenter(GEN al)
.
algcentralproj(
al,z,{
maps = 0})
Given a table algebra al output by algtableinit
and a
t_VEC
z = [z_1,...,z_n]
of orthogonal central idempotents,
returns a t_VEC
[al_1,...,al_n]
of algebras such that
al_i = z_i al
. If maps = 1
, each al_i
is a t_VEC
[quo,proj,lift]
where quo is the quotient algebra, proj is a
t_MAT
representing the projection onto this quotient and lift is a
t_MAT
representing a lift.
A simple example: F_2\oplus
F_4
, generated by 1 = (1,1)
, e = (1,0)
and x
such that x^2+x+1 = 0
. We have e^2 = e
, x^2 = x+1
and ex = 0
.
? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]]; ? A = algtableinit(mt,2); ? e = [0,1,0]~; ? e2 = algsub(A,[1,0,0]~,e); ? [a,a2] = algcentralproj(A,[e,e2]); ? algdim(a) %6 = 1 ? algdim(a2) %7 = 2
The library syntax is GEN
alg_centralproj(GEN al, GEN z, long maps)
.
(
al)
Given an algebra al output by alginit
or algtableinit
,
returns the characteristic of al.
? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]]; ? A = algtableinit(mt,13); ? algchar(A) %3 = 13
The library syntax is GEN
algchar(GEN al)
.
(
al,b,{v = 'x})
Given an element b
in al, returns its characteristic polynomial
as a polynomial in the variable v
. If al is a table algebra output
by algtableinit
, returns the absolute characteristic polynomial of
b, which is an element of F_p[v]
or Q[v]
; if al is a
central simple algebra output by alginit
, returns the reduced
characteristic polynomial of b, which is an element of K[v]
where K
is the center of al.
? al = alginit(nfinit(y), [-1,-1]); \\ (-1,-1)_Q ? algcharpoly(al, [0,1]~) %2 = x^2 + 1
Also accepts a square matrix with coefficients in al.
The library syntax is GEN
algcharpoly(GEN al, GEN b, long v = -1)
where v
is a variable number.
(
al)
al being a table algebra output by algtableinit
, returns
[J,[al_1,...,al_n]]
where J
is a basis of the Jacobson radical of
al and al_1,...,al_n
are the simple factors of the semisimple
algebra al/J
.
The library syntax is GEN
alg_decomposition(GEN al)
.
(
al)
Given a central simple algebra al output by alginit
, returns
the degree of al.
? nf = nfinit(y^3-y+1); ? A = alginit(nf, [-1,-1]); ? algdegree(A) %3 = 2
The library syntax is long
algdegree(GEN al)
.
(
al)
Given a central simple algebra al output by alginit
, returns
the dimension of al over its center. Given a table algebra al
output by algtableinit
, returns the dimension of al over its prime
subfield (Q or F_p
).
? nf = nfinit(y^3-y+1); ? A = alginit(nf, [-1,-1]); ? algdim(A) %3 = 4
The library syntax is long
algdim(GEN al)
.
(
al)
Given a central simple algebra al output by alginit
, computes
the discriminant of the order O_0
stored in al, that is the
determinant of the trace form \rm{Tr} :
O_0 x
O_0 \to
Z.
? nf = nfinit(y^2-5); ? A = alginit(nf, [-3,1-y]); ? [PR,h] = alghassef(A); %3 = [[[2, [2, 0]~, 1, 2, 1], [3, [3, 0]~, 1, 2, 1]], Vecsmall([0, 1])] ? n = algdegree(A); ? D = algabsdim(A); ? h = vector(#h, i, n - gcd(n,h[i])); ? n^D * nf.disc^(n^2) * idealnorm(nf, idealfactorback(nf,PR,h))^n %4 = 12960000 ? algdisc(A) %5 = 12960000
The library syntax is GEN
algdisc(GEN al)
.
(
al,x,y)
Given two elements x
and y
in al, computes their left quotient
x\y
in the algebra al: an element z
such that xz = y
(such
an element is not unique when x
is a zerodivisor). If x
is invertible, this
is the same as x^{-1}y
. Assumes that y
is left divisible by x
(i.e. that
z
exists). Also accepts matrices with coefficients in al.
The library syntax is GEN
algdivl(GEN al, GEN x, GEN y)
.
(
al,x,y)
Given two elements x
and y
in al, return xy^{-1}
. Also accepts
matrices with coefficients in al.
The library syntax is GEN
algdivr(GEN al, GEN x, GEN y)
.
(
gal, {p = 0})
Initialize the group algebra K[G]
over K =
Q (p
omitted) or F_p
where G
is the underlying group of the galoisinit
structure gal.
The input gal is also allowed to be a t_VEC
of permutations that is
closed under products.
Example:
? K = nfsplitting(x^3-x+1); ? gal = galoisinit(K); ? al = alggroup(gal); ? algissemisimple(al) %4 = 1 ? G = [Vecsmall([1,2,3]), Vecsmall([1,3,2])]; ? al2 = alggroup(G, 2); ? algissemisimple(al2) %8 = 0
The library syntax is GEN
alggroup(GEN gal, GEN p = NULL)
.
(
al,
pl)
Given a central simple algebra al output by alginit
and a prime
ideal or an integer between 1
and r_1+r_2
, returns a t_FRAC
h
: the
local Hasse invariant of al at the place specified by pl.
? nf = nfinit(y^2-5); ? A = alginit(nf, [-1,y]); ? alghasse(A, 1) %3 = 1/2 ? alghasse(A, 2) %4 = 0 ? alghasse(A, idealprimedec(nf,2)[1]) %5 = 1/2 ? alghasse(A, idealprimedec(nf,5)[1]) %6 = 0
The library syntax is GEN
alghasse(GEN al, GEN pl)
.
(
al)
Given a central simple algebra al output by alginit
, returns
a t_VEC
[PR, h_f]
describing the local Hasse invariants at the
finite places of the center: PR
is a t_VEC
of primes and h_f
is a
t_VECSMALL
of integers modulo the degree d
of al.
? nf = nfinit(y^2-5); ? A = alginit(nf, [-1,2*y-1]); ? [PR,hf] = alghassef(A); ? PR %4 = [[19, [10, 2]~, 1, 1, [-8, 2; 2, -10]], [2, [2, 0]~, 1, 2, 1]] ? hf %5 = Vecsmall([1, 0])
The library syntax is GEN
alghassef(GEN al)
.
(
al)
Given a central simple algebra al output by alginit
, returns
a t_VECSMALL
h_i
of r_1
integers modulo the degree d
of al,
where r_1
is the number of real places of the center: the local Hasse
invariants of al at infinite places.
? nf = nfinit(y^2-5); ? A = alginit(nf, [-1,y]); ? alghassei(A) %3 = Vecsmall([1, 0])
The library syntax is GEN
alghassei(GEN al)
.
algindex(
al,{
pl})
Return the index of the central simple algebra A
over K
(as output by
alginit), that is the degree e
of the unique central division algebra D
over K
such that A
is isomorphic to some matrix algebra M_d(D)
. If
pl is set, it should be a prime ideal of K
or an integer between 1
and r_1+r_2
, and in that case return the local index at the place pl
instead.
? nf = nfinit(y^2-5); ? A = alginit(nf, [-1,y]); ? algindex(A, 1) %3 = 2 ? algindex(A, 2) %4 = 1 ? algindex(A, idealprimedec(nf,2)[1]) %5 = 2 ? algindex(A, idealprimedec(nf,5)[1]) %6 = 1 ? algindex(A) %7 = 2
The library syntax is long
algindex(GEN al, GEN pl = NULL)
.
(B, C, {v}, {
flag = 1})
Initialize the central simple algebra defined by data B
, C
and
variable v
, as follows.
@3* (multiplication table) B
is the base number field K
in nfinit
form, C
is a ``multiplication table'' over K
.
As a K
-vector space, the algebra is generated by a basis
(e_1 = 1,..., e_n)
; the table is given as a t_VEC
of n
matrices in
M_n(K)
, giving the left multiplication by the basis elements e_i
, in the
given basis.
Assumes that e_1 = 1
, that the multiplication table is integral, and that
K[e_1,...,e_n]
describes a central simple algebra over K
.
{ m_i = [0,-1,0, 0; 1, 0,0, 0; 0, 0,0,-1; 0, 0,1, 0]; m_j = [0, 0,-1,0; 0, 0, 0,1; 1, 0, 0,0; 0,-1, 0,0]; m_k = [0, 0, 0, 0; 0, 0,-1, 0; 0, 1, 0, 0; 1, 0, 0,-1]; A = alginit(nfinit(y), [matid(4), m_i,m_j,m_k], 0); }
represents (in a complicated way) the quaternion algebra (-1,-1)_
Q.
See below for a simpler solution.
@3* (cyclic algebra) B
is an rnf
structure attached to a cyclic
number field extension L/K
of degree d
, C
is a t_VEC
[sigma,b]
with 2 components: sigma
is a t_POLMOD
representing
an automorphism generating Gal(L/K)
, b
is an element in K^*
. This
represents the cyclic algebra (L/K,
sigma,b)
. Currently the element b
has
to be integral.
? Q = nfinit(y); T = polcyclo(5, 'x); F = rnfinit(Q, T); ? A = alginit(F, [Mod(x^2,T), 3]);
defines the cyclic algebra (L/
Q,
sigma, 3)
, where
L =
Q(
zeta_5)
and sigma:
zeta:--->
zeta^2
generates
Gal(L/
Q)
.
@3* (quaternion algebra, special case of the above) B
is an nf
structure attached to a number field K
, C = [a,b]
is a vector
containing two elements of K^*
with a
not a square in K
, returns the quaternion algebra (a,b)_K
.
The variable v
('x
by default) must have higher priority than the
variable of K
.pol
and is used to represent elements in the splitting
field L = K[x]/(x^2-a)
.
? Q = nfinit(y); A = alginit(Q, [-1,-1]); \\ (-1,-1)_B<Q>
@3* (algebra/K
defined by local Hasse invariants)
B
is an nf
structure attached to a number field K
,
C = [d, [PR,h_f], h_i]
is a triple
containing an integer d > 1
, a pair [PR, h_f]
describing the
Hasse invariants at finite places, and h_i
the Hasse invariants
at archimedean (real) places. A local Hasse invariant belongs to (1/d)
Z/
Z
\subset
Q/
Z, and is given either as a t_FRAC
(lift to (1/d)
Z),
a t_INT
or t_INTMOD
modulo d
(lift to Z/d
Z); a whole vector
of local invariants can also be given as a t_VECSMALL
, whose
entries are handled as t_INT
s. PR
is a list of prime ideals
(prid
structures), and h_f
is a vector of the same length giving the
local invariants at those maximal ideals. The invariants at infinite real
places are indexed by the real roots K
.roots
: if the Archimedean
place v
is attached to the j
-th root, the value of
h_v
is given by h_i[j]
, must be 0
or 1/2
(or d/2
modulo d
), and
can be nonzero only if d
is even.
By class field theory, provided the local invariants h_v
sum to 0
, up
to Brauer equivalence, there is a unique central simple algebra over K
with given local invariants and trivial invariant elsewhere. In particular,
up to isomorphism, there is a unique such algebra A
of degree d
.
We realize A
as a cyclic algebra through class field theory. The variable v
('x
by default) must have higher priority than the variable of
K
.pol
and is used to represent elements in the (cyclic) splitting
field extension L/K
for A
.
? nf = nfinit(y^2+1); ? PR = idealprimedec(nf,5); #PR %2 = 2 ? hi = []; ? hf = [PR, [1/3,-1/3]]; ? A = alginit(nf, [3,hf,hi]); ? algsplittingfield(A).pol %6 = x^3 - 21*x + 7
@3* (matrix algebra, toy example) B
is an nf
structure attached
to a number field K
, C = d
is a positive integer. Returns a cyclic
algebra isomorphic to the matrix algebra M_d(K)
.
In all cases, this function computes a maximal order for the algebra by default,
which may require a lot of time. Setting flag = 0
prevents this computation.
The pari object representing such an algebra A
is a t_VEC
with the
following data:
@3* A splitting field L
of A
of the same degree over K
as A
, in
rnfinit
format, accessed with algsplittingfield
.
@3* The same splitting field L
in nfinit
format.
@3* The Hasse invariants at the real places of K
, accessed with
alghassei
.
@3* The Hasse invariants of A
at the finite primes of K
that ramify in
the natural order of A
, accessed with alghassef
.
@3* A basis of an order O_0
expressed on the basis of the natural
order, accessed with algord
.
@3* A basis of the natural order expressed on the basis of O_0
,
accessed with alginvord
.
@3* The left multiplication table of O_0
on the previous basis,
accessed with algmultable
.
@3* The characteristic of A
(always 0
), accessed with algchar
.
@3* The absolute traces of the elements of the basis of O_0
.
@3* If A
was constructed as a cyclic algebra (L/K,
sigma,b)
of degree
d
, a t_VEC
[
sigma,
sigma^2,...,
sigma^{d-1}]
. The function
algaut
returns sigma.
@3* If A
was constructed as a cyclic algebra (L/K,
sigma,b)
, the
element b
, accessed with algb
.
@3* If A
was constructed with its multiplication table mt
over K
,
the t_VEC
of t_MAT
mt
, accessed with algrelmultable
.
@3* If A
was constructed with its multiplication table mt
over K
,
a t_VEC
with three components: a t_COL
representing an element of A
generating the splitting field L
as a maximal subfield of A
, a t_MAT
representing an L
-basis B of A
expressed on the Z-basis of
O_0
, and a t_MAT
representing the Z-basis of O_0
expressed on B. This data is accessed with algsplittingdata
.
The library syntax is GEN
alginit(GEN B, GEN C, long v = -1, long flag)
where v
is a variable number.
(
al,x)
Given an element x
in al, computes its inverse x^{-1}
in the
algebra al. Assumes that x
is invertible.
? A = alginit(nfinit(y), [-1,-1]); ? alginv(A,[1,1,0,0]~) %2 = [1/2, 1/2, 0, 0]~
Also accepts matrices with coefficients in al.
The library syntax is GEN
alginv(GEN al, GEN x)
.
(
al)
Given an central simple algebra al output by alginit
, returns
a Z-basis of the natural order in al with respect to the
order O_0
stored in al.
A = alginit(nfinit(y), [-1,-1]); ? alginvbasis(A) %2 = [1 0 0 -1]
[0 1 0 -1]
[0 0 1 -1]
[0 0 0 2]
The library syntax is GEN
alginvbasis(GEN al)
.
(
mt,p = 0)
Returns 1 if the multiplication table mt
is suitable for
algtableinit(mt,p)
, 0 otherwise. More precisely, mt
should be
a t_VEC
of n
matrices in M_n(K)
, giving the left multiplications
by the basis elements e_1,..., e_n
(structure constants).
We check whether the first basis element e_1
is 1
and e_i(e_je_k) =
(e_ie_j)e_k
for all i,j,k
.
? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]]; ? algisassociative(mt) %2 = 1
May be used to check a posteriori an algebra: we also allow mt
as
output by algtableinit
(p
is ignored in this case).
The library syntax is GEN
algisassociative(GEN mt, GEN p)
.
(
al)
al being a table algebra output by algtableinit
or a central
simple algebra output by alginit
, tests whether the algebra al is
commutative.
? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]]; ? A = algtableinit(mt); ? algiscommutative(A) %3 = 0 ? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]]; ? A = algtableinit(mt,2); ? algiscommutative(A) %6 = 1
The library syntax is GEN
algiscommutative(GEN al)
.
algisdivision(
al,{
pl})
Given a central simple algebra al output by alginit
, test
whether al is a division algebra. If pl is set, it should be a
prime ideal of K
or an integer between 1
and r_1+r_2
, and in that case
test whether al is locally a division algebra at the place pl
instead.
? nf = nfinit(y^2-5); ? A = alginit(nf, [-1,y]); ? algisdivision(A, 1) %3 = 1 ? algisdivision(A, 2) %4 = 0 ? algisdivision(A, idealprimedec(nf,2)[1]) %5 = 1 ? algisdivision(A, idealprimedec(nf,5)[1]) %6 = 0 ? algisdivision(A) %7 = 1
The library syntax is GEN
algisdivision(GEN al, GEN pl = NULL)
.
(
al,x,y,{&z})
Given two elements x
and y
in al, tests whether y
is left
divisible by x
, that is whether there exists z
in al such
that xz = y
, and sets z
to this element if it exists.
? A = alginit(nfinit(y), [-1,1]); ? algisdivl(A,[x+2,-x-2]~,[x,1]~) %2 = 0 ? algisdivl(A,[x+2,-x-2]~,[-x,x]~,&z) %3 = 1 ? z %4 = [Mod(-2/5*x - 1/5, x^2 + 1), 0]~
Also accepts matrices with coefficients in al.
The library syntax is GEN
algisdivl(GEN al, GEN x, GEN y, GEN *z = NULL)
.
algisinv(
al,x,{&
ix})
Given an element x
in al, tests whether x
is invertible, and sets
ix
to the inverse of x
.
? A = alginit(nfinit(y), [-1,1]); ? algisinv(A,[-1,1]~) %2 = 0 ? algisinv(A,[1,2]~,&ix) %3 = 1 ? ix %4 = [Mod(Mod(-1/3, y), x^2 + 1), Mod(Mod(2/3, y), x^2 + 1)]~
Also accepts matrices with coefficients in al.
The library syntax is GEN
algisinv(GEN al, GEN x, GEN *ix = NULL)
.
algisramified(
al,{
pl})
Given a central simple algebra al output by alginit
, test
whether al is ramified, i.e. not isomorphic to a matrix algebra over its
center. If pl is set, it should be a prime ideal of K
or an integer
between 1
and r_1+r_2
, and in that case test whether al is locally
ramified at the place pl instead.
? nf = nfinit(y^2-5); ? A = alginit(nf, [-1,y]); ? algisramified(A, 1) %3 = 1 ? algisramified(A, 2) %4 = 0 ? algisramified(A, idealprimedec(nf,2)[1]) %5 = 1 ? algisramified(A, idealprimedec(nf,5)[1]) %6 = 0 ? algisramified(A) %7 = 1
The library syntax is GEN
algisramified(GEN al, GEN pl = NULL)
.
(
al)
al being a table algebra output by algtableinit
or a central
simple algebra output by alginit
, tests whether the algebra al is
semisimple.
? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]]; ? A = algtableinit(mt); ? algissemisimple(A) %3 = 0 ? m_i=[0,-1,0,0;1,0,0,0;0,0,0,-1;0,0,1,0]; \\ quaternion algebra (-1,-1) ? m_j=[0,0,-1,0;0,0,0,1;1,0,0,0;0,-1,0,0]; ? m_k=[0,0,0,-1;0,0,-1,0;0,1,0,0;1,0,0,0]; ? mt = [matid(4), m_i, m_j, m_k]; ? A = algtableinit(mt); ? algissemisimple(A) %9 = 1
The library syntax is GEN
algissemisimple(GEN al)
.
algissimple(
al, {
ss = 0})
al being a table algebra output by algtableinit
or a central
simple algebra output by alginit
, tests whether the algebra al is
simple. If ss = 1
, assumes that the algebra al is semisimple
without testing it.
? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]]; ? A = algtableinit(mt); \\ matrices [*,*; 0,*] ? algissimple(A) %3 = 0 ? algissimple(A,1) \\ incorrectly assume that A is semisimple %4 = 1 ? m_i=[0,-1,0,0;1,0,0,0;0,0,0,-1;0,0,1,0]; ? m_j=[0,0,-1,0;0,0,0,1;1,0,0,0;0,-1,0,0]; ? m_k=[0,0,0,-1;0,0,b,0;0,1,0,0;1,0,0,0]; ? mt = [matid(4), m_i, m_j, m_k]; ? A = algtableinit(mt); \\ quaternion algebra (-1,-1) ? algissimple(A) %10 = 1 ? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]]; ? A = algtableinit(mt,2); \\ direct sum F_4+F_2 ? algissimple(A) %13 = 0
The library syntax is GEN
algissimple(GEN al, long ss)
.
algissplit(
al,{
pl})
Given a central simple algebra al output by alginit
, test
whether al is split, i.e. isomorphic to a matrix algebra over its center.
If pl is set, it should be a prime ideal of K
or an integer between 1
and r_1+r_2
, and in that case test whether al is locally split at the
place pl instead.
? nf = nfinit(y^2-5); ? A = alginit(nf, [-1,y]); ? algissplit(A, 1) %3 = 0 ? algissplit(A, 2) %4 = 1 ? algissplit(A, idealprimedec(nf,2)[1]) %5 = 0 ? algissplit(A, idealprimedec(nf,5)[1]) %6 = 1 ? algissplit(A) %7 = 0
The library syntax is GEN
algissplit(GEN al, GEN pl = NULL)
.
(
al,m)
Given an algebra al and a square invertible matrix m with size the dimension of al, returns the lattice generated by the columns of m.
? al = alginit(nfinit(y^2+7), [-1,-1]); ? a = [1,1,-1/2,1,1/3,-1,1,1]~; ? mt = algleftmultable(al,a); ? lat = alglathnf(al,mt); ? lat[2] %5 = 1/6
The library syntax is GEN
alglathnf(GEN al, GEN m)
.
(
al,x)
Given an element x in al, computes its left multiplication
table. If x is given in basis form, returns its multiplication table on
the integral basis; if x is given in algebraic form, returns its
multiplication table on the basis corresponding to the algebraic form of
elements of al. In every case, if x is a t_COL
of length n
,
then the output is a n x n
t_MAT
.
Also accepts a square matrix with coefficients in al.
? A = alginit(nfinit(y), [-1,-1]); ? algleftmultable(A,[0,1,0,0]~) %2 = [0 -1 1 0]
[1 0 1 1]
[0 0 1 1]
[0 0 -2 -1]
The library syntax is GEN
algleftmultable(GEN al, GEN x)
.
(
al,x,y)
Given two elements x
and y
in al, computes their product x*y
in the algebra al.
? A = alginit(nfinit(y), [-1,-1]); ? algmul(A,[1,1,0,0]~,[0,0,2,1]~) %2 = [2, 3, 5, -4]~
Also accepts matrices with coefficients in al.
The library syntax is GEN
algmul(GEN al, GEN x, GEN y)
.
(
al)
Returns a multiplication table of al over its
prime subfield (Q or F_p
), as a t_VEC
of t_MAT
: the left
multiplication tables of basis elements. If al was output by
algtableinit
, returns the multiplication table used to define al.
If al was output by alginit
, returns the multiplication table of
the order O_0
stored in al.
? A = alginit(nfinit(y), [-1,-1]); ? M = algmultable(A); ? #M %3 = 4 ? M[1] \\ multiplication by e_1 = 1 %4 = [1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
? M[2] %5 = [0 -1 1 0]
[1 0 1 1]
[0 0 1 1]
[0 0 -2 -1]
The library syntax is GEN
algmultable(GEN al)
.
(
al,x)
Given an element x
in al, computes its opposite -x
in the
algebra al.
? A = alginit(nfinit(y), [-1,-1]); ? algneg(A,[1,1,0,0]~) %2 = [-1, -1, 0, 0]~
Also accepts matrices with coefficients in al.
The library syntax is GEN
algneg(GEN al, GEN x)
.
(
al,x)
Given an element x in al, computes its norm. If al is
a table algebra output by algtableinit
, returns the absolute norm of
x, which is an element of F_p
of Q; if al is a central
simple algebra output by alginit
, returns the reduced norm of x,
which is an element of the center of al.
? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]]; ? A = algtableinit(mt,19); ? algnorm(A,[0,-2,3]~) %3 = 18
Also accepts a square matrix with coefficients in al.
The library syntax is GEN
algnorm(GEN al, GEN x)
.
(
al,T,b)
Given an element b
in al and a polynomial T
in K[X]
,
computes T(b)
in al.
The library syntax is GEN
algpoleval(GEN al, GEN T, GEN b)
.
(
al,x,n)
Given an element x
in al and an integer n
, computes the
power x^n
in the algebra al.
? A = alginit(nfinit(y), [-1,-1]); ? algpow(A,[1,1,0,0]~,7) %2 = [8, -8, 0, 0]~
Also accepts a square matrix with coefficients in al.
The library syntax is GEN
algpow(GEN al, GEN x, GEN n)
.
(
al)
al being the output of algtableinit
representing a semisimple
algebra of positive characteristic, returns a basis of the prime subalgebra
of al. The prime subalgebra of al is the subalgebra fixed by the
Frobenius automorphism of the center of al. It is abstractly isomorphic
to a product of copies of F_p
.
? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]]; ? A = algtableinit(mt,2); ? algprimesubalg(A) %3 = [1 0]
[0 1]
[0 0]
The library syntax is GEN
algprimesubalg(GEN al)
.
(
al,I,{
flag = 0})
al being a table algebra output by algtableinit
and I
being a basis of a two-sided ideal of al represented by a matrix,
returns the quotient al/
I. When flag = 1
, returns a
t_VEC
[
al/
I,
proj,
lift]
where proj and
lift are matrices respectively representing the projection map and a
section of it.
? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]]; ? A = algtableinit(mt,2); ? AQ = algquotient(A,[0;1;0]); ? algdim(AQ) %4 = 2
The library syntax is GEN
alg_quotient(GEN al, GEN I, long flag)
.
(
al)
al being a table algebra output by algtableinit
, returns a
basis of the Jacobson radical of the algebra al over its prime field
(Q or F_p
).
Here is an example with A =
Q[x]/(x^2)
, generated by (1,x)
:
? mt = [matid(2),[0,0;1,0]]; ? A = algtableinit(mt); ? algradical(A) \\ = (x) %3 = [0]
[1]
Another one with 2 x 2
upper triangular matrices over Q, generated
by I_2
, a = [0,1;0,0]
and b = [0,0;0,1]
, such that a^2 =
0
, ab = a
, ba = 0
, b^2 = b
:
? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]]; ? A = algtableinit(mt); ? algradical(A) \\ = (a) %6 = [0]
[1]
[0]
The library syntax is GEN
algradical(GEN al)
.
(
al)
Given a central simple algebra al output by alginit
, return a
t_VEC
containing the list of places of the center of al that are
ramified in al. Each place is described as an integer between 1
and r_1
or as a prime ideal.
? nf = nfinit(y^2-5); ? A = alginit(nf, [-1,y]); ? algramifiedplaces(A) %3 = [1, [2, [2, 0]~, 1, 2, 1]]
The library syntax is GEN
algramifiedplaces(GEN al)
.
(
al,b)
Given an algebra al and an integer b, returns a random
element in al with coefficients in [-b,b]
.
The library syntax is GEN
algrandom(GEN al, GEN b)
.
(
al)
Given a central simple algebra al output by alginit
defined by a multiplication table over its center (a number field), returns this multiplication table.
? nf = nfinit(y^3-5); a = y; b = y^2; ? {m_i = [0,a,0,0; 1,0,0,0; 0,0,0,a; 0,0,1,0];} ? {m_j = [0, 0,b, 0; 0, 0,0,-b; 1, 0,0, 0; 0,-1,0, 0];} ? {m_k = [0, 0,0,-a*b; 0, 0,b, 0; 0,-a,0, 0; 1, 0,0, 0];} ? mt = [matid(4), m_i, m_j, m_k]; ? A = alginit(nf,mt,'x); ? M = algrelmultable(A); ? M[2] == m_i %8 = 1 ? M[3] == m_j %9 = 1 ? M[4] == m_k %10 = 1
The library syntax is GEN
algrelmultable(GEN al)
.
(
al,{
flag = 0})
al being the output of algtableinit
representing a semisimple
algebra, returns a t_VEC
[
al_1,
al_2,...,
al_n]
such
that al is isomorphic to the direct sum of the simple algebras
al_i
. When flag = 1
, each component is instead a t_VEC
[
al_i,
proj_i,
lift_i]
where proj_i
and lift_i
are matrices respectively representing the projection map
on the i
-th factor and a section of it. The factors are sorted by
increasing dimension, then increasing dimension of the center. This ensures
that the ordering of the isomorphism classes of the factors is deterministic
over finite fields, but not necessarily over Q.
@3Warning. The images of the lift_i
are not guaranteed to form a direct sum.
The library syntax is GEN
algsimpledec(GEN al, long flag)
.
(
al)
Given a central simple algebra al output by alginit
defined
by a multiplication table over its center K
(a number field), returns data
stored to compute a splitting of al over an extension. This data is a
t_VEC
[t,Lbas,Lbasinv]
with 3
components:
@3* an element t
of al such that L = K(t)
is a maximal subfield
of al;
@3* a matrix Lbas
expressing a L
-basis of al (given an
L
-vector space structure by multiplication on the right) on the integral
basis of al;
@3* a matrix Lbasinv
expressing the integral basis of al on
the previous L
-basis.
? nf = nfinit(y^3-5); a = y; b = y^2; ? {m_i = [0,a,0,0; 1,0,0,0; 0,0,0,a; 0,0,1,0];} ? {m_j = [0, 0,b, 0; 0, 0,0,-b; 1, 0,0, 0; 0,-1,0, 0];} ? {m_k = [0, 0,0,-a*b; 0, 0,b, 0; 0,-a,0, 0; 1, 0,0, 0];} ? mt = [matid(4), m_i, m_j, m_k]; ? A = alginit(nf,mt,'x); ? [t,Lb,Lbi] = algsplittingdata(A); ? t %8 = [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]~; ? matsize(Lb) %9 = [12, 2] ? matsize(Lbi) %10 = [2, 12]
The library syntax is GEN
algsplittingdata(GEN al)
.
(
al)
Given a central simple algebra al output by alginit
, returns
an rnf
structure: the splitting field of al that is stored in
al, as a relative extension of the center.
nf = nfinit(y^3-5); a = y; b = y^2; {m_i = [0,a,0,0; 1,0,0,0; 0,0,0,a; 0,0,1,0];} {m_j = [0, 0,b, 0; 0, 0,0,-b; 1, 0,0, 0; 0,-1,0, 0];} {m_k = [0, 0,0,-a*b; 0, 0,b, 0; 0,-a,0, 0; 1, 0,0, 0];} mt = [matid(4), m_i, m_j, m_k]; A = alginit(nf,mt,'x); algsplittingfield(A).pol %8 = x^2 - y
The library syntax is GEN
algsplittingfield(GEN al)
.
(
al,x)
A central simple algebra al output by alginit
contains data
describing an isomorphism phi : A\otimes_K L \to M_d(L)
, where d
is the
degree of the algebra and L
is an extension of L
with [L:K] = d
. Returns
the matrix phi(x)
.
? A = alginit(nfinit(y), [-1,-1]); ? algsplittingmatrix(A,[0,0,0,2]~) %2 = [Mod(x + 1, x^2 + 1) Mod(Mod(1, y)*x + Mod(-1, y), x^2 + 1)]
[Mod(x + 1, x^2 + 1) Mod(-x + 1, x^2 + 1)]
Also accepts matrices with coefficients in al.
The library syntax is GEN
algsplittingmatrix(GEN al, GEN x)
.
(
al,x)
Given an element x
in al, computes its square x^2
in the
algebra al.
? A = alginit(nfinit(y), [-1,-1]); ? algsqr(A,[1,0,2,0]~) %2 = [-3, 0, 4, 0]~
Also accepts a square matrix with coefficients in al.
The library syntax is GEN
algsqr(GEN al, GEN x)
.
(
al,x,y)
Given two elements x
and y
in al, computes their difference
x-y
in the algebra al.
? A = alginit(nfinit(y), [-1,-1]); ? algsub(A,[1,1,0,0]~,[1,0,1,0]~) %2 = [0, 1, -1, 0]~
Also accepts matrices with coefficients in al.
The library syntax is GEN
algsub(GEN al, GEN x, GEN y)
.
(
al,B)
al being a table algebra output by algtableinit
and B
being a basis of a subalgebra of al represented by a matrix, returns an
algebra isomorphic to B.
? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]]; ? A = algtableinit(mt,2); ? B = algsubalg(A,[1,0; 0,0; 0,1]); ? algdim(A) %4 = 3 ? algdim(B) %5 = 2
The library syntax is GEN
algsubalg(GEN al, GEN B)
.
(
mt, {p = 0})
Initialize the associative algebra over K =
Q (p omitted) or F_p
defined by the multiplication table mt.
As a K
-vector space, the algebra is generated by a basis
(e_1 = 1, e_2,..., e_n)
; the table is given as a t_VEC
of n
matrices in
M_n(K)
, giving the left multiplication by the basis elements e_i
, in the
given basis.
Assumes that e_1 = 1
, that K e_1\oplus...\oplus K e_n]
describes an
associative algebra over K
, and in the case K =
Q that the multiplication
table is integral. If the algebra is already known to be central
and simple, then the case K =
F_p
is useless, and one should use
alginit
directly.
The point of this function is to input a finite dimensional K
-algebra, so
as to later compute its radical, then to split the quotient algebra as a
product of simple algebras over K
.
The pari object representing such an algebra A
is a t_VEC
with the
following data:
@3* The characteristic of A
, accessed with algchar
.
@3* The multiplication table of A
, accessed with algmultable
.
@3* The traces of the elements of the basis.
A simple example: the 2 x 2
upper triangular matrices over Q,
generated by I_2
, a = [0,1;0,0]
and b = [0,0;0,1]
,
such that a^2 = 0
, ab = a
, ba = 0
, b^2 = b
:
? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]]; ? A = algtableinit(mt); ? algradical(A) \\ = (a) %6 = [0]
[1]
[0] ? algcenter(A) \\ = (I_2) %7 = [1]
[0]
[0]
The library syntax is GEN
algtableinit(GEN mt, GEN p = NULL)
.
algtensor(
al1,
al2,{
maxord = 1})
Given two algebras al1 and al2, computes their tensor
product. For table algebras output by algtableinit
, the flag
maxord is ignored. For central simple algebras output by alginit
,
computes a maximal order by default. Prevent this computation by setting
maxord = 0
.
Currently only implemented for cyclic algebras of coprime degree over the same
center K
, and the tensor product is over K
.
The library syntax is GEN
algtensor(GEN al1, GEN al2, long maxord)
.
(
al,x)
Given an element x in al, computes its trace. If al is
a table algebra output by algtableinit
, returns the absolute trace of
x, which is an element of F_p
or Q; if al is the output of
alginit
, returns the reduced trace of x, which is an element of
the center of al.
? A = alginit(nfinit(y), [-1,-1]); ? algtrace(A,[5,0,0,1]~) %2 = 11
Also accepts a square matrix with coefficients in al.
The library syntax is GEN
algtrace(GEN al, GEN x)
.
(
al)
Given an algebra al output by alginit
or by algtableinit
, returns an integer indicating the type of algebra:
@3* 0
: not a valid algebra.
@3* 1
: table algebra output by algtableinit
.
@3* 2
: central simple algebra output by alginit
and represented by
a multiplication table over its center.
@3* 3
: central simple algebra output by alginit
and represented by
a cyclic algebra.
? algtype([]) %1 = 0 ? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]]; ? A = algtableinit(mt,2); ? algtype(A) %4 = 1 ? nf = nfinit(y^3-5); ? a = y; b = y^2; ? {m_i = [0,a,0,0; 1,0,0,0; 0,0,0,a; 0,0,1,0];} ? {m_j = [0, 0,b, 0; 0, 0,0,-b; 1, 0,0, 0; 0,-1,0, 0];} ? {m_k = [0, 0,0,-a*b; 0, 0,b, 0; 0,-a,0, 0; 1, 0,0, 0];} ? mt = [matid(4), m_i, m_j, m_k]; ? A = alginit(nf,mt,'x); ? algtype(A) %12 = 2 ? A = alginit(nfinit(y), [-1,-1]); ? algtype(A) %14 = 3
The library syntax is long
algtype(GEN al)
.
We group here all functions which are specific to polynomials or power series. Many other functions which can be applied on these objects are described in the other sections. Also, some of the functions described here can be applied to other types.
O(p^e)
If p
is an integer
greater than 2
, returns a p
-adic 0
of precision e
. In all other
cases, returns a power series zero with precision given by e v
, where v
is the X
-adic valuation of p
with respect to its main variable.
The library syntax is GEN
ggrando()
.
GEN
zeropadic(GEN p, long e)
for a p
-adic and
GEN
zeroser(long v, long e)
for a power series zero in variable v
.
(A,B,{v})
Deprecated alias for polresultantext
The library syntax is GEN
polresultantext0(GEN A, GEN B, long v = -1)
where v
is a variable number.
(x,{v})
Derivative of x
with respect to the main
variable if v
is omitted, and with respect to v
otherwise. The derivative
of a scalar type is zero, and the derivative of a vector or matrix is done
componentwise. One can use x'
as a shortcut if the derivative is with
respect to the main variable of x
.
By definition, the main variable of a t_POLMOD
is the main variable among
the coefficients from its two polynomial components (representative and
modulus); in other words, assuming a polmod represents an element of
R[X]/(T(X))
, the variable X
is a mute variable and the derivative is
taken with respect to the main variable used in the base ring R
.
The library syntax is GEN
deriv(GEN x, long v = -1)
where v
is a variable number.
(x,v,d,{n = 1})
Let v
be a vector of variables, and d
a vector of the same length,
return the image of x
by the n
-power (1
if n is not given) of the differential
operator D
that assumes the value d[i]
on the variable v[i]
.
The value of D
on a scalar type is zero, and D
applies componentwise to a vector
or matrix. When applied to a t_POLMOD
, if no value is provided for the variable
of the modulus, such value is derived using the implicit function theorem.
Some examples:
This function can be used to differentiate formal expressions:
If E =
exp (X^2)
then we have E' = 2*X*E
. We can derivate X*exp(X^2)
as follow:
? diffop(E*X,[X,E],[1,2*X*E]) %1 = (2*X^2 + 1)*E
Let Sin
and Cos
be two function such that Sin^2+Cos^2 = 1
and Cos' = -Sin
. We can differentiate Sin/Cos
as follow,
PARI inferring the value of Sin'
from the equation:
? diffop(Mod('Sin/'Cos,'Sin^2+'Cos^2-1),['Cos],[-'Sin]) %1 = Mod(1/Cos^2, Sin^2 + (Cos^2 - 1))
Compute the Bell polynomials (both complete and partial) via the Faa di Bruno formula:
Bell(k,n=-1)= { my(var(i)=eval(Str("X",i))); my(x,v,dv); v=vector(k,i,if(i==1,'E,var(i-1))); dv=vector(k,i,if(i==1,'X*var(1)*'E,var(i))); x=diffop('E,v,dv,k)/'E; if(n<0,subst(x,'X,1),polcoeff(x,n,'X)) }
The library syntax is GEN
diffop0(GEN x, GEN v, GEN d, long n)
.
For n = 1
, the function GEN
diffop(GEN x, GEN v, GEN d)
is also available.
(x)
Replaces in x
the formal variables by the values that
have been assigned to them after the creation of x
. This is mainly useful
in GP, and not in library mode. Do not confuse this with substitution (see
subst
).
If x
is a character string, eval(x)
executes x
as a GP
command, as if directly input from the keyboard, and returns its
output.
? x1 = "one"; x2 = "two"; ? n = 1; eval(Str("x", n)) %2 = "one" ? f = "exp"; v = 1; ? eval(Str(f, "(", v, ")")) %4 = 2.7182818284590452353602874713526624978
@3Note that the first construct could be implemented in a
simpler way by using a vector x = ["one","two"]; x[n]
, and the second
by using a closure f = exp; f(v)
. The final example is more interesting:
? genmat(u,v) = matrix(u,v,i,j, eval( Str("x",i,j) )); ? genmat(2,3) \\ generic 2 x 3 matrix %2 = [x11 x12 x13]
[x21 x22 x23]
A syntax error in the evaluation expression raises an e_SYNTAX
exception, which can be trapped as usual:
? 1a *** syntax error, unexpected variable name, expecting $end or {\tt ';'}: 1a *** ^- ? E(expr) = { iferr(eval(expr), e, print("syntax error"), errname(e) == "e_SYNTAX"); } ? E("1+1") %1 = 2 ? E("1a") syntax error
The library syntax is geval(GEN x)
.
(
pol,p,r)
p
-adic factorization
of the polynomial pol to precision r
, the result being a
two-column matrix as in factor
. Note that this is not the same
as a factorization over Z/p^r
Z (polynomials over that ring do not form a
unique factorization domain, anyway), but approximations in Q/p^r
Z of
the true factorization in Q_p[X]
.
? factorpadic(x^2 + 9, 3,5) %1 = [(1 + O(3^5))*x^2 + O(3^5)*x + (3^2 + O(3^5)) 1] ? factorpadic(x^2 + 1, 5,3) %2 = [ (1 + O(5^3))*x + (2 + 5 + 2*5^2 + O(5^3)) 1]
[(1 + O(5^3))*x + (3 + 3*5 + 2*5^2 + O(5^3)) 1]
@3
The factors are normalized so that their leading coefficient is a power of
p
. The method used is a modified version of the round 4 algorithm of
Zassenhaus.
If pol has inexact t_PADIC
coefficients, this is not always
well-defined; in this case, the polynomial is first made integral by dividing
out the p
-adic content, then lifted to Z using truncate
coefficientwise.
Hence we actually factor exactly a polynomial which is only p
-adically
close to the input. To avoid pitfalls, we advise to only factor polynomials
with exact rational coefficients.
The library syntax is factorpadic(GEN f,GEN p, long r)
. The function factorpadic0
is
deprecated, provided for backward compatibility.
(x,{v})
formal integration of x
with respect to the variable v
(wrt.
the main variable if v
is omitted). Since PARI cannot represent
logarithmic or arctangent terms, any such term in the result will yield an
error:
? intformal(x^2) %1 = 1/3*x^3 ? intformal(x^2, y) %2 = y*x^2 ? intformal(1/x) *** at top-level: intformal(1/x) *** ^-------------- *** intformal: domain error in intformal: residue(series, pole) != 0
The argument x
can be of any type. When x
is a rational function, we
assume that the base ring is an integral domain of characteristic zero.
By definition, the main variable of a t_POLMOD
is the main variable
among the coefficients from its two polynomial components
(representative and modulus); in other words, assuming a polmod represents an
element of R[X]/(T(X))
, the variable X
is a mute variable and the
integral is taken with respect to the main variable used in the base ring R
.
In particular, it is meaningless to integrate with respect to the main
variable of x.mod
:
? intformal(Mod(1,x^2+1), 'x) *** intformal: incorrect priority in intformal: variable x = x
The library syntax is GEN
integ(GEN x, long v = -1)
where v
is a variable number.
(
pol,a)
Vector of p
-adic roots of the
polynomial pol
congruent to the p
-adic number a
modulo p
, and with
the same p
-adic precision as a
. The number a
can be an ordinary
p
-adic number (type t_PADIC
, i.e. an element of Z_p
) or can be an
integral element of a finite extension of Q_p
, given as a t_POLMOD
at least one of whose coefficients is a t_PADIC
. In this case, the result
is the vector of roots belonging to the same extension of Q_p
as a
.
The library syntax is GEN
padicappr(GEN pol, GEN a)
.
Also available is GEN
Zp_appr(GEN f, GEN a)
when a
is a
t_PADIC
.
(p, N, {
flag = 0})
Returns a vector of polynomials generating all the extensions of degree
N
of the field Q_p
of p
-adic rational numbers; N
is
allowed to be a 2-component vector [n,d]
, in which case we return the
extensions of degree n
and discriminant p^d
.
The list is minimal in the sense that two different polynomials generate
non-isomorphic extensions; in particular, the number of polynomials is the
number of classes of non-isomorphic extensions. If P
is a polynomial in this
list, alpha is any root of P
and K =
Q_p(
alpha)
, then alpha
is the sum of a uniformizer and a (lift of a) generator of the residue field
of K
; in particular, the powers of alpha generate the ring of p
-adic
integers of K
.
If flag = 1
, replace each polynomial P
by a vector [P, e, f, d, c]
where e
is the ramification index, f
the residual degree, d
the
valuation of the discriminant, and c
the number of conjugate fields.
If flag = 2
, only return the number of extensions in a fixed
algebraic closure (Krasner's formula), which is much faster.
The library syntax is GEN
padicfields0(GEN p, GEN N, long flag)
.
Also available is
GEN
padicfields(GEN p, long n, long d, long flag)
, which computes
extensions of Q_p
of degree n
and discriminant p^d
.
(n,{
flag = 1},{a = 'x})
Returns the n-th
Chebyshev polynomial of the first kind T_n
(flag = 1
) or the second
kind U_n
(flag = 2
), evaluated at a
('x
by default). Both series of
polynomials satisfy the 3-term relation
P_{n+1} = 2xP_n - P_{n-1},
and are determined by the initial conditions U_0 = T_0 = 1
, T_1 = x
,
U_1 = 2x
. In fact T_n' = n U_{n-1}
and, for all complex numbers z
, we
have T_n(
cos z) =
cos (nz)
and U_{n-1}(
cos z) =
sin (nz)/
sin z
.
If n >= 0
, then these polynomials have degree n
. For n < 0
,
T_n
is equal to T_{-n}
and U_n
is equal to -U_{-2-n}
.
In particular, U_{-1} = 0
.
The library syntax is GEN
polchebyshev_eval(long n, long flag, GEN a = NULL)
.
Also available are
GEN
polchebyshev(long n, long flag, long v)
,
GEN
polchebyshev1(long n, long v)
and
GEN
polchebyshev2(long n, long v)
for T_n
and U_n
respectively.
polclass(D, {
inv = 0}, {x = 'x})
Return a polynomial in Z[x]
generating the Hilbert class field for the
imaginary quadratic discriminant D
. If inv
is 0 (the default),
use the modular j
-function and return the classical Hilbert polynomial,
otherwise use a class invariant. The following invariants correspond to
the different values of inv
, where f
denotes Weber's function
weber
, and w_{p,q}
the double eta quotient given by
w_{p,q} = (
eta(x/p)
eta(x/q) )/(
eta(x)
eta(x/{pq}) )
The invariants w_{p,q}
are not allowed unless they satisfy the following
technical conditions ensuring they do generate the Hilbert class
field and not a strict subfield:
@3* if p != q
, we need them both non-inert, prime to the conductor of
Z[
sqrt {D}]
. Let P, Q
be prime ideals above p
and q
; if both are
unramified, we further require that P^{+- 1} Q^{+- 1}
be all distinct in
the class group of Z[
sqrt {D}]
; if both are ramified, we require that PQ
!= 1
in the class group.
@3* if p = q
, we want it split and prime to the conductor and
the prime ideal above it must have order != 1, 2, 4
in the class group.
@3Invariants are allowed under the additional conditions on D
listed below.
@3* 0 : j
@3* 1 : f
, D = 1 mod 8
and D = 1,2 mod 3
;
@3* 2 : f^2
, D = 1 mod 8
and D = 1,2 mod 3
;
@3* 3 : f^3
, D = 1 mod 8
;
@3* 4 : f^4
, D = 1 mod 8
and D = 1,2 mod 3
;
@3* 5 : gamma_2 = j^{1/3}
, D = 1,2 mod 3
;
@3* 6 : w_{2,3}
, D = 1 mod 8
and D = 1,2 mod 3
;
@3* 8 : f^8
, D = 1 mod 8
and D = 1,2 mod 3
;
@3* 9 : w_{3,3}
, D = 1 mod 2
and D = 1,2 mod 3
;
@3* 10: w_{2,5}
, D != 60 mod 80
and D = 1,2 mod 3
;
@3* 14: w_{2,7}
, D = 1 mod 8
;
@3* 15: w_{3,5}
, D = 1,2 mod 3
;
@3* 21: w_{3,7}
, D = 1 mod 2
and 21
does not divide D
@3* 23: w_{2,3}^2
, D = 1,2 mod 3
;
@3* 24: w_{2,5}^2
, D = 1,2 mod 3
;
@3* 26: w_{2,13}
, D != 156 mod 208
;
@3* 27: w_{2,7}^2
, D != 28 mod 112
;
@3* 28: w_{3,3}^2
, D = 1,2 mod 3
;
@3* 35: w_{5,7}
, D = 1,2 mod 3
;
@3* 39: w_{3,13}
, D = 1 mod 2
and D = 1,2 mod 3
;
The algorithm for computing the polynomial does not use the floating point
approach, which would evaluate a precise modular function in a precise
complex argument. Instead, it relies on a faster Chinese remainder based
approach modulo small primes, in which the class invariant is only defined
algebraically by the modular polynomial relating the modular function to j
.
So in fact, any of the several roots of the modular polynomial may actually
be the class invariant, and more precise assertions cannot be made.
For instance, while polclass(D)
returns the minimal polynomial of
j(
tau)
with tau (any) quadratic integer for the discriminant D
,
the polynomial returned by polclass(D, 5)
can be the minimal polynomial
of any of gamma_2 (
tau)
, zeta_3
gamma_2 (
tau)
or
zeta_3^2
gamma_2 (
tau)
, the three roots of the modular polynomial
j =
gamma_2^3
, in which j
has been specialised to j (
tau)
.
The modular polynomial is given by
j = ((f^{24}-16)^3 )/(f^{24})
for Weber's function f
.
For the double eta quotients of level N = p q
, all functions are covered
such that the modular curve X_0^+ (N)
, the function field of which is
generated by the functions invariant under Gamma^0 (N)
and the
Fricke--Atkin--Lehner involution, is of genus 0
with function field
generated by (a power of) the double eta quotient w
.
This ensures that the full Hilbert class field (and not a proper subfield)
is generated by class invariants from these double eta quotients.
Then the modular polynomial is of degree 2
in j
, and
of degree psi (N) = (p+1)(q+1)
in w
.
? polclass(-163) %1 = x + 262537412640768000 ? polclass(-51, , 'z) %2 = z^2 + 5541101568*z + 6262062317568 ? polclass(-151,1) x^7 - x^6 + x^5 + 3*x^3 - x^2 + 3*x + 1
The library syntax is GEN
polclass(GEN D, long inv, long x = -1)
where x
is a variable number.
(x,n,{v})
Coefficient of degree n
of the polynomial x
, with respect to the
main variable if v
is omitted, with respect to v
otherwise. If n
is greater than the degree, the result is zero.
Naturally applies to scalars (polynomial of degree 0
), as well as to
rational functions whose denominator is a monomial.
It also applies to power series: if n
is less than the valuation, the result
is zero. If it is greater than the largest significant degree, then an error
message is issued.
For greater flexibility, x
can be a vector or matrix type and the
function then returns component(x,n)
.
The library syntax is GEN
polcoeff0(GEN x, long n, long v = -1)
where v
is a variable number.
(n,{a = 'x})
n
-th cyclotomic polynomial, evaluated at a
('x
by default). The
integer n
must be positive.
Algorithm used: reduce to the case where n
is squarefree; to compute the
cyclotomic polynomial, use Phi_{np}(x) =
Phi_n(x^p)/
Phi(x)
; to compute
it evaluated, use Phi_n(x) =
prod_{d | n} (x^d-1)^{
mu(n/d)}
. In the
evaluated case, the algorithm assumes that a^d - 1
is either 0
or
invertible, for all d | n
. If this is not the case (the base ring has
zero divisors), use subst(polcyclo(n),x,a)
.
The library syntax is GEN
polcyclo_eval(long n, GEN a = NULL)
.
The variant GEN
polcyclo(long n, long v)
returns the n
-th
cyclotomic polynomial in variable v
.
(f)
Returns a vector of polynomials, whose product is the product of
distinct cyclotomic polynomials dividing f
.
? f = x^10+5*x^8-x^7+8*x^6-4*x^5+8*x^4-3*x^3+7*x^2+3; ? v = polcyclofactors(f) %2 = [x^2 + 1, x^2 + x + 1, x^4 - x^3 + x^2 - x + 1] ? apply(poliscycloprod, v) %3 = [1, 1, 1] ? apply(poliscyclo, v) %4 = [4, 3, 10]
@3In general, the polynomials are products of cyclotomic polynomials and not themselves irreducible:
? g = x^8+2*x^7+6*x^6+9*x^5+12*x^4+11*x^3+10*x^2+6*x+3; ? polcyclofactors(g) %2 = [x^6 + 2*x^5 + 3*x^4 + 3*x^3 + 3*x^2 + 2*x + 1] ? factor(%[1]) %3 = [ x^2 + x + 1 1]
[x^4 + x^3 + x^2 + x + 1 1]
The library syntax is GEN
polcyclofactors(GEN f)
.
(x,{v})
Degree of the polynomial x
in the main variable if v
is omitted, in
the variable v
otherwise.
The degree of 0
is -oo
. The degree of a non-zero scalar is 0
.
Finally, when x
is a non-zero polynomial or rational function, returns the
ordinary degree of x
. Raise an error otherwise.
The library syntax is GEN
gppoldegree(GEN x, long v = -1)
where v
is a variable number.
Also available is
long
poldegree(GEN x, long v)
, which returns -LONG_MAX
if x = 0
and the degree as a long
integer.
(
pol,{v})
Discriminant of the polynomial
pol in the main variable if v
is omitted, in v
otherwise. Uses a
modular algorithm over Z or Q, and the subresultant algorithm
otherwise.
? T = x^4 + 2*x+1; ? poldisc(T) %2 = -176 ? poldisc(T^2) %3 = 0
For convenience, the function also applies to types t_QUAD
and
t_QFI
/t_QFR
:
? z = 3*quadgen(8) + 4; ? poldisc(z) %2 = 8 ? q = Qfb(1,2,3); ? poldisc(q) %4 = -8
The library syntax is GEN
poldisc0(GEN pol, long v = -1)
where v
is a variable number.
(f)
Reduced discriminant vector of the
(integral, monic) polynomial f
. This is the vector of elementary divisors
of Z[
alpha]/f'(
alpha)
Z[
alpha]
, where alpha is a root of the
polynomial f
. The components of the result are all positive, and their
product is equal to the absolute value of the discriminant of f
.
The library syntax is GEN
reduceddiscsmith(GEN f)
.
(f)
Returns the Graeffe transform g
of f
, such that g(x^2) = f(x)
f(-x)
.
The library syntax is GEN
polgraeffe(GEN f)
.
(A, B, p, e)
Given a prime p
, an integral polynomial A
whose leading coefficient
is a p
-unit, a vector B
of integral polynomials that are monic and
pairwise relatively prime modulo p
, and whose product is congruent to
A/lc(A)
modulo p
, lift the elements of B
to polynomials whose
product is congruent to A
modulo p^e
.
More generally, if T
is an integral polynomial irreducible mod p
, and
B
is a factorization of A
over the finite field F_p[t]/(T)
, you can
lift it to Z_p[t]/(T, p^e)
by replacing the p
argument with [p,T]
:
? { T = t^3 - 2; p = 7; A = x^2 + t + 1; B = [x + (3*t^2 + t + 1), x + (4*t^2 + 6*t + 6)]; r = polhensellift(A, B, [p, T], 6) } %1 = [x + (20191*t^2 + 50604*t + 75783), x + (97458*t^2 + 67045*t + 41866)] ? liftall( r[1] * r[2] * Mod(Mod(1,p^6),T) ) %2 = x^2 + (t + 1)
The library syntax is GEN
polhensellift(GEN A, GEN B, GEN p, long e)
.
(n,{a = 'x})
n-th
Hermite polynomial H_n
evaluated at a
('x
by default), i.e.
H_n(x) = (-1)^n e^{x^2} (d^n)/(dx^n)e^{-x^2}.
The library syntax is GEN
polhermite_eval(long n, GEN a = NULL)
.
The variant GEN
polhermite(long n, long v)
returns the n
-th
Hermite polynomial in variable v
.
(X,{Y},{t = 'x},{&e})
Given the data vectors
X
and Y
of the same length n
(X
containing the x
-coordinates,
and Y
the corresponding y
-coordinates), this function finds the
interpolating polynomial P
of minimal degree passing through these
points and evaluates it at t
. If Y
is omitted, the polynomial P
interpolates the (i,X[i])
. If present, e
will contain an error estimate
on the returned value.
The library syntax is GEN
polint(GEN X, GEN Y = NULL, GEN t = NULL, GEN *e = NULL)
.
(f)
Returns 0 if f
is not a cyclotomic polynomial, and n > 0
if f =
Phi_n
, the n
-th cyclotomic polynomial.
? poliscyclo(x^4-x^2+1) %1 = 12 ? polcyclo(12) %2 = x^4 - x^2 + 1 ? poliscyclo(x^4-x^2-1) %3 = 0
The library syntax is long
poliscyclo(GEN f)
.
(f)
Returns 1 if f
is a product of cyclotomic polynomial, and 0
otherwise.
? f = x^6+x^5-x^3+x+1; ? poliscycloprod(f) %2 = 1 ? factor(f) %3 = [ x^2 + x + 1 1]
[x^4 - x^2 + 1 1] ? [ poliscyclo(T) | T <- %[,1] ] %4 = [3, 12] ? polcyclo(3) * polcyclo(12) %5 = x^6 + x^5 - x^3 + x + 1
The library syntax is long
poliscycloprod(GEN f)
.
(
pol)
pol being a polynomial (univariate in the present version 2.9.1), returns 1 if pol is non-constant and irreducible, 0 otherwise. Irreducibility is checked over the smallest base field over which pol seems to be defined.
The library syntax is long
isirreducible(GEN pol)
.
(x,{v})
Leading coefficient of the polynomial or power series x
. This is
computed with respect to the main variable of x
if v
is omitted, with
respect to the variable v
otherwise.
The library syntax is GEN
pollead(GEN x, long v = -1)
where v
is a variable number.
(n,{a = 'x})
n-th
Legendre polynomial evaluated at a
('x
by
default).
The library syntax is GEN
pollegendre_eval(long n, GEN a = NULL)
.
To obtain the n
-th Legendre polynomial in variable v
,
use GEN
pollegendre(long n, long v)
.
polmodular(L, {
inv = 0}, {x = 'x}, {y = 'y}, {
derivs = 0})
Return the modular polynomial of prime level L
in variables x
and y
for the modular function specified by inv
. If inv
is 0 (the
default), use the modular j
function, if inv
is 1 use the
Weber-f
function, and if inv
is 5 use gamma_2 =
sqrt [3]{j}
.
See polclass
for the full list of invariants.
If x
is given as Mod(j, p)
or an element j
of
a finite field (as a t_FFELT
), then return the modular polynomial of
level L
evaluated at j
. If j
is from a finite field and
derivs
is non-zero, then return a triple where the
last two elements are the first and second derivatives of the modular
polynomial evaluated at j
.
? polmodular(3) %1 = x^4 + (-y^3 + 2232*y^2 - 1069956*y + 36864000)*x^3 + ... ? polmodular(7, 1, , 'J) %2 = x^8 - J^7*x^7 + 7*J^4*x^4 - 8*J*x + J^8 ? polmodular(7, 5, 7*ffgen(19)^0, 'j) %3 = j^8 + 4*j^7 + 4*j^6 + 8*j^5 + j^4 + 12*j^2 + 18*j + 18 ? polmodular(7, 5, Mod(7,19), 'j) %4 = Mod(1, 19)*j^8 + Mod(4, 19)*j^7 + Mod(4, 19)*j^6 + ...
? u = ffgen(5)^0; T = polmodular(3,0,,'j)*u; ? polmodular(3, 0, u,'j,1) %6 = [j^4 + 3*j^2 + 4*j + 1, 3*j^2 + 2*j + 4, 3*j^3 + 4*j^2 + 4*j + 2] ? subst(T,x,u) %7 = j^4 + 3*j^2 + 4*j + 1 ? subst(T',x,u) %8 = 3*j^2 + 2*j + 4 ? subst(T'',x,u) %9 = 3*j^3 + 4*j^2 + 4*j + 2
The library syntax is GEN
polmodular(long L, long inv, GEN x = NULL, long y = -1, long derivs)
where y
is a variable number.
(
pol)
Reciprocal polynomial of pol, i.e. the coefficients are in reverse order. pol must be a polynomial.
The library syntax is GEN
polrecip(GEN pol)
.
(x,y,{v},{
flag = 0})
Resultant of the two
polynomials x
and y
with exact entries, with respect to the main
variables of x
and y
if v
is omitted, with respect to the variable v
otherwise. The algorithm assumes the base ring is a domain. If you also need
the u
and v
such that x*u + y*v = Res(x,y)
, use the
polresultantext
function.
If flag = 0
(default), uses the algorithm best suited to the inputs,
either the subresultant algorithm (Lazard/Ducos variant, generic case),
a modular algorithm (inputs in Q[X]
) or Sylvester's matrix (inexact
inputs).
If flag = 1
, uses the determinant of Sylvester's matrix instead; this should
always be slower than the default.
The library syntax is GEN
polresultant0(GEN x, GEN y, long v = -1, long flag)
where v
is a variable number.
(A,B,{v})
Finds polynomials U
and V
such that A*U + B*V = R
, where R
is
the resultant of U
and V
with respect to the main variables of A
and
B
if v
is omitted, and with respect to v
otherwise. Returns the row
vector [U,V,R]
. The algorithm used (subresultant) assumes that the base
ring is a domain.
? A = x*y; B = (x+y)^2; ? [U,V,R] = polresultantext(A, B) %2 = [-y*x - 2*y^2, y^2, y^4] ? A*U + B*V %3 = y^4 ? [U,V,R] = polresultantext(A, B, y) %4 = [-2*x^2 - y*x, x^2, x^4] ? A*U+B*V %5 = x^4
The library syntax is GEN
polresultantext0(GEN A, GEN B, long v = -1)
where v
is a variable number.
Also available is
GEN
polresultantext(GEN x, GEN y)
.
(x)
Complex roots of the polynomial
x, given as a column vector where each root is repeated according to
its multiplicity. The precision is given as for transcendental functions: in
GP it is kept in the variable realprecision
and is transparent to the
user, but it must be explicitly given as a second argument in library mode.
The algorithm used is a modification of A. Schönhagenage>'s root-finding algorithm, due to and originally implemented by X. Gourdon. Barring bugs, it is guaranteed to converge and to give the roots to the required accuracy.
The library syntax is GEN
roots(GEN x, long prec)
.
(
pol,p,{
flag = 0})
Row vector of roots modulo p
of the polynomial pol.
Multiple roots are not repeated.
? polrootsmod(x^2-1,2) %1 = [Mod(1, 2)]~
If p
is very small, you may set flag = 1
, which uses a naive search.
The library syntax is GEN
rootmod0(GEN pol, GEN p, long flag)
.
(x,p,r)
Vector of p
-adic roots of the polynomial pol, given to
p
-adic precision r
p
is assumed to be a prime. Multiple roots are
not repeated. Note that this is not the same as the roots in
Z/p^r
Z, rather it gives approximations in Z/p^r
Z of the true roots
living in Q_p
.
? polrootspadic(x^3 - x^2 + 64, 2, 5) %1 = [2^3 + O(2^5), 2^3 + 2^4 + O(2^5), 1 + O(2^5)]~
If pol has inexact t_PADIC
coefficients, this is not always
well-defined; in this case, the polynomial is first made integral by dividing
out the p
-adic content, then lifted
to Z using truncate
coefficientwise. Hence the roots given are
approximations of the roots of an exact polynomial which is p
-adically
close to the input. To avoid pitfalls, we advise to only factor polynomials
with eact rational coefficients.
The library syntax is GEN
rootpadic(GEN x, GEN p, long r)
.
polrootsreal(T, {
ab})
Real roots of the polynomial T
with rational coefficients, multiple
roots being included according to their multiplicity. The roots are given
to a relative accuracy of realprecision
. If argument ab is
present, it must be a vector [a,b]
with two components (of type
t_INT
, t_FRAC
or t_INFINITY
) and we restrict to roots belonging
to that closed interval.
? \p9 ? polrootsreal(x^2-2) %1 = [-1.41421356, 1.41421356]~ ? polrootsreal(x^2-2, [1,+oo]) %2 = [1.41421356]~ ? polrootsreal(x^2-2, [2,3]) %3 = []~ ? polrootsreal((x-1)*(x-2), [2,3]) %4 = [2.00000000]~
The algorithm used is a modification of Uspensky's method (relying on Descartes's rule of sign), following Rouillier and Zimmerman's article ``Efficient isolation of a polynomial real roots'' (http://hal.inria.fr/inria-00072518/). Barring bugs, it is guaranteed to converge and to give the roots to the required accuracy.
@3Remark. If the polynomial T
is of the
form Q(x^h)
for some h >= 2
and ab is omitted, the routine will
apply the algorithm to Q
(restricting to non-negative roots when h
is
even), then take h
-th roots. On the other hand, if you want to specify
ab, you should apply the routine to Q
yourself and a suitable
interval [a',b']
using approximate h
-th roots adapted to your problem:
the function will not perform this change of variables if ab is present.
The library syntax is GEN
realroots(GEN T, GEN ab = NULL, long prec)
.
polsturm(T,{
ab})
Number of real roots of the real squarefree polynomial T. If
the argument ab is present, it must be a vector [a,b]
with
two real components (of type t_INT
, t_REAL
, t_FRAC
or t_INFINITY
) and we count roots belonging to that closed interval.
If possible, you should stick to exact inputs, that is avoid t_REAL
s in
T
and the bounds a,b
: the result is then guaranteed and we use a fast
algorithm (Uspensky's method, relying on Descartes's rule of sign, see
polrootsreal
); otherwise, we use Sturm's algorithm and the result
may be wrong due to round-off errors.
? T = (x-1)*(x-2)*(x-3); ? polsturm(T) %2 = 3 ? polsturm(T, [-oo,2]) %3 = 2 ? polsturm(T, [1/2,+oo]) %4 = 3 ? polsturm(T, [1, Pi]) \\ Pi inexact: not recommended ! %5 = 3 ? polsturm(T*1., [0, 4]) \\ T*1. inexact: not recommended ! %6 = 3 ? polsturm(T^2, [0, 4]) \\ not squarefree *** at top-level: polsturm(T^2,[0,4]) *** ^------------------- *** polsturm: domain error in polsturm: issquarefree(pol) = 0 ? polsturm((T*1.)^2, [0, 4]) \\ not squarefree AND inexact *** at top-level: polsturm((T*1.)^2,[0 *** ^-------------------- *** polsturm: precision too low in polsturm.
@3In the last example, the input polynomial is not squarefree but there is no way to ascertain it from the given floating point approximation: we get a precision error in this case.
The library syntax is long
RgX_sturmpart(GEN T, GEN ab)
or
long
sturm(GEN T)
(for the case ab = NULL
). The function
long
sturmpart(GEN T, GEN a, GEN b)
is obsolete and deprecated.
(n,d,{v = 'x})
Gives polynomials (in variable v
) defining the sub-Abelian extensions
of degree d
of the cyclotomic field Q(
zeta_n)
, where d |
phi(n)
.
If there is exactly one such extension the output is a polynomial, else it is
a vector of polynomials, possibly empty. To get a vector in all cases,
use concat([], polsubcyclo(n,d))
.
The function galoissubcyclo
allows to specify exactly which
sub-Abelian extension should be computed.
The library syntax is GEN
polsubcyclo(long n, long d, long v = -1)
where v
is a variable number.
(x,y)
Forms the Sylvester matrix
corresponding to the two polynomials x
and y
, where the coefficients of
the polynomials are put in the columns of the matrix (which is the natural
direction for solving equations afterwards). The use of this matrix can be
essential when dealing with polynomials with inexact entries, since
polynomial Euclidean division doesn't make much sense in this case.
The library syntax is GEN
sylvestermatrix(GEN x, GEN y)
.
(x,n)
Creates the column vector of the symmetric powers of the roots of the
polynomial x
up to power n
, using Newton's formula.
The library syntax is GEN
polsym(GEN x, long n)
.
(n,{v = 'x})
Deprecated alias for polchebyshev
The library syntax is GEN
polchebyshev1(long n, long v = -1)
where v
is a variable number.
(n,m)
Creates Zagier's polynomial P_n^{(m)}
used in
the functions sumalt
and sumpos
(with flag = 1
), see
``Convergence acceleration of alternating series'', Cohen et al.,
Experiment. Math., vol. 9, 2000, pp. 3--12.
If m < 0
or m >= n
, P_n^{(m)} = 0
.
We have
P_n := P_n^{(0)}
is T_n(2x-1)
, where T_n
is the Legendre polynomial of
the second kind. For n > m > 0
, P_n^{(m)}
is the m
-th difference with
step 2
of the sequence n^{m+1}P_n
; in this case, it satisfies
2 P_n^{(m)}(sin^2 t) = (d^{m+1})/(dt^{m+1})(
sin (2t)^m
sin (2(n-m)t)).
The library syntax is GEN
polzag(long n, long m)
.
(x,y)
Convolution (or Hadamard product) of the
two power series x
and y
; in other words if x =
sum a_k*X^k
and y =
sum
b_k*X^k
then serconvol(x,y) =
sum a_k*b_k*X^k
.
The library syntax is GEN
convol(GEN x, GEN y)
.
(x)
x
must be a power series with non-negative
exponents or a polynomial. If x =
sum (a_k/k!)*X^k
then the result is sum
a_k*X^k
.
The library syntax is GEN
laplace(GEN x)
.
(s)
Reverse power series of s
, i.e. the series t
such that t(s) = x
;
s
must be a power series whose valuation is exactly equal to one.
? \ps 8 ? t = serreverse(tan(x)) %2 = x - 1/3*x^3 + 1/5*x^5 - 1/7*x^7 + O(x^8) ? tan(t) %3 = x + O(x^8)
The library syntax is GEN
serreverse(GEN s)
.
(x,y,z)
Replace the simple variable y
by the argument z
in the ``polynomial''
expression x
. Every type is allowed for x
, but if it is not a genuine
polynomial (or power series, or rational function), the substitution will be
done as if the scalar components were polynomials of degree zero. In
particular, beware that:
? subst(1, x, [1,2; 3,4]) %1 = [1 0]
[0 1]
? subst(1, x, Mat([0,1])) *** at top-level: subst(1,x,Mat([0,1]) *** ^-------------------- *** subst: forbidden substitution by a non square matrix.
@3
If x
is a power series, z
must be either a polynomial, a power
series, or a rational function. Finally, if x
is a vector,
matrix or list, the substitution is applied to each individual entry.
Use the function substvec
to replace several variables at once,
or the function substpol
to replace a polynomial expression.
The library syntax is GEN
gsubst(GEN x, long y, GEN z)
where y
is a variable number.
(x,y,z)
Replace the ``variable'' y
by the argument z
in the ``polynomial''
expression x
. Every type is allowed for x
, but the same behavior
as subst
above apply.
The difference with subst
is that y
is allowed to be any polynomial
here. The substitution is done moding out all components of x
(recursively) by y - t
, where t
is a new free variable of lowest
priority. Then substituting t
by z
in the resulting expression. For
instance
? substpol(x^4 + x^2 + 1, x^2, y) %1 = y^2 + y + 1 ? substpol(x^4 + x^2 + 1, x^3, y) %2 = x^2 + y*x + 1 ? substpol(x^4 + x^2 + 1, (x+1)^2, y) %3 = (-4*y - 6)*x + (y^2 + 3*y - 3)
The library syntax is GEN
gsubstpol(GEN x, GEN y, GEN z)
.
Further, GEN
gdeflate(GEN T, long v, long d)
attempts to
write T(x)
in the form t(x^d)
, where x =
pol_x
(v)
, and returns
NULL
if the substitution fails (for instance in the example %2
above).
(x,v,w)
v
being a vector of monomials of degree 1 (variables),
w
a vector of expressions of the same length, replace in the expression
x
all occurrences of v_i
by w_i
. The substitutions are done
simultaneously; more precisely, the v_i
are first replaced by new
variables in x
, then these are replaced by the w_i
:
? substvec([x,y], [x,y], [y,x]) %1 = [y, x] ? substvec([x,y], [x,y], [y,x+y]) %2 = [y, x + y] \\ not [y, 2*y]
The library syntax is GEN
gsubstvec(GEN x, GEN v, GEN w)
.
(f,{v})
formal sum of the polynomial expression f
with respect to the
main variable if v
is omitted, with respect to the variable v
otherwise;
it is assumed that the base ring has characteristic zero. In other words,
considering f
as a polynomial function in the variable v
,
returns F
, a polynomial in v
vanishing at 0
, such that F(b) - F(a)
= sum_{v = a+1}^b f(v)
:
? sumformal(n) \\ 1 + ... + n %1 = 1/2*n^2 + 1/2*n ? f(n) = n^3+n^2+1; ? F = sumformal(f(n)) \\ f(1) + ... + f(n) %3 = 1/4*n^4 + 5/6*n^3 + 3/4*n^2 + 7/6*n ? sum(n = 1, 2000, f(n)) == subst(F, n, 2000) %4 = 1 ? sum(n = 1001, 2000, f(n)) == subst(F, n, 2000) - subst(F, n, 1000) %5 = 1 ? sumformal(x^2 + x*y + y^2, y) %6 = y*x^2 + (1/2*y^2 + 1/2*y)*x + (1/3*y^3 + 1/2*y^2 + 1/6*y) ? x^2 * y + x * sumformal(y) + sumformal(y^2) == % %7 = 1
The library syntax is GEN
sumformal(GEN f, long v = -1)
where v
is a variable number.
taylor(x,t,{d =
seriesprecision})
Taylor expansion around 0
of x
with respect to
the simple variable t
. x
can be of any reasonable type, for example a
rational function. Contrary to Ser
, which takes the valuation into
account, this function adds O(t^d)
to all components of x
.
? taylor(x/(1+y), y, 5) %1 = (y^4 - y^3 + y^2 - y + 1)*x + O(y^5) ? Ser(x/(1+y), y, 5) *** at top-level: Ser(x/(1+y),y,5) *** ^---------------- *** Ser: main variable must have higher priority in gtoser.
The library syntax is GEN
tayl(GEN x, long t, long precdl)
where t
is a variable number.
thue(
tnf,a,{
sol})
Returns all solutions of the equation
P(x,y) = a
in integers x
and y
, where tnf was created with
thueinit(P)
. If present, sol must contain the solutions of
Norm (x) = a
modulo units of positive norm in the number field
defined by P
(as computed by bnfisintnorm
). If there are infinitely
many solutions, an error is issued.
It is allowed to input directly the polynomial P
instead of a tnf,
in which case, the function first performs thueinit(P,0)
. This is
very wasteful if more than one value of a
is required.
If tnf was computed without assuming GRH (flag 1
in thueinit
),
then the result is unconditional. Otherwise, it depends in principle of the
truth of the GRH, but may still be unconditionally correct in some
favorable cases. The result is conditional on the GRH if
a != +- 1
and, P
has a single irreducible rational factor, whose
attached tentative class number h
and regulator R
(as computed
assuming the GRH) satisfy
@3* h > 1
,
@3* R/0.2 > 1.5
.
Here's how to solve the Thue equation x^{13} - 5y^{13} = - 4
:
? tnf = thueinit(x^13 - 5); ? thue(tnf, -4) %1 = [[1, 1]]
@3In this case, one checks that bnfinit(x^13 -5).no
is 1
. Hence, the only solution is (x,y) = (1,1)
, and the result is
unconditional. On the other hand:
? P = x^3-2*x^2+3*x-17; tnf = thueinit(P); ? thue(tnf, -15) %2 = [[1, 1]] \\ a priori conditional on the GRH. ? K = bnfinit(P); K.no %3 = 3 ? K.reg %4 = 2.8682185139262873674706034475498755834
This time the result is conditional. All results computed using this
particular tnf are likewise conditional, except for a right-hand
side of +- 1
.
The above result is in fact correct, so we did not just disprove the GRH:
? tnf = thueinit(x^3-2*x^2+3*x-17, 1 /*unconditional*/); ? thue(tnf, -15) %4 = [[1, 1]]
Note that reducible or non-monic polynomials are allowed:
? tnf = thueinit((2*x+1)^5 * (4*x^3-2*x^2+3*x-17), 1); ? thue(tnf, 128) %2 = [[-1, 0], [1, 0]]
@3Reducible polynomials are in fact much easier to handle.
The library syntax is GEN
thue(GEN tnf, GEN a, GEN sol = NULL)
.
(P,{
flag = 0})
Initializes the tnf corresponding to P
, a non-constant
univariate polynomial with integer coefficients.
The result is meant to be used in conjunction with thue
to solve Thue
equations P(X / Y)Y^{
deg P} = a
, where a
is an integer. Accordingly,
P
must either have at least two distinct irreducible factors over Q,
or have one irreducible factor T
with degree > 2
or two conjugate
complex roots: under these (necessary and sufficient) conditions, the
equation has finitely many integer solutions.
? S = thueinit(t^2+1); ? thue(S, 5) %2 = [[-2, -1], [-2, 1], [-1, -2], [-1, 2], [1, -2], [1, 2], [2, -1], [2, 1]] ? S = thueinit(t+1); *** at top-level: thueinit(t+1) *** ^------------- *** thueinit: domain error in thueinit: P = t + 1
@3The hardest case is when deg P > 2
and P
is irreducible
with at least one real root. The routine then uses Bilu-Hanrot's algorithm.
If flag is non-zero, certify results unconditionally. Otherwise, assume
GRH, this being much faster of course. In the latter case, the result
may still be unconditionally correct, see thue
. For instance in most
cases where P
is reducible (not a pure power of an irreducible), or
conditional computed class groups are trivial or the right hand side
is +-1
, then results are unconditional.
@3Note. The general philosophy is to disprove the existence of large
solutions then to enumerate bounded solutions naively. The implementation
will overflow when there exist huge solutions and the equation has degree
> 2
(the quadratic imaginary case is special, since we can use
bnfisintnorm
):
? thue(t^3+2, 10^30) *** at top-level: L=thue(t^3+2,10^30) *** ^----------------- *** thue: overflow in thue (SmallSols): y <= 80665203789619036028928. ? thue(x^2+2, 10^30) \\ quadratic case much easier %1 = [[-1000000000000000, 0], [1000000000000000, 0]]
@3Note. It is sometimes possible to circumvent the above, and in any
case obtain an important speed-up, if you can write P = Q(x^d)
for some d >
1
and Q
still satisfying the thueinit
hypotheses. You can then solve
the equation attached to Q
then eliminate all solutions (x,y)
such that
either x
or y
is not a d
-th power.
? thue(x^4+1, 10^40); \\ stopped after 10 hours ? filter(L,d) = my(x,y); [[x,y] | v<-L, ispower(v[1],d,&x)&&ispower(v[2],d,&y)]; ? L = thue(x^2+1, 10^40); ? filter(L, 2) %4 = [[0, 10000000000], [10000000000, 0]]
@3The last 2 commands use less than 20ms.
The library syntax is GEN
thueinit(GEN P, long flag, long prec)
.
Note that most linear algebra functions operating on subspaces defined by
generating sets (such as mathnf
, qflll
, etc.) take matrices as
arguments. As usual, the generating vectors are taken to be the
columns of the given matrix.
Since PARI does not have a strong typing system, scalars live in
unspecified commutative base rings. It is very difficult to write
robust linear algebra routines in such a general setting. We thus
assume that the base ring is a domain and work over its field of
fractions. If the base ring is not a domain, one gets an error as soon
as a non-zero pivot turns out to be non-invertible. Some functions,
e.g. mathnf
or mathnfmod
, specifically assume that the base ring is
Z.
(z,k,{
flag = 0})
z
being real/complex, or p
-adic, finds a polynomial (in the variable
'x
) of degree at most
k
, with integer coefficients, having z
as approximate root. Note that the
polynomial which is obtained is not necessarily the ``correct'' one. In fact
it is not even guaranteed to be irreducible. One can check the closeness
either by a polynomial evaluation (use subst
), or by computing the
roots of the polynomial given by algdep
(use polroots
or
polrootspadic
).
Internally, lindep
([1,z,...,z^k],
flag)
is used. A non-zero value of
flag may improve on the default behavior if the input number is known to a
huge accuracy, and you suspect the last bits are incorrect: if flag > 0
the computation is done with an accuracy of flag decimal digits; to get
meaningful results, the parameter flag should be smaller than the number of
correct decimal digits in the input.
But default values are usually sufficient, so try without flag first:
? \p200 ? z = 2^(1/6)+3^(1/5); ? algdep(z, 30); \\ right in 280ms ? algdep(z, 30, 100); \\ wrong in 169ms ? algdep(z, 30, 170); \\ right in 288ms ? algdep(z, 30, 200); \\ wrong in 320ms ? \p250 ? z = 2^(1/6)+3^(1/5); \\ recompute to new, higher, accuracy ! ? algdep(z, 30); \\ right in 329ms ? algdep(z, 30, 200); \\ right in 324ms ? \p500 ? algdep(2^(1/6)+3^(1/5), 30); \\ right in 677ms ? \p1000 ? algdep(2^(1/6)+3^(1/5), 30); \\ right in 1.5s
The changes in realprecision
only affect the quality of the
initial approximation to 2^{1/6} + 3^{1/5}
, algdep
itself uses
exact operations. The size of its operands depend on the accuracy of the
input of course: more accurate input means slower operations.
Proceeding by increments of 5 digits of accuracy, algdep
with default
flag produces its first correct result at 195 digits, and from then on a
steady stream of correct results:
\\ assume T contains the correct result, for comparison forstep(d=100, 250, 5, localprec(d);\ print(d, " ", algdep(2^(1/6)+3^(1/5),30) == T))
The above example is the test case studied in a 2000 paper by Borwein and Lisonek: Applications of integer relation algorithms, Discrete Math., 217, p. 65--82. The version of PARI tested there was 1.39, which succeeded reliably from precision 265 on, in about 200 as much time as the current version.
The library syntax is GEN
algdep0(GEN z, long k, long flag)
.
Also available is GEN
algdep(GEN z, long k)
(flag = 0
).
(A,{v = 'x},{
flag = 5})
characteristic polynomial
of A
with respect to the variable v
, i.e. determinant of v*I-A
if A
is a square matrix.
? charpoly([1,2;3,4]); %1 = x^2 - 5*x - 2 ? charpoly([1,2;3,4],, 't) %2 = t^2 - 5*t - 2
If A
is not a square matrix, the function returns the characteristic
polynomial of the map ``multiplication by A
'' if A
is a scalar:
? charpoly(Mod(x+2, x^3-2)) %1 = x^3 - 6*x^2 + 12*x - 10 ? charpoly(I) %2 = x^2 + 1 ? charpoly(quadgen(5)) %3 = x^2 - x - 1 ? charpoly(ffgen(ffinit(2,4))) %4 = Mod(1, 2)*x^4 + Mod(1, 2)*x^3 + Mod(1, 2)*x^2 + Mod(1, 2)*x + Mod(1, 2)
The value of flag is only significant for matrices, and we advise to stick
to the default value. Let n
be the dimension of A
.
If flag = 0
, same method (Le Verrier's) as for computing the adjoint matrix,
i.e. using the traces of the powers of A
. Assumes that n!
is
invertible; uses O(n^4)
scalar operations.
If flag = 1
, uses Lagrange interpolation which is usually the slowest method.
Assumes that n!
is invertible; uses O(n^4)
scalar operations.
If flag = 2
, uses the Hessenberg form. Assumes that the base ring is a field.
Uses O(n^3)
scalar operations, but suffers from coefficient explosion
unless the base field is finite or R.
If flag = 3
, uses Berkowitz's division free algorithm, valid over any
ring (commutative, with unit). Uses O(n^4)
scalar operations.
If flag = 4
, x
must be integral. Uses a modular algorithm: Hessenberg form
for various small primes, then Chinese remainders.
If flag = 5
(default), uses the ``best'' method given x
.
This means we use Berkowitz unless the base ring is Z (use flag = 4
)
or a field where coefficient explosion does not occur,
e.g. a finite field or the reals (use flag = 2
).
The library syntax is GEN
charpoly0(GEN A, long v = -1, long flag)
where v
is a variable number.
Also available are
GEN
charpoly(GEN x, long v)
(flag = 5
),
GEN
caract(GEN A, long v)
(flag = 1
),
GEN
carhess(GEN A, long v)
(flag = 2
),
GEN
carberkowitz(GEN A, long v)
(flag = 3
) and
GEN
caradj(GEN A, long v, GEN *pt)
. In this
last case, if pt is not NULL
, *pt
receives the address of
the adjoint matrix of A
(see matadjoint
), so both can be obtained at
once.
(x,{y})
Concatenation of x
and y
. If x
or y
is
not a vector or matrix, it is considered as a one-dimensional vector. All
types are allowed for x
and y
, but the sizes must be compatible. Note
that matrices are concatenated horizontally, i.e. the number of rows stays
the same. Using transpositions, one can concatenate them vertically,
but it is often simpler to use matconcat
.
? x = matid(2); y = 2*matid(2); ? concat(x,y) %2 = [1 0 2 0]
[0 1 0 2] ? concat(x~,y~)~ %3 = [1 0]
[0 1]
[2 0]
[0 2] ? matconcat([x;y]) %4 = [1 0]
[0 1]
[2 0]
[0 2]
@3
To concatenate vectors sideways (i.e. to obtain a two-row or two-column
matrix), use Mat
instead, or matconcat
:
? x = [1,2]; ? y = [3,4]; ? concat(x,y) %3 = [1, 2, 3, 4]
? Mat([x,y]~) %4 = [1 2]
[3 4] ? matconcat([x;y]) %5 = [1 2]
[3 4]
Concatenating a row vector to a matrix having the same number of columns will
add the row to the matrix (top row if the vector is x
, i.e. comes first, and
bottom row otherwise).
The empty matrix [;]
is considered to have a number of rows compatible
with any operation, in particular concatenation. (Note that this is
not the case for empty vectors [ ]
or [ ]~
.)
If y
is omitted, x
has to be a row vector or a list, in which case its
elements are concatenated, from left to right, using the above rules.
? concat([1,2], [3,4]) %1 = [1, 2, 3, 4] ? a = [[1,2]~, [3,4]~]; concat(a) %2 = [1 3]
[2 4]
? concat([1,2; 3,4], [5,6]~) %3 = [1 2 5]
[3 4 6] ? concat([%, [7,8]~, [1,2,3,4]]) %5 = [1 2 5 7]
[3 4 6 8]
[1 2 3 4]
The library syntax is GEN
gconcat(GEN x, GEN y = NULL)
.
GEN
gconcat1(GEN x)
is a shortcut for gconcat(x,NULL)
.
(v,q,b,
expr)
q
being a square and symmetric integral matrix representing a positive
definite
quadratic form, evaluate expr
for all vector v
such that q(v) <= b
.
The formal variable v
runs through all such vectors in turn.
? forqfvec(v, [3,2;2,3], 3, print(v)) [0, 1]~ [1, 0]~ [-1, 1]~
The library syntax is void
forqfvec0(GEN v, GEN q = NULL, GEN b)
.
The following function is also available:
void
forqfvec(void *E, long (*fun)(void *, GEN, GEN, double), GEN q, GEN b)
:
Evaluate fun(E,w,v,m)
on all v
such that q(v) < b
, where v
is a
t_VECSMALL
and m = q(v)
is a C double. The function fun
must
return 0
, unless forqfvec
should stop, in which case, it should
return 1
.
(v,{
flag = 0})
finds a small non-trivial integral linear
combination between components of v
. If none can be found return an empty
vector.
If v
is a vector with real/complex entries we use a floating point
(variable precision) LLL algorithm. If flag = 0
the accuracy is chosen
internally using a crude heuristic. If flag > 0
the computation is done with
an accuracy of flag decimal digits. To get meaningful results in the latter
case, the parameter flag should be smaller than the number of correct
decimal digits in the input.
? lindep([sqrt(2), sqrt(3), sqrt(2)+sqrt(3)]) %1 = [-1, -1, 1]~
If v
is p
-adic, flag is ignored and the algorithm LLL-reduces a
suitable (dual) lattice.
? lindep([1, 2 + 3 + 3^2 + 3^3 + 3^4 + O(3^5)]) %2 = [1, -2]~
If v
is a matrix (or a vector of column vectors, or a vector of row
vectors), flag is ignored and the function returns a non trivial kernel
vector if one exists, else an empty vector.
? lindep([1,2,3;4,5,6;7,8,9]) %3 = [1, -2, 1]~ ? lindep([[1,0], [2,0]]) %4 = [2, -1]~ ? lindep([[1,0], [0,1]]) %5 = []~
If v
contains polynomials or power series over some base field, finds a
linear relation with coefficients in the field.
? lindep([x*y, x^2 + y, x^2*y + x*y^2, 1]) %4 = [y, y, -1, -y^2]~
@3For better control, it is preferable to use t_POL
rather
than t_SER
in the input, otherwise one gets a linear combination which is
t
-adically small, but not necessarily 0
. Indeed, power series are first
converted to the minimal absolute accuracy occurring among the entries of v
(which can cause some coefficients to be ignored), then truncated to
polynomials:
? v = [t^2+O(t^4), 1+O(t^2)]; L=lindep(v) %1 = [1, 0]~ ? v*L %2 = t^2+O(t^4) \\ small but not 0
The library syntax is GEN
lindep0(GEN v, long flag)
.
Also available are GEN
lindep(GEN v)
(real/complex entries,
flag = 0
), GEN
lindep2(GEN v, long flag)
(real/complex entries)
GEN
padic_lindep(GEN v)
(p
-adic entries) and
GEN
Xadic_lindep(GEN v)
(polynomial entries).
Finally GEN
deplin(GEN v)
returns a non-zero kernel vector for a
t_MAT
input.
(M,{
flag = 0})
adjoint matrix of M
, i.e. a matrix N
of cofactors of M
, satisfying M*N =
det (M)*
Id . M
must be a
(non-necessarily invertible) square matrix of dimension n
.
If flag is 0 or omitted, we try to use Leverrier-Faddeev's algorithm,
which assumes that n!
invertible. If it fails or flag = 1
,
compute T = charpoly(M)
independently first and return
(-1)^{n-1} (T(x)-T(0))/x
evaluated at M
.
? a = [1,2,3;3,4,5;6,7,8] * Mod(1,4); %2 = [Mod(1, 4) Mod(2, 4) Mod(3, 4)]
[Mod(3, 4) Mod(0, 4) Mod(1, 4)]
[Mod(2, 4) Mod(3, 4) Mod(0, 4)]
@3
Both algorithms use O(n^4)
operations in the base ring, and are usually
slower than computing the characteristic polynomial or the inverse of M
directly.
The library syntax is GEN
matadjoint0(GEN M, long flag)
.
Also available are
GEN
adj(GEN x)
(flag = 0) and
GEN
adjsafe(GEN x)
(flag = 1).
(x)
The left companion matrix to the non-zero polynomial x
.
The library syntax is GEN
matcompanion(GEN x)
.
(v)
Returns a t_MAT
built from the entries of v
, which may
be a t_VEC
(concatenate horizontally), a t_COL
(concatenate
vertically), or a t_MAT
(concatenate vertically each column, and
concatenate vertically the resulting matrices). The entries of v
are always
considered as matrices: they can themselves be t_VEC
(seen as a row
matrix), a t_COL
seen as a column matrix), a t_MAT
, or a scalar (seen
as an 1 x 1
matrix).
? A=[1,2;3,4]; B=[5,6]~; C=[7,8]; D=9; ? matconcat([A, B]) \\ horizontal %1 = [1 2 5]
[3 4 6] ? matconcat([A, C]~) \\ vertical %2 = [1 2]
[3 4]
[7 8] ? matconcat([A, B; C, D]) \\ block matrix %3 = [1 2 5]
[3 4 6]
[7 8 9]
@3 If the dimensions of the entries to concatenate do not match up, the above rules are extended as follows:
@3* each entry v_{i,j}
of v
has a natural length and height: 1 x
1
for a scalar, 1 x n
for a t_VEC
of length n
, n x 1
for a t_COL
, m x n
for an m x n
t_MAT
@3* let H_i
be the maximum over j
of the lengths of the v_{i,j}
,
let L_j
be the maximum over i
of the heights of the v_{i,j}
.
The dimensions of the (i,j)
-th block in the concatenated matrix are
H_i x L_j
.
@3* a scalar s = v_{i,j}
is considered as s
times an identity matrix
of the block dimension min (H_i,L_j)
@3* blocks are extended by 0 columns on the right and 0 rows at the bottom, as needed.
? matconcat([1, [2,3]~, [4,5,6]~]) \\ horizontal %4 = [1 2 4]
[0 3 5]
[0 0 6] ? matconcat([1, [2,3], [4,5,6]]~) \\ vertical %5 = [1 0 0]
[2 3 0]
[4 5 6] ? matconcat([B, C; A, D]) \\ block matrix %6 = [5 0 7 8]
[6 0 0 0]
[1 2 9 0]
[3 4 0 9] ? U=[1,2;3,4]; V=[1,2,3;4,5,6;7,8,9]; ? matconcat(matdiagonal([U, V])) \\ block diagonal %7 = [1 2 0 0 0]
[3 4 0 0 0]
[0 0 1 2 3]
[0 0 4 5 6]
[0 0 7 8 9]
The library syntax is GEN
matconcat(GEN v)
.
(x,{
flag = 0})
Determinant of the square matrix x
.
If flag = 0
, uses an appropriate algorithm depending on the coefficients:
@3* integer entries: modular method due to Dixon, Pernet and Stein.
@3* real or p
-adic entries: classical Gaussian elimination using maximal
pivot.
@3* intmod entries: classical Gaussian elimination using first non-zero pivot.
@3* other cases: Gauss-Bareiss.
If flag = 1
, uses classical Gaussian elimination with appropriate pivoting
strategy (maximal pivot for real or p
-adic coefficients). This is usually
worse than the default.
The library syntax is GEN
det0(GEN x, long flag)
.
Also available are GEN
det(GEN x)
(flag = 0
),
GEN
det2(GEN x)
(flag = 1
) and GEN
ZM_det(GEN x)
for integer
entries.
(B)
Let B
be an m x n
matrix with integer coefficients. The
determinant D
of the lattice generated by the columns of B
is
the square root of det (B^T B)
if B
has maximal rank m
, and 0
otherwise.
This function uses the Gauss-Bareiss algorithm to compute a positive
multiple of D
. When B
is square, the function actually returns
D = |
det B|
.
This function is useful in conjunction with mathnfmod
, which needs to
know such a multiple. If the rank is maximal and the matrix non-square,
you can obtain D
exactly using
matdet( mathnfmod(B, matdetint(B)) )
Note that as soon as one of the dimensions gets large (m
or n
is larger
than 20, say), it will often be much faster to use mathnf(B, 1)
or
mathnf(B, 4)
directly.
The library syntax is GEN
detint(GEN B)
.
(x)
x
being a vector, creates the diagonal matrix
whose diagonal entries are those of x
.
? matdiagonal([1,2,3]); %1 = [1 0 0]
[0 2 0]
[0 0 3]
@3Block diagonal matrices are easily created using
matconcat
:
? U=[1,2;3,4]; V=[1,2,3;4,5,6;7,8,9]; ? matconcat(matdiagonal([U, V])) %1 = [1 2 0 0 0]
[3 4 0 0 0]
[0 0 1 2 3]
[0 0 4 5 6]
[0 0 7 8 9]
The library syntax is GEN
diagonal(GEN x)
.
(x,{
flag = 0})
Returns the (complex) eigenvectors of x
as columns of a matrix.
If flag = 1
, return [L,H]
, where L
contains the
eigenvalues and H
the corresponding eigenvectors; multiple eigenvalues are
repeated according to the eigenspace dimension (which may be less
than the eigenvalue multiplicity in the characteristic polynomial).
This function first computes the characteristic polynomial of x
and
approximates its complex roots (
lambda_i)
, then tries to compute the
eigenspaces as kernels of the x -
lambda_i
. This algorithm is
ill-conditioned and is likely to miss kernel vectors if some roots of the
characteristic polynomial are close, in particular if it has multiple roots.
? A = [13,2; 10,14]; mateigen(A) %1 = [-1/2 2/5]
[ 1 1] ? [L,H] = mateigen(A, 1); ? L %3 = [9, 18] ? H %4 = [-1/2 2/5]
[ 1 1]
@3
For symmetric matrices, use qfjacobi
instead; for Hermitian matrices,
compute
A = real(x); B = imag(x); y = matconcat([A, -B; B, A]);
@3and apply qfjacobi
to y
.
The library syntax is GEN
mateigen(GEN x, long flag, long prec)
.
Also available is GEN
eigen(GEN x, long prec)
(flag = 0
)
(M,{
flag},{v = 'x})
Returns the Frobenius form of
the square matrix M
. If flag = 1
, returns only the elementary divisors as
a vector of polynomials in the variable v
. If flag = 2
, returns a
two-components vector [F,B] where F
is the Frobenius form and B
is
the basis change so that M = B^{-1}FB
.
The library syntax is GEN
matfrobenius(GEN M, long flag, long v = -1)
where v
is a variable number.
(x)
Returns a matrix similar to the square matrix x
, which is in upper Hessenberg
form (zero entries below the first subdiagonal).
The library syntax is GEN
hess(GEN x)
.
(n)
x
being a long
, creates the
Hilbert matrixof order x
, i.e. the matrix whose coefficient
(i
,j
) is 1/ (i+j-1)
.
The library syntax is GEN
mathilbert(long n)
.
(M,{
flag = 0})
Let R
be a Euclidean ring, equal to Z or to K[X]
for some field
K
. If M
is a (not necessarily square) matrix with entries in R
, this
routine finds the upper triangular Hermite normal form of M
.
If the rank of M
is equal to its number of rows, this is a square
matrix. In general, the columns of the result form a basis of the R
-module
spanned by the columns of M
.
The values 0,1,2,3
of flag have a binary meaning, analogous to the one
in matsnf
; in this case, binary digits of flag mean:
@3* 1 (complete output): if set, outputs [H,U]
, where H
is the Hermite
normal form of M
, and U
is a transformation matrix such that MU = [0|H]
.
The matrix U
belongs to GL(R)
. When M
has a large kernel, the
entries of U
are in general huge.
@3* 2 (generic input): Deprecated. If set, assume that R = K[X]
is
a polynomial ring; otherwise, assume that R =
Z. This flag is now useless
since the routine always checks whether the matrix has integral entries.
@3For these 4 values, we use a naive algorithm, which behaves well
in small dimension only. Larger values correspond to different algorithms,
are restricted to integer matrices, and all output the unimodular
matrix U
. From now on all matrices have integral entries.
@3* flag = 4
, returns [H,U]
as in ``complete output'' above, using a
variant of LLL reduction along the way. The matrix U
is provably
small in the L_2
sense, and in general close to optimal; but the
reduction is in general slow, although provably polynomial-time.
If flag = 5
, uses Batut's algorithm and output [H,U,P]
, such that H
and
U
are as before and P
is a permutation of the rows such that P
applied
to MU
gives H
. This is in general faster than flag = 4
but the matrix U
is usually worse; it is heuristically smaller than with the default algorithm.
When the matrix is dense and the dimension is large (bigger than 100, say),
flag = 4
will be fastest. When M
has maximal rank, then
H = mathnfmod(M, matdetint(M))
@3will be even faster. You can then recover U
as M^{-1}H
.
? M = matrix(3,4,i,j,random([-5,5])) %1 = [ 0 2 3 0]
[-5 3 -5 -5]
[ 4 3 -5 4]
? [H,U] = mathnf(M, 1); ? U %3 = [-1 0 -1 0]
[ 0 5 3 2]
[ 0 3 1 1]
[ 1 0 0 0]
? H %5 = [19 9 7]
[ 0 9 1]
[ 0 0 1]
? M*U %6 = [0 19 9 7]
[0 0 9 1]
[0 0 0 1]
For convenience, M
is allowed to be a t_VEC
, which is then
automatically converted to a t_MAT
, as per the Mat
function.
For instance to solve the generalized extended gcd problem, one may use
? v = [116085838, 181081878, 314252913,10346840]; ? [H,U] = mathnf(v, 1); ? U %2 = [ 103 -603 15 -88]
[-146 13 -1208 352]
[ 58 220 678 -167]
[-362 -144 381 -101] ? v*U %3 = [0, 0, 0, 1]
@3This also allows to input a matrix as a t_VEC
of
t_COL
s of the same length (which Mat
would concatenate to
the t_MAT
having those columns):
? v = [[1,0,4]~, [3,3,4]~, [0,-4,-5]~]; mathnf(v) %1 = [47 32 12]
[ 0 1 0]
[ 0 0 1]
The library syntax is GEN
mathnf0(GEN M, long flag)
.
Also available are GEN
hnf(GEN M)
(flag = 0
) and
GEN
hnfall(GEN M)
(flag = 1
). To reduce huge relation matrices
(sparse with small entries, say dimension 400
or more), you can use the
pair hnfspec
/ hnfadd
. Since this is quite technical and the
calling interface may change, they are not documented yet. Look at the code
in basemath/hnf_snf.c
.
(x,d)
If x
is a (not necessarily square) matrix of
maximal rank with integer entries, and d
is a multiple of the (non-zero)
determinant of the lattice spanned by the columns of x
, finds the
upper triangular Hermite normal form of x
.
If the rank of x
is equal to its number of rows, the result is a square
matrix. In general, the columns of the result form a basis of the lattice
spanned by the columns of x
. Even when d
is known, this is in general
slower than mathnf
but uses much less memory.
The library syntax is GEN
hnfmod(GEN x, GEN d)
.
(x,d)
Outputs the (upper triangular)
Hermite normal form of x
concatenated with the diagonal
matrix with diagonal d
. Assumes that x
has integer entries.
Variant: if d
is an integer instead of a vector, concatenate d
times the
identity matrix.
? m=[0,7;-1,0;-1,-1] %1 = [ 0 7]
[-1 0]
[-1 -1] ? mathnfmodid(m, [6,2,2]) %2 = [2 1 1]
[0 1 0]
[0 0 1] ? mathnfmodid(m, 10) %3 = [10 7 3]
[ 0 1 0]
[ 0 0 1]
The library syntax is GEN
hnfmodid(GEN x, GEN d)
.
(Q,v)
applies a sequence Q
of Householder
transforms, as returned by matqr
(M,1)
to the vector or matrix v
.
The library syntax is GEN
mathouseholder(GEN Q, GEN v)
.
(n)
Creates the n x n
identity matrix.
The library syntax is GEN
matid(long n)
.
(x,{
flag = 0})
Gives a basis for the image of the
matrix x
as columns of a matrix. A priori the matrix can have entries of
any type. If flag = 0
, use standard Gauss pivot. If flag = 1
, use
matsupplement
(much slower: keep the default flag!).
The library syntax is GEN
matimage0(GEN x, long flag)
.
Also available is GEN
image(GEN x)
(flag = 0
).
(x)
Gives the vector of the column indices which
are not extracted by the function matimage
, as a permutation
(t_VECSMALL
). Hence the number of
components of matimagecompl(x)
plus the number of columns of
matimage(x)
is equal to the number of columns of the matrix x
.
The library syntax is GEN
imagecompl(GEN x)
.
(x)
x
being a matrix of rank r
, returns a vector with two
t_VECSMALL
components y
and z
of length r
giving a list of rows
and columns respectively (starting from 1) such that the extracted matrix
obtained from these two vectors using vecextract(x,y,z)
is
invertible.
The library syntax is GEN
indexrank(GEN x)
.
(x,y)
x
and y
being two matrices with the same
number of rows each of whose columns are independent, finds a basis of the
Q-vector space equal to the intersection of the spaces spanned by the
columns of x
and y
respectively. The faster function
idealintersect
can be used to intersect fractional ideals (projective
Z_K
modules of rank 1
); the slower but much more general function
nfhnf
can be used to intersect general Z_K
-modules.
The library syntax is GEN
intersect(GEN x, GEN y)
.
(x,y)
Given a matrix x
and
a column vector or matrix y
, returns a preimage z
of y
by x
if one
exists (i.e such that x z = y
), an empty vector or matrix otherwise. The
complete inverse image is z + Ker x
, where a basis of the kernel of
x
may be obtained by matker
.
? M = [1,2;2,4]; ? matinverseimage(M, [1,2]~) %2 = [1, 0]~ ? matinverseimage(M, [3,4]~) %3 = []~ \\ no solution ? matinverseimage(M, [1,3,6;2,6,12]) %4 = [1 3 6]
[0 0 0] ? matinverseimage(M, [1,2;3,4]) %5 = [;] \\ no solution ? K = matker(M) %6 = [-2]
[1]
The library syntax is GEN
inverseimage(GEN x, GEN y)
.
(x)
Returns true (1) if x
is a diagonal matrix, false (0) if not.
The library syntax is GEN
isdiagonal(GEN x)
.
(x,{
flag = 0})
Gives a basis for the kernel of the matrix x
as columns of a matrix.
The matrix can have entries of any type, provided they are compatible with
the generic arithmetic operations (+
, x
and /
).
If x
is known to have integral entries, set flag = 1
.
The library syntax is GEN
matker0(GEN x, long flag)
.
Also available are GEN
ker(GEN x)
(flag = 0
),
GEN
keri(GEN x)
(flag = 1
).
(x,{
flag = 0})
Gives an LLL-reduced Z-basis
for the lattice equal to the kernel of the matrix x
with rational entries.
flag is deprecated, kept for backward compatibility.
The library syntax is GEN
matkerint0(GEN x, long flag)
.
Use directly GEN
kerint(GEN x)
if x
is known to have
integer entries, and Q_primpart
first otherwise.
(x,d)
Product of the matrix x
by the diagonal
matrix whose diagonal entries are those of the vector d
. Equivalent to,
but much faster than x*matdiagonal(d)
.
The library syntax is GEN
matmuldiagonal(GEN x, GEN d)
.
(x,y)
Product of the matrices x
and y
assuming that the result is a
diagonal matrix. Much faster than x*y
in that case. The result is
undefined if x*y
is not diagonal.
The library syntax is GEN
matmultodiagonal(GEN x, GEN y)
.
(n,{q})
Creates as a matrix the lower triangular
Pascal triangle of order x+1
(i.e. with binomial coefficients
up to x
). If q
is given, compute the q
-Pascal triangle (i.e. using
q
-binomial coefficients).
The library syntax is GEN
matqpascal(long n, GEN q = NULL)
.
Also available is GEN
matpascal(GEN x)
.
(M,{
flag = 0})
Returns [Q,R]
, the QR-decomposition of the square invertible
matrix M
with real entries: Q
is orthogonal and R
upper triangular. If
flag = 1
, the orthogonal matrix is returned as a sequence of Householder
transforms: applying such a sequence is stabler and faster than
multiplication by the corresponding Q
matrix.
More precisely, if
[Q,R] = matqr(M); [q,r] = matqr(M, 1);
@3then r = R
and mathouseholder
(q, M)
is
(close to) R
; furthermore
mathouseholder(q, matid(#M)) == Q~
@3the inverse of Q
. This function raises an error if the
precision is too low or x
is singular.
The library syntax is GEN
matqr(GEN M, long flag, long prec)
.
(x)
Rank of the matrix x
.
The library syntax is long
rank(GEN x)
.
matrix(m,n,{X},{Y},{
expr = 0})
Creation of the
m x n
matrix whose coefficients are given by the expression
expr. There are two formal parameters in expr, the first one
(X
) corresponding to the rows, the second (Y
) to the columns, and X
goes from 1 to m
, Y
goes from 1 to n
. If one of the last 3 parameters
is omitted, fill the matrix with zeroes.
(A,{p = 0})
A
being an m x n
matrix in M_{m,n}(
Q)
, let
Im_
Q A
(resp. Im_
Z A
) the Q-vector space
(resp. the Z-module) spanned by the columns of A
. This function has
varying behavior depending on the sign of p
:
If p >= 0
, A
is assumed to have maximal rank n <= m
. The function
returns a matrix B\in M_{m,n}(
Z)
, with Im_
Q B = Im_
Q A
,
such that the GCD of all its n x n
minors is coprime to
p
; in particular, if p = 0
(default), this GCD is 1
.
? minors(x) = vector(#x[,1], i, matdet(x[^i,])); ? A = [3,1/7; 5,3/7; 7,5/7]; minors(A) %1 = [4/7, 8/7, 4/7] \\ determinants of all 2x2 minors ? B = matrixqz(A) %2 = [3 1]
[5 2]
[7 3] ? minors(%) %3 = [1, 2, 1] \\ B integral with coprime minors
If p = -1
, returns the HNF basis of the lattice Z^n
cap Im_
Z A
.
If p = -2
, returns the HNF basis of the lattice Z^n
cap Im_
Q A
.
? matrixqz(A,-1) %4 = [8 5]
[4 3]
[0 1]
? matrixqz(A,-2) %5 = [2 -1]
[1 0]
[0 1]
The library syntax is GEN
matrixqz0(GEN A, GEN p = NULL)
.
(x)
x
being a vector or matrix, returns a row vector
with two components, the first being the number of rows (1 for a row vector),
the second the number of columns (1 for a column vector).
The library syntax is GEN
matsize(GEN x)
.
(X,{
flag = 0})
If X
is a (singular or non-singular) matrix outputs the vector of
elementary divisors of X
, i.e. the diagonal of the
Smith normal form of X
, normalized so that d_n | d_{n-1} |
... | d_1
.
The binary digits of flag mean:
1 (complete output): if set, outputs [U,V,D]
, where U
and V
are two
unimodular matrices such that UXV
is the diagonal matrix D
. Otherwise
output only the diagonal of D
. If X
is not a square matrix, then D
will be a square diagonal matrix padded with zeros on the left or the top.
2 (generic input): if set, allows polynomial entries, in which case the
input matrix must be square. Otherwise, assume that X
has integer
coefficients with arbitrary shape.
4 (cleanup): if set, cleans up the output. This means that elementary
divisors equal to 1
will be deleted, i.e. outputs a shortened vector D'
instead of D
. If complete output was required, returns [U',V',D']
so
that U'XV' = D'
holds. If this flag is set, X
is allowed to be of the
form `vector of elementary divisors' or [U,V,D]
as would normally be output with the cleanup flag
unset.
The library syntax is GEN
matsnf0(GEN X, long flag)
.
(M,B)
M
being an invertible matrix and B
a column
vector, finds the solution X
of MX = B
, using Dixon p
-adic lifting method
if M
and B
are integral and Gaussian elimination otherwise. This
has the same effect as, but is faster, than M^{-1}*B
.
The library syntax is GEN
gauss(GEN M, GEN B)
.
For integral input, the function
GEN
ZM_gauss(GEN M,GEN B)
is also available.
(M,D,B,{
flag = 0})
M
being any integral matrix,
D
a column vector of non-negative integer moduli, and B
an integral
column vector, gives a small integer solution to the system of congruences
sum_i m_{i,j}x_j = b_i (mod d_i)
if one exists, otherwise returns
zero. Shorthand notation: B
(resp. D
) can be given as a single integer,
in which case all the b_i
(resp. d_i
) above are taken to be equal to B
(resp. D
).
? M = [1,2;3,4]; ? matsolvemod(M, [3,4]~, [1,2]~) %2 = [-2, 0]~ ? matsolvemod(M, 3, 1) \\ M X = [1,1]~ over F_3 %3 = [-1, 1]~ ? matsolvemod(M, [3,0]~, [1,2]~) \\ x + 2y = 1 (mod 3), 3x + 4y = 2 (in Z) %4 = [6, -4]~
If flag = 1
, all solutions are returned in the form of a two-component row
vector [x,u]
, where x
is a small integer solution to the system of
congruences and u
is a matrix whose columns give a basis of the homogeneous
system (so that all solutions can be obtained by adding x
to any linear
combination of columns of u
). If no solution exists, returns zero.
The library syntax is GEN
matsolvemod0(GEN M, GEN D, GEN B, long flag)
.
Also available are GEN
gaussmodulo(GEN M, GEN D, GEN B)
(flag = 0
) and GEN
gaussmodulo2(GEN M, GEN D, GEN B)
(flag = 1
).
(x)
Assuming that the columns of the matrix x
are linearly independent (if they are not, an error message is issued), finds
a square invertible matrix whose first columns are the columns of x
,
i.e. supplement the columns of x
to a basis of the whole space.
? matsupplement([1;2]) %1 = [1 0]
[2 1]
Raises an error if x
has 0 columns, since (due to a long standing design
bug), the dimension of the ambient space (the number of rows) is unknown in
this case:
? matsupplement(matrix(2,0)) *** at top-level: matsupplement(matrix *** ^-------------------- *** matsupplement: sorry, suppl [empty matrix] is not yet implemented.
The library syntax is GEN
suppl(GEN x)
.
(x)
Transpose of x
(also x~
).
This has an effect only on vectors and matrices.
The library syntax is GEN
gtrans(GEN x)
.
(A,{v = 'x})
minimal polynomial
of A
with respect to the variable v
., i.e. the monic polynomial P
of minimal degree (in the variable v
) such that P(A) = 0
.
The library syntax is GEN
minpoly(GEN A, long v = -1)
where v
is a variable number.
(x)
Square of the L^2
-norm of x
. More precisely,
if x
is a scalar, norml2(x)
is defined to be the square
of the complex modulus of x
(real t_QUAD
s are not supported).
If x
is a polynomial, a (row or column) vector or a matrix, norml2(x)
is
defined recursively as sum_i norml2(x_i)
, where (x_i)
run through
the components of x
. In particular, this yields the usual sum |x_i|^2
(resp. sum |x_{i,j}|^2
) if x
is a polynomial or vector (resp. matrix) with
complex components.
? norml2( [ 1, 2, 3 ] ) \\ vector %1 = 14 ? norml2( [ 1, 2; 3, 4] ) \\ matrix %2 = 30 ? norml2( 2*I + x ) %3 = 5 ? norml2( [ [1,2], [3,4], 5, 6 ] ) \\ recursively defined %4 = 91
The library syntax is GEN
gnorml2(GEN x)
.
normlp(x,{p =
oo})
L^p
-norm of x
; sup norm if p
is omitted or +oo
. More precisely,
if x
is a scalar, normlp
(x, p)
is defined to be abs
(x)
.
If x
is a polynomial, a (row or column) vector or a matrix:
@3* if p
is omitted or +oo
, then normlp(x)
is defined
recursively as max _i normlp(x_i))
, where (x_i)
run through the
components of x
. In particular, this yields the usual sup norm if x
is a
polynomial or vector with complex components.
@3* otherwise, normlp(x, p)
is defined recursively as (
sum_i
normlp^p(x_i,p))^{1/p}
. In particular, this yields the usual (
sum
|x_i|^p)^{1/p}
if x
is a polynomial or vector with complex components.
? v = [1,-2,3]; normlp(v) \\ vector %1 = 3 ? normlp(v, +oo) \\ same, more explicit %2 = 3 ? M = [1,-2;-3,4]; normlp(M) \\ matrix %3 = 4 ? T = (1+I) + I*x^2; normlp(T) %4 = 1.4142135623730950488016887242096980786 ? normlp([[1,2], [3,4], 5, 6]) \\ recursively defined %5 = 6
? normlp(v, 1) %6 = 6 ? normlp(M, 1) %7 = 10 ? normlp(T, 1) %8 = 2.4142135623730950488016887242096980786
The library syntax is GEN
gnormlp(GEN x, GEN p = NULL, long prec)
.
qfauto(G,{
fl})
G
being a square and symmetric matrix with integer entries representing a
positive definite quadratic form, outputs the automorphism group of the
associate lattice.
Since this requires computing the minimal vectors, the computations can
become very lengthy as the dimension grows. G
can also be given by an
qfisominit
structure.
See qfisominit
for the meaning of fl.
The output is a two-components vector [o,g]
where o
is the group order
and g
is the list of generators (as a vector). For each generator H
,
the equality G = {^t}H G H
holds.
The interface of this function is experimental and will likely change in the future.
This function implements an algorithm of Plesken and Souvignier, following Souvignier's implementation.
The library syntax is GEN
qfauto0(GEN G, GEN fl = NULL)
.
The function GEN
qfauto(GEN G, GEN fl)
is also available
where G
is a vector of zm
matrices.
(
qfa,{
flag})
qfa being an automorphism group as output by
qfauto
, export the underlying matrix group as a string suitable
for (no flags or flag = 0
) GAP or (flag = 1
) Magma. The following example
computes the size of the matrix group using GAP:
? G = qfauto([2,1;1,2]) %1 = [12, [[-1, 0; 0, -1], [0, -1; 1, 1], [1, 1; 0, -1]]] ? s = qfautoexport(G) %2 = "Group([[-1, 0], [0, -1]], [[0, -1], [1, 1]], [[1, 1], [0, -1]])" ? extern("echo \"Order("s");\" | gap -q") %3 = 12
The library syntax is GEN
qfautoexport(GEN qfa, long flag)
.
(x,y,{q})
This function is obsolete, use qfeval
.
The library syntax is GEN
qfbil(GEN x, GEN y, GEN q = NULL)
.
({q},x,{y})
Evaluate the binary quadratic form q
(given by a symmetric matrix)
at the vector x
; if y
is present, evaluate the polar form at (x,y)
;
if q
omitted, use the standard Euclidean scalar product, corresponding to
the identity matrix.
Roughly equivalent to x~ * q * y
, but a little faster and
more convenient (does not distinguish between column and row vectors):
? x = [1,2,3]~; y = [-1,3,1]~; q = [1,2,3;2,2,-1;3,-1,9]; ? qfeval(q,x,y) %2 = 23 ? for(i=1,10^6, qfeval(q,x,y)) time = 661ms ? for(i=1,10^6, x~*q*y) time = 697ms
@3The speedup is noticeable for the quadratic form,
compared to x~ * q * x
, since we save almost half the
operations:
? for(i=1,10^6, qfeval(q,x)) time = 487ms
@3The special case q = Id
is handled faster if we
omit q
altogether:
? qfeval(,x,y) %1 = 2 ? q = matid(#x); ? for(i=1,10^6, qfeval(q,x,y)) time = 529 ms. ? for(i=1,10^6, qfeval(,x,y)) time = 228 ms. ? for(i=1,10^6, x~*y) time = 274 ms.
We also allow t_MAT
s of compatible dimensions for x
,
and return x~ * q * x
in this case as well:
? M = [1,2,3;4,5,6;7,8,9]; qfeval(,M) \\ Gram matrix %5 = [66 78 90]
[78 93 108]
[90 108 126]
? q = [1,2,3;2,2,-1;3,-1,9]; ? for(i=1,10^6, qfeval(q,M)) time = 2,008 ms. ? for(i=1,10^6, M~*q*M) time = 2,368 ms.
? for(i=1,10^6, qfeval(,M)) time = 1,053 ms. ? for(i=1,10^6, M~*M) time = 1,171 ms.
If q
is a t_QFI
or t_QFR
, it is implicitly converted to the
attached symmetric t_MAT
. This is done more
efficiently than by direct conversion, since we avoid introducing a
denominator 2
and rational arithmetic:
? q = Qfb(2,3,4); x = [2,3]; ? qfeval(q, x) %2 = 62 ? Q = Mat(q) %3 = [ 2 3/2]
[3/2 4] ? qfeval(Q, x) %4 = 62 ? for (i=1, 10^6, qfeval(q,x)) time = 758 ms. ? for (i=1, 10^6, qfeval(Q,x)) time = 1,110 ms.
Finally, when x
is a t_MAT
with integral coefficients, we allow
a t_QFI
or t_QFR
for q
and return the binary
quadratic form q o M
. Again, the conversion to t_MAT
is less
efficient in this case:
? q = Qfb(2,3,4); Q = Mat(q); x = [1,2;3,4]; ? qfeval(q, x) %2 = Qfb(47, 134, 96) ? qfeval(Q,x) %3 = [47 67]
[67 96] ? for (i=1, 10^6, qfeval(q,x)) time = 701 ms. ? for (i=1, 10^6, qfeval(Q,x)) time = 1,639 ms.
The library syntax is GEN
qfeval0(GEN q = NULL, GEN x, GEN y = NULL)
.
(q)
decomposition into squares of the
quadratic form represented by the symmetric matrix q
. The result is a
matrix whose diagonal entries are the coefficients of the squares, and the
off-diagonal entries on each line represent the bilinear forms. More
precisely, if (a_{ij})
denotes the output, one has
q(x) =
sum_i a_{ii} (x_i +
sum_{j != i} a_{ij} x_j)^2
? qfgaussred([0,1;1,0]) %1 = [1/2 1]
[-1 -1/2]
@3This means that 2xy = (1/2)(x+y)^2 - (1/2)(x-y)^2
.
Singular matrices are supported, in which case some diagonal coefficients
will vanish:
? qfgaussred([1,1;1,1]) %1 = [1 1]
[1 0]
@3This means that x^2 + 2xy + y^2 = (x+y)^2
.
The library syntax is GEN
qfgaussred(GEN q)
.
GEN
qfgaussred_positive(GEN q)
assumes that q
is
positive definite and is a little faster; returns NULL
if a vector
with negative norm occurs (non positive matrix or too many rounding errors).
qfisom(G,H,{
fl})
G
, H
being square and symmetric matrices with integer entries representing
positive definite quadratic forms, return an invertible matrix S
such that
G = {^t}S H S
. This defines a isomorphism between the corresponding lattices.
Since this requires computing the minimal vectors, the computations can
become very lengthy as the dimension grows.
See qfisominit
for the meaning of fl.
G
can also be given by an qfisominit
structure which is preferable if
several forms H
need to be compared to G
.
This function implements an algorithm of Plesken and Souvignier, following Souvignier's implementation.
The library syntax is GEN
qfisom0(GEN G, GEN H, GEN fl = NULL)
.
Also available is GEN
qfisom(GEN G, GEN H, GEN fl)
where G
is a vector of zm
, and H
is a zm
.
qfisominit(G,{
fl},{m})
G
being a square and symmetric matrix with integer entries representing a
positive definite quadratic form, return an isom
structure allowing to
compute isomorphisms between G
and other quadratic forms faster.
The interface of this function is experimental and will likely change in future release.
If present, the optional parameter fl must be a t_VEC
with two
components. It allows to specify the invariants used, which can make the
computation faster or slower. The components are
@3* fl[1]
Depth of scalar product combination to use.
@3* fl[2]
Maximum level of Bacher polynomials to use.
If present, m
must be the set of vectors of norm up to the maximal of the
diagonal entry of G
, either as a matrix or as given by qfminim
.
Otherwise this function computes the minimal vectors so it become very
lengthy as the dimension of G
grows.
The library syntax is GEN
qfisominit0(GEN G, GEN fl = NULL, GEN m = NULL)
.
Also available is
GEN
qfisominit(GEN F, GEN fl)
where F
is a vector of zm
.
(A)
Apply Jacobi's eigenvalue algorithm to the real symmetric matrix A
.
This returns [L, V]
, where
@3* L
is the vector of (real) eigenvalues of A
, sorted in increasing
order,
@3* V
is the corresponding orthogonal matrix of eigenvectors of A
.
? \p19 ? A = [1,2;2,1]; mateigen(A) %1 = [-1 1]
[ 1 1] ? [L, H] = qfjacobi(A); ? L %3 = [-1.000000000000000000, 3.000000000000000000]~ ? H %4 = [ 0.7071067811865475245 0.7071067811865475244]
[-0.7071067811865475244 0.7071067811865475245] ? norml2( (A-L[1])*H[,1] ) \\ approximate eigenvector %5 = 9.403954806578300064 E-38 ? norml2(H*H~ - 1) %6 = 2.350988701644575016 E-38 \\ close to orthogonal
The library syntax is GEN
jacobi(GEN A, long prec)
.
(x,{
flag = 0})
LLL algorithm applied to the
columns of the matrix x
. The columns of x
may be linearly
dependent. The result is a unimodular transformation matrix T
such that x
.T
is an LLL-reduced basis of the lattice generated by the column
vectors of x
. Note that if x
is not of maximal rank T
will not be
square. The LLL parameters are (0.51,0.99)
, meaning that the Gram-Schmidt
coefficients for the final basis satisfy mu_{i,j} <= |0.51|
, and the
Lovász's constant is 0.99
.
If flag = 0
(default), assume that x
has either exact (integral or
rational) or real floating point entries. The matrix is rescaled, converted
to integers and the behavior is then as in flag = 1
.
If flag = 1
, assume that x
is integral. Computations involving Gram-Schmidt
vectors are approximate, with precision varying as needed (Lehmer's trick,
as generalized by Schnorr). Adapted from Nguyen and Stehlé's algorithm
and Stehlé's code (fplll-1.3
).
If flag = 2
, x
should be an integer matrix whose columns are linearly
independent. Returns a partially reduced basis for x
, using an unpublished
algorithm by Peter Montgomery: a basis is said to be partially reduced
if |v_i +- v_j| >= |v_i|
for any two distinct basis vectors v_i,
v_j
.
This is faster than flag = 1
, esp. when one row is huge compared
to the other rows (knapsack-style), and should quickly produce relatively
short vectors. The resulting basis is not LLL-reduced in general.
If LLL reduction is eventually desired, avoid this partial reduction:
applying LLL to the partially reduced matrix is significantly slower
than starting from a knapsack-type lattice.
If flag = 4
, as flag = 1
, returning a vector [K, T]
of matrices: the
columns of K
represent a basis of the integer kernel of x
(not LLL-reduced in general) and T
is the transformation
matrix such that x.T
is an LLL-reduced Z-basis of the image
of the matrix x
.
If flag = 5
, case as case 4
, but x
may have polynomial coefficients.
If flag = 8
, same as case 0
, but x
may have polynomial coefficients.
The library syntax is GEN
qflll0(GEN x, long flag)
.
Also available are GEN
lll(GEN x)
(flag = 0
),
GEN
lllint(GEN x)
(flag = 1
), and GEN
lllkerim(GEN x)
(flag = 4
).
(G,{
flag = 0})
Same as qflll
, except that the
matrix G = x~ * x
is the Gram matrix of some lattice vectors x
,
and not the coordinates of the vectors themselves. In particular, G
must
now be a square symmetric real matrix, corresponding to a positive
quadratic form (not necessarily definite: x
needs not have maximal rank).
The result is a unimodular
transformation matrix T
such that x.T
is an LLL-reduced basis of
the lattice generated by the column vectors of x
. See qflll
for
further details about the LLL implementation.
If flag = 0
(default), assume that G
has either exact (integral or
rational) or real floating point entries. The matrix is rescaled, converted
to integers and the behavior is then as in flag = 1
.
If flag = 1
, assume that G
is integral. Computations involving Gram-Schmidt
vectors are approximate, with precision varying as needed (Lehmer's trick,
as generalized by Schnorr). Adapted from Nguyen and Stehlé's algorithm
and Stehlé's code (fplll-1.3
).
flag = 4
: G
has integer entries, gives the kernel and reduced image of x
.
flag = 5
: same as 4
, but G
may have polynomial coefficients.
The library syntax is GEN
qflllgram0(GEN G, long flag)
.
Also available are GEN
lllgram(GEN G)
(flag = 0
),
GEN
lllgramint(GEN G)
(flag = 1
), and GEN
lllgramkerim(GEN G)
(flag = 4
).
(x,{b},{m},{
flag = 0})
x
being a square and symmetric matrix representing a positive definite
quadratic form, this function deals with the vectors of x
whose norm is
less than or equal to b
, enumerated using the Fincke-Pohst algorithm,
storing at most m
vectors (no limit if m
is omitted). The function
searches for the minimal non-zero vectors if b
is omitted. The behavior is
undefined if x
is not positive definite (a ``precision too low'' error is
most likely, although more precise error messages are possible). The precise
behavior depends on flag.
If flag = 0
(default), returns at most 2m
vectors. The result is a
three-component vector, the first component being the number of vectors
enumerated (which may be larger than 2m
), the second being the maximum
norm found, and the last vector
is a matrix whose columns are found vectors, only one being given for each
pair +- v
(at most m
such pairs, unless m
was omitted). The vectors
are returned in no particular order.
If flag = 1
, ignores m
and returns [N,v]
, where v
is a non-zero vector
of length N <= b
, or []
if no non-zero vector has length <= b
.
If no explicit b
is provided, return a vector of smallish norm
(smallest vector in an LLL-reduced basis).
In these two cases, x
must have integral entries. The
implementation uses low precision floating point computations for maximal
speed, which gives incorrect result when x
has large entries. (The
condition is checked in the code and the routine raises an error if
large rounding errors occur.) A more robust, but much slower,
implementation is chosen if the following flag is used:
If flag = 2
, x
can have non integral real entries. In this case, if b
is omitted, the ``minimal'' vectors only have approximately the same norm.
If b
is omitted, m
is an upper bound for the number of vectors that
will be stored and returned, but all minimal vectors are nevertheless
enumerated. If m
is omitted, all vectors found are stored and returned;
note that this may be a huge vector!
? x = matid(2); ? qfminim(x) \\ 4 minimal vectors of norm 1: F<+->[0,1], F<+->[1,0] %2 = [4, 1, [0, 1; 1, 0]] ? { x = [4, 2, 0, 0, 0,-2, 0, 0, 0, 0, 0, 0, 1,-1, 0, 0, 0, 1, 0,-1, 0, 0, 0,-2; 2, 4,-2,-2, 0,-2, 0, 0, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0,-1, 0, 1,-1,-1; 0,-2, 4, 0,-2, 0, 0, 0, 0, 0, 0, 0,-1, 1, 0, 0, 1, 0, 0, 1,-1,-1, 0, 0; 0,-2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1,-1, 0, 0, 0, 1,-1, 0, 1,-1, 1, 0; 0, 0,-2, 0, 4, 0, 0, 0, 1,-1, 0, 0, 1, 0, 0, 0,-2, 0, 0,-1, 1, 1, 0, 0; -2, -2,0, 0, 0, 4,-2, 0,-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0,-1, 1, 1; 0, 0, 0, 0, 0,-2, 4,-2, 0, 0, 0, 0, 0, 1, 0, 0, 0,-1, 0, 0, 0, 1,-1, 0; 0, 0, 0, 0, 0, 0,-2, 4, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0,-1,-1,-1, 0, 1, 0; 0, 0, 0, 0, 1,-1, 0, 0, 4, 0,-2, 0, 1, 1, 0,-1, 0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 0, 0,-1, 0, 0, 0, 0, 4, 0, 0, 1, 1,-1, 1, 0, 0, 0, 1, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 0,-2, 0, 4,-2, 0,-1, 0, 0, 0,-1, 0,-1, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-2, 4,-1, 1, 0, 0,-1, 1, 0, 1, 1, 1,-1, 0; 1, 0,-1, 1, 1, 0, 0,-1, 1, 1, 0,-1, 4, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1,-1; -1,-1, 1,-1, 0, 0, 1, 0, 1, 1,-1, 1, 0, 4, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1; 0, 0, 0, 0, 0, 0, 0, 0, 0,-1, 0, 0, 0, 1, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 0, 0, 0, 0, 0,-1, 1, 0, 0, 1, 1, 0, 4, 0, 0, 0, 0, 1, 1, 0, 0; 0, 0, 1, 0,-2, 0, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 4, 1, 1, 1, 0, 0, 1, 1; 1, 0, 0, 1, 0, 0,-1, 0, 1, 0,-1, 1, 1, 0, 0, 0, 1, 4, 0, 1, 1, 0, 1, 0; 0, 0, 0,-1, 0, 1, 0,-1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 4, 0, 1, 1, 0, 1; -1, -1,1, 0,-1, 1, 0,-1, 0, 1,-1, 1, 0, 1, 0, 0, 1, 1, 0, 4, 0, 0, 1, 1; 0, 0,-1, 1, 1, 0, 0,-1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 4, 1, 0, 1; 0, 1,-1,-1, 1,-1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 4, 0, 1; 0,-1, 0, 1, 0, 1,-1, 1, 0, 1, 0,-1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 4, 1; -2,-1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,-1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 4]; } ? qfminim(x,,0) \\ the Leech lattice has 196560 minimal vectors of norm 4 time = 648 ms. %4 = [196560, 4, [;]] ? qfminim(x,,0,2); \\ safe algorithm. Slower and unnecessary here. time = 18,161 ms. %5 = [196560, 4.000061035156250000, [;]]
@3
In the last example, we store 0 vectors to limit memory use. All minimal
vectors are nevertheless enumerated. Provided parisize
is about 50MB,
qfminim(x)
succeeds in 2.5 seconds.
The library syntax is GEN
qfminim0(GEN x, GEN b = NULL, GEN m = NULL, long flag, long prec)
.
Also available are
GEN
minim(GEN x, GEN b = NULL, GEN m = NULL)
(flag = 0
),
GEN
minim2(GEN x, GEN b = NULL, GEN m = NULL)
(flag = 1
).
GEN
minim_raw(GEN x, GEN b = NULL, GEN m = NULL)
(do not perform LLL
reduction on x and return NULL
on accuracy error).
(x,{q})
This function is obsolete, use qfeval
.
The library syntax is GEN
qfnorm(GEN x, GEN q = NULL)
.
(G,V)
Return the orbits of V
under the action of the group
of linear transformation generated by the set G
.
It is assumed that G
contains minus identity, and only one vector
in {v, -v}
should be given.
If G
does not stabilize V
, the function return 0
.
In the example below, we compute representatives and lengths of the orbits of
the vectors of norm <= 3
under the automorphisms of the lattice A_1^6
.
? Q=matid(6); G=qfauto(Q); V=qfminim(Q,3); ? apply(x->[x[1],#x],qforbits(G,V)) %2 = [[[0,0,0,0,0,1]~,6],[[0,0,0,0,1,-1]~,30],[[0,0,0,1,-1,-1]~,80]]
The library syntax is GEN
qforbits(GEN G, GEN V)
.
(G,
sol, {
flag = 0})
Coefficients of binary quadratic forms that parametrize the
solutions of the ternary quadratic form G
, using the particular
solution sol.
flag is optional and can be 1, 2, or 3, in which case the flag-th form is
reduced. The default is flag = 0 (no reduction).
? G = [1,0,0;0,1,0;0,0,-34]; ? M = qfparam(G, qfsolve(G)) %2 = [ 3 -10 -3]
[-5 -6 5]
[ 1 0 1]
Indeed, the solutions can be parametrized as
(3x^2 - 10xy - 3y^2)^2 + (-5x^2 - 6xy + 5y^2)^2 -34(x^2 + y^2)^2 = 0.
? v = y^2 * M*[1,x/y,(x/y)^2]~ %3 = [3*x^2 - 10*y*x - 3*y^2, -5*x^2 - 6*y*x + 5*y^2, -x^2 - y^2]~ ? v~*G*v %4 = 0
The library syntax is GEN
qfparam(GEN G, GEN sol, long flag)
.
(G)
G
being a square and symmetric matrix with
integer entries representing a positive definite quadratic form, outputs the
perfection rank of the form. That is, gives the rank of the family of the s
symmetric matrices v_iv_i^t
, where s
is half the number of minimal
vectors and the v_i
(1 <= i <= s
) are the minimal vectors.
Since this requires computing the minimal vectors, the computations can
become very lengthy as the dimension of x
grows.
The library syntax is GEN
perf(GEN G)
.
(q,B,{
flag = 0})
q
being a square and symmetric matrix with integer entries representing a
positive definite quadratic form, count the vectors representing successive
integers.
@3* If flag = 0
, count all vectors. Outputs the vector whose i
-th
entry, 1 <= i <= B
is half the number of vectors v
such that q(v) = i
.
@3* If flag = 1
, count vectors of even norm. Outputs the vector
whose i
-th entry, 1 <= i <= B
is half the number of vectors such
that q(v) = 2i
.
? q = [2, 1; 1, 3]; ? qfrep(q, 5) %2 = Vecsmall([0, 1, 2, 0, 0]) \\ 1 vector of norm 2, 2 of norm 3, etc. ? qfrep(q, 5, 1) %3 = Vecsmall([1, 0, 0, 1, 0]) \\ 1 vector of norm 2, 0 of norm 4, etc.
This routine uses a naive algorithm based on qfminim
, and
will fail if any entry becomes larger than 2^{31}
(or 2^{63}
).
The library syntax is GEN
qfrep0(GEN q, GEN B, long flag)
.
(x)
Returns [p,m]
the signature of the quadratic form represented by the
symmetric matrix x
. Namely, p
(resp. m
) is the number of positive
(resp. negative) eigenvalues of x
. The result is computed using Gaussian
reduction.
The library syntax is GEN
qfsign(GEN x)
.
(G)
Given a square symmetric matrix G
of dimension n >= 1
, solve over
Q the quadratic equation X^tGX = 0
. The matrix G
must have rational
coefficients. The solution might be a single non-zero vector (vectorv) or a
matrix (whose columns generate a totally isotropic subspace).
If no solution exists, returns an integer, that can be a prime p
such that
there is no local solution at p
, or -1
if there is no real solution,
or -2
if n = 2
and -
det G
is positive but not a square (which implies
there is a real solution, but no local solution at some p
dividing det G
).
? G = [1,0,0;0,1,0;0,0,-34]; ? qfsolve(G) %1 = [-3, -5, 1]~ ? qfsolve([1,0; 0,2]) %2 = -1 \\ no real solution ? qfsolve([1,0,0;0,3,0; 0,0,-2]) %3 = 3 \\ no solution in Q_3 ? qfsolve([1,0; 0,-2]) %4 = -2 \\ no solution, n = 2
The library syntax is GEN
qfsolve(GEN G)
.
(s,p,r)
finds a linear relation between powers (1,s,
..., s^p)
of the series s
, with polynomial coefficients of degree
<= r
. In case no relation is found, return 0
.
? s = 1 + 10*y - 46*y^2 + 460*y^3 - 5658*y^4 + 77740*y^5 + O(y^6); ? seralgdep(s, 2, 2) %2 = -x^2 + (8*y^2 + 20*y + 1) ? subst(%, x, s) %3 = O(y^6) ? seralgdep(s, 1, 3) %4 = (-77*y^2 - 20*y - 1)*x + (310*y^3 + 231*y^2 + 30*y + 1) ? seralgdep(s, 1, 2) %5 = 0
@3The series main variable must not be x
, so as to be able
to express the result as a polynomial in x
.
The library syntax is GEN
seralgdep(GEN s, long p, long r)
.
(f,X,{Y})
The set whose elements are the f(x,y), where x,y run through X,Y.
respectively. If Y
is omitted, assume that X = Y
and that f
is symmetric:
f(x,y) = f(y,x)
for all x,y
in X
.
? X = [1,2,3]; Y = [2,3,4]; ? setbinop((x,y)->x+y, X,Y) \\ set X + Y %2 = [3, 4, 5, 6, 7] ? setbinop((x,y)->x-y, X,Y) \\ set X - Y %3 = [-3, -2, -1, 0, 1] ? setbinop((x,y)->x+y, X) \\ set 2X = X + X %2 = [2, 3, 4, 5, 6]
The library syntax is GEN
setbinop(GEN f, GEN X, GEN Y = NULL)
.
(x,y)
Intersection of the two sets x
and y
(see setisset
).
If x
or y
is not a set, the result is undefined.
The library syntax is GEN
setintersect(GEN x, GEN y)
.
(x)
Returns true (1) if x
is a set, false (0) if
not. In PARI, a set is a row vector whose entries are strictly
increasing with respect to a (somewhat arbitrary) universal comparison
function. To convert any object into a set (this is most useful for
vectors, of course), use the function Set
.
? a = [3, 1, 1, 2]; ? setisset(a) %2 = 0 ? Set(a) %3 = [1, 2, 3]
The library syntax is long
setisset(GEN x)
.
(x,y)
Difference of the two sets x
and y
(see setisset
),
i.e. set of elements of x
which do not belong to y
.
If x
or y
is not a set, the result is undefined.
The library syntax is GEN
setminus(GEN x, GEN y)
.
(S,x,{
flag = 0})
Determines whether x
belongs to the set S
(see setisset
).
We first describe the default behaviour, when flag is zero or omitted. If x
belongs to the set S
, returns the index j
such that S[j] = x
, otherwise
returns 0.
? T = [7,2,3,5]; S = Set(T); ? setsearch(S, 2) %2 = 1 ? setsearch(S, 4) \\ not found %3 = 0 ? setsearch(T, 7) \\ search in a randomly sorted vector %4 = 0 \\ WRONG !
If S
is not a set, we also allow sorted lists with
respect to the cmp
sorting function, without repeated entries,
as per listsort
(L,1)
; otherwise the result is undefined.
? L = List([1,4,2,3,2]); setsearch(L, 4) %1 = 0 \\ WRONG ! ? listsort(L, 1); L \\ sort L first %2 = List([1, 2, 3, 4]) ? setsearch(L, 4) %3 = 4 \\ now correct
If flag is non-zero, this function returns the index j
where x
should be
inserted, and 0
if it already belongs to S
. This is meant to be used for
dynamically growing (sorted) lists, in conjunction with listinsert
.
? L = List([1,5,2,3,2]); listsort(L,1); L %1 = List([1,2,3,5]) ? j = setsearch(L, 4, 1) \\ 4 should have been inserted at index j %2 = 4 ? listinsert(L, 4, j); L %3 = List([1, 2, 3, 4, 5])
The library syntax is long
setsearch(GEN S, GEN x, long flag)
.
(x,y)
Union of the two sets x
and y
(see setisset
).
If x
or y
is not a set, the result is undefined.
The library syntax is GEN
setunion(GEN x, GEN y)
.
(x)
This applies to quite general x
. If x
is not a
matrix, it is equal to the sum of x
and its conjugate, except for polmods
where it is the trace as an algebraic number.
For x
a square matrix, it is the ordinary trace. If x
is a
non-square matrix (but not a vector), an error occurs.
The library syntax is GEN
gtrace(GEN x)
.
(x,y,{z})
Extraction of components of the vector or matrix x
according to y
.
In case x
is a matrix, its components are the columns of x
. The
parameter y
is a component specifier, which is either an integer, a string
describing a range, or a vector.
If y
is an integer, it is considered as a mask: the binary bits of y
are
read from right to left, but correspond to taking the components from left to
right. For example, if y = 13 = (1101)_2
then the components 1,3 and 4 are
extracted.
If y
is a vector (t_VEC
, t_COL
or t_VECSMALL
), which must have
integer entries, these entries correspond to the component numbers to be
extracted, in the order specified.
If y
is a string, it can be
@3* a single (non-zero) index giving a component number (a negative index means we start counting from the end).
@3* a range of the form "a..b"
, where a
and b
are
indexes as above. Any of a
and b
can be omitted; in this case, we take
as default values a = 1
and b = -1
, i.e. the first and last components
respectively. We then extract all components in the interval [a,b]
, in
reverse order if b < a
.
In addition, if the first character in the string is ^
, the
complement of the given set of indices is taken.
If z
is not omitted, x
must be a matrix. y
is then the row
specifier, and z
the column specifier, where the component specifier
is as explained above.
? v = [a, b, c, d, e]; ? vecextract(v, 5) \\ mask %1 = [a, c] ? vecextract(v, [4, 2, 1]) \\ component list %2 = [d, b, a] ? vecextract(v, "2..4") \\ interval %3 = [b, c, d] ? vecextract(v, "-1..-3") \\ interval + reverse order %4 = [e, d, c] ? vecextract(v, "^2") \\ complement %5 = [a, c, d, e] ? vecextract(matid(3), "2..", "..") %6 = [0 1 0]
[0 0 1]
The range notations v[i..j]
and v[^i]
(for t_VEC
or
t_COL
) and M[i..j, k..l]
and friends (for t_MAT
) implement a
subset of the above, in a simpler and faster way, hence should be
preferred in most common situations. The following features are not
implemented in the range notation:
@3* reverse order,
@3* omitting either a
or b
in a..b
.
The library syntax is GEN
extract0(GEN x, GEN y, GEN z = NULL)
.
vecsearch(v,x,{
cmpf})
Determines whether x
belongs to the sorted vector or list v
: return
the (positive) index where x
was found, or 0
if it does not belong to
v
.
If the comparison function cmpf is omitted, we assume that v
is sorted in
increasing order, according to the standard comparison function lex
,
thereby restricting the possible types for x
and the elements of v
(integers, fractions, reals, and vectors of such).
If cmpf
is present, it is understood as a comparison function and we
assume that v
is sorted according to it, see vecsort
for how to
encode comparison functions.
? v = [1,3,4,5,7]; ? vecsearch(v, 3) %2 = 2 ? vecsearch(v, 6) %3 = 0 \\ not in the list ? vecsearch([7,6,5], 5) \\ unsorted vector: result undefined %4 = 0
By abuse of notation, x
is also allowed to be a matrix, seen as a vector
of its columns; again by abuse of notation, a t_VEC
is considered
as part of the matrix, if its transpose is one of the matrix columns.
? v = vecsort([3,0,2; 1,0,2]) \\ sort matrix columns according to lex order %1 = [0 2 3]
[0 2 1] ? vecsearch(v, [3,1]~) %2 = 3 ? vecsearch(v, [3,1]) \\ can search for x or x~ %3 = 3 ? vecsearch(v, [1,2]) %4 = 0 \\ not in the list
@3
The library syntax is long
vecsearch(GEN v, GEN x, GEN cmpf = NULL)
.
vecsort(x,{
cmpf},{
flag = 0})
Sorts the vector x
in ascending order, using a mergesort method.
x
must be a list, vector or matrix (seen as a vector of its columns).
Note that mergesort is stable, hence the initial ordering of ``equal''
entries (with respect to the sorting criterion) is not changed.
If cmpf
is omitted, we use the standard comparison function
lex
, thereby restricting the possible types for the elements of x
(integers, fractions or reals and vectors of those). If cmpf
is
present, it is understood as a comparison function and we sort according to
it. The following possibilities exist:
@3* an integer k
: sort according to the value of the k
-th
subcomponents of the components of x
.
@3* a vector: sort lexicographically according to the components listed in
the vector. For example, if cmpf = [2,1,3]
, sort with respect to
the second component, and when these are equal, with respect to the first,
and when these are equal, with respect to the third.
@3* a comparison function (t_CLOSURE
), with two arguments x
and y
,
and returning an integer which is < 0
, > 0
or = 0
if x < y
, x > y
or
x = y
respectively. The sign
function is very useful in this context:
? vecsort([3,0,2; 1,0,2]) \\ sort columns according to lex order %1 = [0 2 3]
[0 2 1] ? vecsort(v, (x,y)->sign(y-x)) \\ reverse sort ? vecsort(v, (x,y)->sign(abs(x)-abs(y))) \\ sort by increasing absolute value ? cmpf(x,y) = my(dx = poldisc(x), dy = poldisc(y)); sign(abs(dx) - abs(dy)) ? vecsort([x^2+1, x^3-2, x^4+5*x+1], cmpf)
@3
The last example used the named cmpf
instead of an anonymous function,
and sorts polynomials with respect to the absolute value of their
discriminant. A more efficient approach would use precomputations to ensure
a given discriminant is computed only once:
? DISC = vector(#v, i, abs(poldisc(v[i]))); ? perm = vecsort(vector(#v,i,i), (x,y)->sign(DISC[x]-DISC[y])) ? vecextract(v, perm)
@3Similar ideas apply whenever we sort according to the values of a function which is expensive to compute.
@3The binary digits of flag mean:
@3* 1: indirect sorting of the vector x
, i.e. if x
is an
n
-component vector, returns a permutation of [1,2,...,n]
which
applied to the components of x
sorts x
in increasing order.
For example, vecextract(x, vecsort(x,,1))
is equivalent to
vecsort(x)
.
@3* 4: use descending instead of ascending order.
@3* 8: remove ``duplicate'' entries with respect to the sorting function (keep the first occurring entry). For example:
? vecsort([Pi,Mod(1,2),z], (x,y)->0, 8) \\ make everything compare equal %1 = [3.141592653589793238462643383] ? vecsort([[2,3],[0,1],[0,3]], 2, 8) %2 = [[0, 1], [2, 3]]
The library syntax is GEN
vecsort0(GEN x, GEN cmpf = NULL, long flag)
.
(v)
Return the sum of the components of the vector v
. Return 0
on an
empty vector.
? vecsum([1,2,3]) %1 = 6 ? vecsum([]) %2 = 0
The library syntax is GEN
vecsum(GEN v)
.
vector(n,{X},{
expr = 0})
Creates a row vector (type
t_VEC
) with n
components whose components are the expression
expr evaluated at the integer points between 1 and n
. If one of the
last two arguments is omitted, fill the vector with zeroes.
? vector(3,i, 5*i) %1 = [5, 10, 15] ? vector(3) %2 = [0, 0, 0]
The variable X
is lexically scoped to each evaluation of expr. Any
change to X
within expr does not affect subsequent evaluations, it
still runs 1 to n
. A local change allows for example different indexing:
vector(10, i, i=i-1; f(i)) \\ i = 0, ..., 9 vector(10, i, i=2*i; f(i)) \\ i = 2, 4, ..., 20
This per-element scope for X
differs from for
loop evaluations,
as the following example shows:
n = 3 v = vector(n); vector(n, i, i++) ----> [2, 3, 4] v = vector(n); for (i = 1, n, v[i] = i++) ----> [2, 0, 4]
vectorsmall(n,{X},{
expr = 0})
Creates a row vector of small integers (type
t_VECSMALL
) with n
components whose components are the expression
expr evaluated at the integer points between 1 and n
. If one of the
last two arguments is omitted, fill the vector with zeroes.
vectorv(n,{X},{
expr = 0})
As vector
, but returns a column vector (type t_COL
).
Although the gp
calculator is programmable, it is useful to have
a number of preprogrammed loops, including sums, products, and a certain
number of recursions. Also, a number of functions from numerical analysis
like numerical integration and summation of series will be described here.
One of the parameters in these loops must be the control variable, hence a
simple variable name. In the descriptions, the letter X
will always denote
any simple variable name, and represents the formal parameter used in the
function. The expression to be summed, integrated, etc. is any legal PARI
expression, including of course expressions using loops.
@3Library mode. Since it is easier to program directly the loops in library mode, these functions are mainly useful for GP programming. On the other hand, numerical routines code a function (to be integrated, summed, etc.) with two parameters named
GEN (*eval)(void*,GEN) void *E; \\ context: eval(E, x) must evaluate your function at x.
see the Libpari manual for details.
@3Numerical integration.
Starting with version 2.2.9 the ``double exponential'' univariate
integration method is implemented in intnum
and its variants. Romberg
integration is still available under the name intnumromb
, but
superseded. It is possible to compute numerically integrals to thousands of
decimal places in reasonable time, as long as the integrand is regular. It is
also reasonable to compute numerically integrals in several variables,
although more than two becomes lengthy. The integration domain may be
non-compact, and the integrand may have reasonable singularities at
endpoints. To use intnum
, you must split the integral into a sum
of subintegrals where the function has no singularities except at the
endpoints. Polynomials in logarithms are not considered singular, and
neglecting these logs, singularities are assumed to be algebraic (asymptotic
to C(x-a)^{-
alpha}
for some alpha > -1
when x
is
close to a
), or to correspond to simple discontinuities of some (higher)
derivative of the function. For instance, the point 0
is a singularity of
abs(x)
.
See also the discrete summation methods below, sharing the prefix sum
.
asympnum(
expr,{k = 20},{
alpha = 1})
Asymptotic expansion of expr, corresponding to a sequence u(n)
,
assuming it has the shape
u(n) ~
sum_{i >= 0} a_i n^{-i
alpha}
with rational coefficients a_i
with reasonable height; the algorithm
is heuristic and performs repeated calls to limitnum, with
k
and alpha
are as in limitnum
? f(n) = n! / (n^n*exp(-n)*sqrt(n)); ? asympnum(f) %2 = [] \\ failure ! ? l = limitnum(f) %3 = 2.5066282746310005024157652848110452530 ? asympnum(n->f(n)/l) \\ normalize %4 = [1, 1/12, 1/288, -139/51840]
@3and we indeed get a few terms of Stirling's expansion. Note that it helps to normalize with a limit computed to higher accuracy:
? \p100 ? L = limitnum(f) ? \p38 ? asympnum(n->f(n)/L) \\ we get more terms! %6 = [1, 1/12, 1/288, -139/51840, -571/2488320, 163879/209018880,\ 5246819/75246796800, -534703531/902961561600]
@3If alpha
is not an integer, loss of accuracy is
expected, so it should be precomputed to double accuracy, say:
? \p38 ? asympnum(n->-log(1-1/n^Pi),,Pi) %1 = [0, 1, 1/2, 1/3] ? asympnum(n->-log(1-1/sqrt(n)),,1/2) %2 = [0, 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, \ 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20, 1/21, 1/22]
? localprec(100); a = Pi; ? asympnum(n->-log(1-1/n^a),,a) \\ better ! %4 = [0, 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12]
The library syntax is asympnum(void *E, GEN (*u)(void *,GEN,long), long muli, GEN alpha, long prec)
, where u(E, n, prec)
must return u(n)
in precision prec
.
Also available is
GEN
asympnum0(GEN u, long muli, GEN alpha, long prec)
, where u
must be a vector of sufficient length as above.
contfraceval(
CF,t,{
lim = -1})
Given a continued fraction CF
output by contfracinit
, evaluate
the first lim
terms of the continued fraction at t
(all
terms if lim
is negative or omitted; if positive, lim
must be
less than or equal to the length of CF
.
The library syntax is GEN
contfraceval(GEN CF, GEN t, long lim)
.
contfracinit(M,{
lim = -1})
Given M
representing the power series S =
sum_{n >= 0} M[n+1]z^n
,
transform it into a continued fraction; restrict to n <= lim
if latter is non-negative. M
can be a vector, a power
series, a polynomial, or a rational function.
The result is a 2-component vector [A,B]
such that
S = M[1] / (1+A[1]z+B[1]z^2/(1+A[2]z+B[2]z^2/(1+...1/(1+A[lim/2]z))))
.
Does not work if any coefficient of M
vanishes, nor for series for
which certain partial denominators vanish.
The library syntax is GEN
contfracinit(GEN M, long lim)
.
(X = a,
expr)
Numerical derivation of expr with respect to X
at X = a
.
? derivnum(x=0,sin(exp(x))) - cos(1) %1 = -1.262177448 E-29
A clumsier approach, which would not work in library mode, is
? f(x) = sin(exp(x)) ? f'(0) - cos(1) %1 = -1.262177448 E-29
When a
is a power series, compute derivnum(t = a,f)
as f'(a) =
(f(a))'/a'
.
The library syntax is derivnum(void *E, GEN (*eval)(void*,GEN), GEN a, long prec)
. Also
available is GEN
derivfun(void *E, GEN (*eval)(void *, GEN), GEN a, long prec)
, which also allows power series for a
.
intcirc(X = a,R,
expr,{
tab})
Numerical
integration of (2i
pi)^{-1}
expr with respect to X
on the circle
|X-a |= R
.
In other words, when expr is a meromorphic
function, sum of the residues in the corresponding disk; tab is as in
intnum
, except that if computed with intnuminit
it should be with
the endpoints [-1, 1]
.
? \p105 ? intcirc(s=1, 0.5, zeta(s)) - 1 time = 496 ms. %1 = 1.2883911040127271720 E-101 + 0.E-118*I
The library syntax is intcirc(void *E, GEN (*eval)(void*,GEN), GEN a,GEN R,GEN tab, long prec)
.
(t = a,b,f,{m = 0})
Initialize tables for use with integral transforms such Fourier, Laplace or Mellin transforms, in order to compute
int_a^b f(t) k(t,z) dt
for some kernel k(t,z)
.
The endpoints a
and b
are coded as in intnum
, f
is the
function to which the integral transform is to be applied and the
non-negative integer m
is as in intnum
: multiply the number of
sampling points roughly by 2^m
, hopefully increasing the accuracy. This
function is particularly useful when the function f
is hard to compute,
such as a gamma product.
@3Limitation. the endpoints a
and b
must be at infinity,
with the same asymptotic behaviour. Oscillating types are not supported.
This is easily overcome by integrating vectors of functions, see example
below.
@3Examples.
@3* numerical Fourier transform
F(z) =
int_{- oo }^{+ oo } f(t)e^{-2i
pi z t} dt.
First the easy case, assume that f
decrease exponentially:
f(t) = exp(-t^2); A = [-oo,1]; B = [+oo,1]; \p200 T = intfuncinit(t = A,B , f(t)); F(z) = { my(a = -2*I*Pi*z); intnum(t = A,B, exp(a*t), T); } ? F(1) - sqrt(Pi)*exp(-Pi^2) %1 = -1.3... E-212
Now the harder case, f
decrease slowly: we must specify the oscillating
behaviour. Thus, we cannot precompute usefully since everything depends on
the point we evaluate at:
f(t) = 1 / (1+ abs(t)); \p200 \\ Fourier cosine transform FC(z) = { my(a = 2*Pi*z); intnum(t = [-oo, a*I], [+oo, a*I], cos(a*t)*f(t)); } FC(1)
@3* Fourier coefficients: we must integrate over a period, but
intfuncinit
does not support finite endpoints.
The solution is to integrate a vector of functions !
FourierSin(f, T, k) = \\ first k sine Fourier coeffs { my (w = 2*Pi/T); my (v = vector(k+1)); intnum(t = -T/2, T/2, my (z = exp(I*w*t)); v[1] = z; for (j = 2, k, v[j] = v[j-1]*z); f(t) * imag(v)) * 2/T; } FourierSin(t->sin(2*t), 2*Pi, 10)
@3The same technique can be used instead of intfuncinit
to integrate f(t) k(t,z)
whenever the list of z
-values is known
beforehand.
Note that the above code includes an unrelated optimization: the
sin (j w t)
are computed as imaginary parts of exp (i j w t)
and the
latter by successive multiplications.
@3* numerical Mellin inversion
F(z) = (2i
pi)^{-1}
int_{c -i oo }^{c+i oo } f(s)z^{-s} ds
= (2
pi)^{-1}
int_{- oo }^{+ oo }
f(c + i t)e^{-
log z(c + it)} dt.
We take c = 2
in the program below:
f(s) = gamma(s)^3; \\ f(c+it) decrease as exp(-3Pi|t|/2) c = 2; \\ arbitrary A = [-oo,3*Pi/2]; B = [+oo,3*Pi/2]; T = intfuncinit(t=A,B, f(c + I*t)); F(z) = { my (a = -log(z)); intnum(t=A,B, exp(a*I*t), T)*exp(a*c) / (2*Pi); }
The library syntax is intfuncinit(void *E, GEN (*eval)(void*,GEN), GEN a,GEN b,long m, long prec)
.
intnum(X = a,b,
expr,{
tab})
Numerical integration
of expr on ]a,b[
with respect to X
, using the
double-exponential method, and thus O(D
log D)
evaluation of
the integrand in precision D
. The integrand may have values
belonging to a vector space over the real numbers; in particular, it can be
complex-valued or vector-valued. But it is assumed that the function is
regular on ]a,b[
. If the endpoints a
and b
are finite and the
function is regular there, the situation is simple:
? intnum(x = 0,1, x^2) %1 = 0.3333333333333333333333333333 ? intnum(x = 0,Pi/2, [cos(x), sin(x)]) %2 = [1.000000000000000000000000000, 1.000000000000000000000000000]
An endpoint equal to +- oo
is coded as +oo
or -oo
, as
expected:
? intnum(x = 1,+oo, 1/x^2) %3 = 1.000000000000000000000000000
In basic usage, it is assumed that the function does not decrease exponentially fast at infinity:
? intnum(x=0,+oo, exp(-x)) *** at top-level: intnum(x=0,+oo,exp(- *** ^-------------------- *** exp: overflow in expo().
We shall see in a moment how to avoid that last problem, after describing the last optional argument tab.
@3The tab. argument
The routine uses weights w_i
, which are mostly independent of the function
being integrated, evaluated at many sampling points x_i
and
approximates the integral by sum w_i f(x_i)
. If tab is
@3* a non-negative integer m
, we multiply the number of sampling points
by 2^m
, hopefully increasing accuracy. Note that the running time
increases roughly by a factor 2^m
. One may try consecutive values of m
until they give the same value up to an accepted error.
@3* a set of integration tables containing precomputed x_i
and w_i
as output by intnuminit
. This is useful if several integrations of
the same type are performed (on the same kind of interval and functions,
for a given accuracy): we skip a precomputation of O(D
log D)
elementary functions in accuracy D
, whose running time has the same order
of magnitude as the evaluation of the integrand. This is in particular
usefule for multivariate integrals.
@3Specifying the behavior at endpoints.
This is done as follows. An endpoint a
is either given as such (a scalar,
real or complex, oo
or -oo
for +- oo
), or as a two
component vector [a,
alpha]
, to indicate the behavior of the integrand in a
neighborhood of a
.
If a
is finite, the code [a,
alpha]
means the function has a
singularity of the form (x-a)^{
alpha}
, up to logarithms. (If alpha >=
0
, we only assume the function is regular, which is the default assumption.)
If a wrong singularity exponent is used, the result will lose a catastrophic
number of decimals:
? intnum(x=0, 1, x^(-1/2)) \\ assume x^{-1/2} is regular at 0 %1 = 1.9999999999999999999999999999827931660 ? intnum(x=[0,-1/2], 1, x^(-1/2)) \\ no, it's not %2 = 2.0000000000000000000000000000000000000 ? intnum(x=[0,-1/10], 1, x^(-1/2)) \\ using a wrong exponent is bad %3 = 1.9999999999999999999999999999999901912
If a
is +- oo
, which is coded as +oo
or -oo
,
the situation is more complicated, and [+-oo,
alpha]
means:
@3* alpha = 0
(or no alpha at all, i.e. simply +-oo
)
assumes that the integrand tends to zero moderately quickly, at least as
O(x^{-2})
but not exponentially fast.
@3* alpha > 0
assumes that the function tends to zero exponentially fast
approximately as exp (-
alpha x)
. This includes oscillating but quickly
decreasing functions such as exp (-x)
sin (x)
.
? intnum(x=0, +oo, exp(-2*x)) *** at top-level: intnum(x=0,+oo,exp(- *** ^-------------------- *** exp: exponent (expo) overflow ? intnum(x=0, [+oo, 2], exp(-2*x)) \\ OK! %1 = 0.50000000000000000000000000000000000000 ? intnum(x=0, [+oo, 3], exp(-2*x)) \\ imprecise exponent, still OK ! %2 = 0.50000000000000000000000000000000000000 ? intnum(x=0, [+oo, 10], exp(-2*x)) \\ wrong exponent ==E<gt> disaster %3 = 0.49999999999952372962457451698256707393
@3As the last exemple shows, the exponential decrease rate must be indicated to avoid overflow, but the method is robust enough for a rough guess to be acceptable.
@3* alpha < -1
assumes that the function tends to 0
slowly, like
x^{
alpha}
. Here the algorithm is less robust and it is essential to give a
sharp alpha, unless alpha <= -2
in which case we use
the default algorithm as if alpha were missing (or equal to 0
).
? intnum(x=1, +oo, x^(-3/2)) \\ default %1 = 1.9999999999999999999999999999646391207 ? intnum(x=1, [+oo,-3/2], x^(-3/2)) \\ precise decrease rate %2 = 2.0000000000000000000000000000000000000 ? intnum(x=1, [+oo,-11/10], x^(-3/2)) \\ worse than default %3 = 2.0000000000000000000000000089298011973
The last two codes are reserved for oscillating functions.
Let k > 0
real, and g(x)
a non-oscillating function tending slowly to 0
(e.g. like a negative power of x
), then
@3* alpha = k * I
assumes that the function behaves like cos (kx)g(x)
.
@3* alpha = -k* I
assumes that the function behaves like sin (kx)g(x)
.
@3Here it is critical to give the exact value of k
. If the
oscillating part is not a pure sine or cosine, one must expand it into a
Fourier series, use the above codings, and sum the resulting contributions.
Otherwise you will get nonsense. Note that cos (kx)
, and similarly
sin (kx)
, means that very function, and not a translated version such as
cos (kx+a)
.
@3Note. If f(x) =
cos (kx)g(x)
where g(x)
tends to zero
exponentially fast as exp (-
alpha x)
, it is up to the user to choose
between [+-oo,
alpha]
and [+-oo,k* I]
, but a good rule of
thumb is that
if the oscillations are weaker than the exponential decrease, choose
[+-oo,
alpha]
, otherwise choose [+-oo,k*I]
, although the
latter can
reasonably be used in all cases, while the former cannot. To take a specific
example, in the inverse Mellin transform, the integrand is almost always a
product of an exponentially decreasing and an oscillating factor. If we
choose the oscillating type of integral we perhaps obtain the best results,
at the expense of having to recompute our functions for a different value of
the variable z
giving the transform, preventing us to use a function such
as intfuncinit
. On the other hand using the exponential type of
integral, we obtain less accurate results, but we skip expensive
recomputations. See intfuncinit
for more explanations.
We shall now see many examples to get a feeling for what the various
parameters achieve. All examples below assume precision is set to 115
decimal digits. We first type
? \p 115
@3Apparent singularities. In many cases, apparent singularities
can be ignored. For instance, if f(x) = 1
/(
exp (x)-1) -
exp (-x)/x
, then int_0^ oo f(x)dx =
gamma, Euler's
constant Euler
. But
? f(x) = 1/(exp(x)-1) - exp(-x)/x ? intnum(x = 0, [oo,1], f(x)) - Euler %1 = 0.E-115
But close to 0
the function f
is computed with an enormous loss of
accuracy, and we are in fact lucky that it get multiplied by weights which are
sufficiently close to 0
to hide this:
? f(1e-200) %2 = -3.885337784451458142 E84
A more robust solution is to define the function differently near special points, e.g. by a Taylor expansion
? F = truncate( f(t + O(t^10)) ); \\ expansion around t = 0 ? poldegree(F) %4 = 7 ? g(x) = if (x > 1e-18, f(x), subst(F,t,x)); \\ note that 7.18 > 105 ? intnum(x = 0, [oo,1], g(x)) - Euler %2 = 0.E-115
@3It is up to the user to determine constants such as the
10^{-18}
and 10
used above.
@3True singularities. With true singularities the result is worse. For instance
? intnum(x = 0, 1, x^(-1/2)) - 2 %1 = -3.5... E-68 \\ only 68 correct decimals
? intnum(x = [0,-1/2], 1, x^(-1/2)) - 2 %2 = 0.E-114 \\ better
@3Oscillating functions.
? intnum(x = 0, oo, sin(x) / x) - Pi/2 %1 = 16.19.. \\ nonsense ? intnum(x = 0, [oo,1], sin(x)/x) - Pi/2 %2 = -0.006.. \\ bad ? intnum(x = 0, [oo,-I], sin(x)/x) - Pi/2 %3 = 0.E-115 \\ perfect ? intnum(x = 0, [oo,-I], sin(2*x)/x) - Pi/2 \\ oops, wrong k %4 = 0.06... ? intnum(x = 0, [oo,-2*I], sin(2*x)/x) - Pi/2 %5 = 0.E-115 \\ perfect
? intnum(x = 0, [oo,-I], sin(x)^3/x) - Pi/4 %6 = -0.0008... \\ bad ? sin(x)^3 - (3*sin(x)-sin(3*x))/4 %7 = O(x^17)
@3
We may use the above linearization and compute two oscillating integrals with
endpoints [oo, -I]
and [oo, -3*I]
respectively, or
notice the obvious change of variable, and reduce to the single integral
(1/2)
int_0^ oo
sin (x)/xdx
. We finish with some more complicated
examples:
? intnum(x = 0, [oo,-I], (1-cos(x))/x^2) - Pi/2 %1 = -0.0003... \\ bad ? intnum(x = 0, 1, (1-cos(x))/x^2) \ + intnum(x = 1, oo, 1/x^2) - intnum(x = 1, [oo,I], cos(x)/x^2) - Pi/2 %2 = 0.E-115 \\ perfect
? intnum(x = 0, [oo, 1], sin(x)^3*exp(-x)) - 0.3 %3 = -7.34... E-55 \\ bad ? intnum(x = 0, [oo,-I], sin(x)^3*exp(-x)) - 0.3 %4 = 8.9... E-103 \\ better. Try higher m ? tab = intnuminit(0,[oo,-I], 1); \\ double number of sampling points ? intnum(x = 0, oo, sin(x)^3*exp(-x), tab) - 0.3 %6 = 0.E-115 \\ perfect
@3Warning. Like sumalt
, intnum
often assigns a
reasonable value to diverging integrals. Use these values at your own risk!
For example:
? intnum(x = 0, [oo, -I], x^2*sin(x)) %1 = -2.0000000000...
Note the formula
int_0^ oo
sin (x)/x^sdx =
cos (
pi s/2)
Gamma(1-s) ,
a priori valid only for 0 <
Re (s) < 2
, but the right hand side provides an
analytic continuation which may be evaluated at s = -2
...
@3Multivariate integration.
Using successive univariate integration with respect to different formal
parameters, it is immediate to do naive multivariate integration. But it is
important to use a suitable intnuminit
to precompute data for the
internal integrations at least!
For example, to compute the double integral on the unit disc x^2+y^2 <= 1
of the function x^2+y^2
, we can write
? tab = intnuminit(-1,1); ? intnum(x=-1,1, intnum(y=-sqrt(1-x^2),sqrt(1-x^2), x^2+y^2, tab),tab) - Pi/2 %2 = -7.1... E-115 \\ OK
@3 The first tab is essential, the second optional. Compare:
? tab = intnuminit(-1,1); time = 4 ms. ? intnum(x=-1,1, intnum(y=-sqrt(1-x^2),sqrt(1-x^2), x^2+y^2)); time = 3,092 ms. \\ slow ? intnum(x=-1,1, intnum(y=-sqrt(1-x^2),sqrt(1-x^2), x^2+y^2, tab), tab); time = 252 ms. \\ faster ? intnum(x=-1,1, intnum(y=-sqrt(1-x^2),sqrt(1-x^2), x^2+y^2, tab)); time = 261 ms. \\ the I<internal> integral matters most
The library syntax is intnum(void *E, GEN (*eval)(void*,GEN), GEN a,GEN b,GEN tab, long prec)
,
where an omitted tab is coded as NULL
.
intnumgauss(X = a,b,
expr,{
tab})
Numerical integration of expr on the compact interval [a,b]
with
respect to X
using Gauss-Legendre quadrature; tab
is either omitted
or precomputed with intnumgaussinit
. As a convenience, it can be an
integer n
in which case we call
intnumgaussinit
(n)
and use n
-point quadrature.
? test(n, b = 1) = T=intnumgaussinit(n);\ intnumgauss(x=-b,b, 1/(1+x^2),T) - 2*atan(b); ? test(0) \\ default %1 = -9.490148553624725335 E-22 ? test(40) %2 = -6.186629001816965717 E-31 ? test(50) %3 = -1.1754943508222875080 E-38 ? test(50, 2) \\ double interval length %4 = -4.891779568527713636 E-21 ? test(90, 2) \\ n must almost be doubled as well! %5 = -9.403954806578300064 E-38
@3On the other hand, we recommend to split the integral
and change variables rather than increasing n
too much:
? f(x) = 1/(1+x^2); ? b = 100; ? intnumgauss(x=0,1, f(x)) + intnumgauss(x=1,1/b, f(1/x)*(-1/x^2)) - atan(b) %3 = -1.0579449157400587572 E-37
The library syntax is GEN
intnumgauss0(GEN X, GEN b, GEN expr, GEN tab = NULL, long prec)
.
({n})
Initialize tables for n
-point Gauss-Legendre integration of
a smooth function f
lon a compact
interval [a,b]
at current realprecision
. If n
is omitted, make a
default choice n ~ realprecision
, suitable for analytic
functions on [-1,1]
. The error is bounded by
((b-a)^{2n+1} (n!)^4)/((2n+1)[(2n)!]^3) f^{(2n)} (
xi) ,
a <
xi < b
so, if the interval length increases, n
should be increased as well.
? T = intnumgaussinit(); ? intnumgauss(t=-1,1,exp(t), T) - exp(1)+exp(-1) %1 = -5.877471754111437540 E-39 ? intnumgauss(t=-10,10,exp(t), T) - exp(10)+exp(-10) %2 = -8.358367809712546836 E-35 ? intnumgauss(t=-1,1,1/(1+t^2), T) - Pi/2 %3 = -9.490148553624725335 E-22
? T = intnumgaussinit(50); ? intnumgauss(t=-1,1,1/(1+t^2), T) - Pi/2 %5 = -1.1754943508222875080 E-38 ? intnumgauss(t=-5,5,1/(1+t^2), T) - 2*atan(5) %6 = -1.2[...]E-8
On the other hand, we recommend to split the integral and change variables
rather than increasing n
too much, see intnumgauss
.
The library syntax is GEN
intnumgaussinit(long n, long prec)
.
(a,b,{m = 0})
Initialize tables for integration from
a
to b
, where a
and b
are coded as in intnum
. Only the
compactness, the possible existence of singularities, the speed of decrease
or the oscillations at infinity are taken into account, and not the values.
For instance intnuminit(-1,1)
is equivalent to intnuminit(0,Pi)
,
and intnuminit([0,-1/2],oo)
is equivalent to
intnuminit([-1,-1/2], -oo)
; on the other hand, the order matters
and
intnuminit([0,-1/2], [1,-1/3])
is not equivalent to
intnuminit([0,-1/3], [1,-1/2])
!
If m
is present, it must be non-negative and we multiply the default
number of sampling points by 2^m
(increasing the running time by a
similar factor).
The result is technical and liable to change in the future, but we document
it here for completeness. Let x =
phi(t)
, t\in ]- oo , oo [
be an
internally chosen change of variable, achieving double exponential decrease of
the integrand at infinity. The integrator intnum
will compute
h
sum_{|n| < N}
phi'(nh) F(
phi(nh))
for some integration step h
and truncation parameter N
.
In basic use, let
[h, x0, w0, xp, wp, xm, wm] = intnuminit(a,b);
@3* h
is the integration step
@3* x_0 =
phi(0)
and w_0 =
phi'(0)
,
@3* xp contains the phi(nh)
, 0 < n < N
,
@3* xm contains the phi(nh)
, 0 < -n < N
, or is empty.
@3* wp contains the phi'(nh)
, 0 < n < N
,
@3* wm contains the phi'(nh)
, 0 < -n < N
, or is empty.
The arrays xm and wm are left empty when phi is an odd
function. In complicated situations when non-default behaviour is specified at
end points, intnuminit
may return up to 3
such arrays, corresponding
to a splitting of up to 3
integrals of basic type.
If the functions to be integrated later are of the form F = f(t) k(t,z)
for some kernel k
(e.g. Fourier, Laplace, Mellin,...), it is
useful to also precompute the values of f(
phi(nh))
, which is accomplished
by intfuncinit
. The hard part is to determine the behaviour
of F
at endpoints, depending on z
.
The library syntax is GEN
intnuminit(GEN a, GEN b, long m, long prec)
.
(X = a,b,
expr,{
flag = 0})
Numerical integration of expr (smooth in ]a,b[
), with respect to
X
. Suitable for low accuracy; if expr is very regular (e.g. analytic
in a large region) and high accuracy is desired, try intnum
first.
Set flag = 0
(or omit it altogether) when a
and b
are not too large, the
function is smooth, and can be evaluated exactly everywhere on the interval
[a,b]
.
If flag = 1
, uses a general driver routine for doing numerical integration,
making no particular assumption (slow).
flag = 2
is tailored for being used when a
or b
are infinite using the
change of variable t = 1/X
. One must have ab > 0
, and in fact if
for example b = + oo
, then it is preferable to have a
as large as
possible, at least a >= 1
.
If flag = 3
, the function is allowed to be undefined
at a
(but right continuous) or b
(left continuous),
for example the function sin (x)/x
between x = 0
and 1
.
The user should not require too much accuracy: realprecision
about
30 decimal digits (realbitprecision
about 100 bits) is OK,
but not much more. In addition, analytical cleanup of the integral must have
been done: there must be no singularities in the interval or at the
boundaries. In practice this can be accomplished with a change of
variable. Furthermore, for improper integrals, where one or both of the
limits of integration are plus or minus infinity, the function must decrease
sufficiently rapidly at infinity, which can often be accomplished through
integration by parts. Finally, the function to be integrated should not be
very small (compared to the current precision) on the entire interval. This
can of course be accomplished by just multiplying by an appropriate constant.
Note that infinity can be represented with essentially no loss of
accuracy by an appropriate huge number. However beware of real underflow
when dealing with rapidly decreasing functions. For example, in order to
compute the int_0^ oo e^{-x^2}dx
to 28 decimal digits, then one can
set infinity equal to 10 for example, and certainly not to 1e1000
.
The library syntax is intnumromb_bitprec(void *E, GEN (*eval)(void*,GEN), GEN a, GEN b, long flag, long bitprec)
,
where eval(x, E)
returns the value of the function at x
.
You may store any additional information required by eval
in E
, or set
it to NULL
. The historical variant
\synx{intnumromb}{..., long prec}{intnumromb}, where prec
is expressed in words,
not bits, is obsolete and should no longer be used.
limitnum(
expr,{k = 20},{
alpha = 1})
Lagrange-Zagier numerical extrapolation of expr, corresponding to a
sequence
u_n
, either given by a closure n- > u(n)
or by a vector of values
I.e., assuming that u_n
tends to a finite limit ell, try to determine
ell. This routine is purely numerical and heuristic, thus may or may not
work on your examples; k
is ignored if u
is given by a vector,
and otherwise is a multiplier such that we extrapolate from u(kn)
.
Assume that u_n
has an asymptotic expansion in n^{-
alpha}
:
u_n =
ell +
sum_{i >= 1} a_i n^{-i
alpha}
for some a_i
.
? limitnum(n -> n*sin(1/n)) %1 = 1.0000000000000000000000000000000000000
? limitnum(n -> (1+1/n)^n) - exp(1) %2 = 0.E-37
? limitnum(n -> 2^(4*n+1)*(n!)^4 / (2*n)! /(2*n+1)! ) %3 = 3.1415926535897932384626433832795028842 ? Pi %4 = 3.1415926535897932384626433832795028842
@3
If u_n
is given by a vector, it must be long enough for the extrapolation
to make sense: at least k
times the current realprecision
. The
preferred format is thus a closure, although it becomes inconvenient
when u_n
cannot be directly computed in time polynomial in log n
,
for instance if it is defined as a sum or by induction. In that case,
passing a vector of values is the best option. It usually pays off to
interpolate u(kn)
for some k > 1
:
? limitnum(vector(10,n,(1+1/n)^n)) *** ^-------------------- *** limitnum: non-existent component in limitnum: index < 20 \\ at this accuracy, we must have at least 20 values ? limitnum(vector(20,n,(1+1/n)^n)) - exp(1) %5 = -2.05... E-20 ? limitnum(vector(20,n, m=10*n;(1+1/m)^m)) - exp(1) \\ better accuracy %6 = 0.E-37
? v = vector(20); s = 0; ? for(i=1,#v, s += 1/i; v[i]= s - log(i)); ? limitnum(v) - Euler %9 = -1.6... E-19
? V = vector(200); s = 0; ? for(i=1,#V, s += 1/i; V[i]= s); ? v = vector(#V \ 10, i, V[10*i] - log(10*i)); ? limitnum(v) - Euler %13 = 6.43... E-29
The library syntax is limitnum(void *E, GEN (*u)(void *,GEN,long), long muli, GEN alpha, long prec)
, where u(E, n, prec)
must return u(n)
in precision prec
.
Also available is
GEN
limitnum0(GEN u, long muli, GEN alpha, long prec)
, where u
must be a vector of sufficient length as above.
(X = a,b,
expr,{x = 1})
Product of expression
expr, initialized at x
, the formal parameter X
going from a
to
b
. As for sum
, the main purpose of the initialization parameter x
is to force the type of the operations being performed. For example if it is
set equal to the integer 1, operations will start being done exactly. If it
is set equal to the real 1.
, they will be done using real numbers having
the default precision. If it is set equal to the power series 1+O(X^k)
for
a certain k
, they will be done using power series of precision at most k
.
These are the three most common initializations.
@3As an extreme example, compare
? prod(i=1, 100, 1 - X^i); \\ this has degree 5050 !! time = 128 ms. ? prod(i=1, 100, 1 - X^i, 1 + O(X^101)) time = 8 ms. %2 = 1 - X - X^2 + X^5 + X^7 - X^12 - X^15 + X^22 + X^26 - X^35 - X^40 + \ X^51 + X^57 - X^70 - X^77 + X^92 + X^100 + O(X^101)
Of course, in this specific case, it is faster to use eta
,
which is computed using Euler's formula.
? prod(i=1, 1000, 1 - X^i, 1 + O(X^1001)); time = 589 ms. ? \ps1000 seriesprecision = 1000 significant terms ? eta(X) - % time = 8ms. %4 = O(X^1001)
The library syntax is produit(GEN a, GEN b, char *expr, GEN x)
.
(X = a,b,
expr)
Product of expression expr,
initialized at 1. (i.e. to a real number equal to 1 to the current
realprecision
), the formal parameter X
ranging over the prime numbers
between a
and b
.
The library syntax is prodeuler(void *E, GEN (*eval)(void*,GEN), GEN a,GEN b, long prec)
.
(X = a,
expr,{
flag = 0})
infinite product of
expression expr, the formal parameter X
starting at a
. The evaluation
stops when the relative error of the expression minus 1 is less than the
default precision. In particular, non-convergent products result in infinite
loops. The expressions must always evaluate to an element of C.
If flag = 1
, do the product of the (1+
expr) instead.
The library syntax is prodinf(void *E, GEN (*eval)(void*,GEN), GEN a, long prec)
(flag = 0
), or prodinf1
with the same arguments (flag = 1
).
(X = a,b,
expr)
Find a real root of expression
expr between a
and b
, under the condition
expr(X = a) *
expr(X = b) <= 0
. (You will get an error message
roots must be bracketed in solve
if this does not hold.)
This routine uses Brent's method and can fail miserably if expr is
not defined in the whole of [a,b]
(try solve(x = 1, 2, tan(x))
).
The library syntax is zbrent(void *E,GEN (*eval)(void*,GEN),GEN a,GEN b,long prec)
.
(X = a,b,
step,
expr,{
flag = 0})
Find zeros of a continuous function in the real interval [a,b]
by naive
interval splitting. This function is heuristic and may or may not find the
intended zeros. Binary digits of flag mean
@3* 1: return as soon as one zero is found, otherwise return all zeros found;
@3* 2: refine the splitting until at least one zero is found (may loop indefinitely if there are no zeros);
@3* 4: do a multiplicative search (we must have a > 0
and step >
1
), otherwise an additive search; step is the multiplicative or
additive step.
@3* 8: refine the splitting until at least one zero is very close to an integer.
? solvestep(X=0,10,1,sin(X^2),1) %1 = 1.7724538509055160272981674833411451828 ? solvestep(X=1,12,2,besselj(4,X),4) %2 = [7.588342434..., 11.064709488...]
The library syntax is solvestep(void *E, GEN (*eval)(void*,GEN), GEN a,GEN b, GEN step,long flag,long prec)
.
(X = a,b,
expr,{x = 0})
Sum of expression expr,
initialized at x
, the formal parameter going from a
to b
. As for
prod
, the initialization parameter x
may be given to force the type
of the operations being performed.
@3As an extreme example, compare
? sum(i=1, 10^4, 1/i); \\ rational number: denominator has 4345 digits. time = 236 ms. ? sum(i=1, 5000, 1/i, 0.) time = 8 ms. %2 = 9.787606036044382264178477904
The library syntax is somme(GEN a, GEN b, char *expr, GEN x)
.
(X = a,
expr,{
flag = 0})
Numerical summation of the series expr, which should be an
alternating series (-1)^k a_k
, the formal variable X
starting at
a
. Use an algorithm of Cohen, Villegas and Zagier (Experiment. Math.
9 (2000), no. 1, 3--12).
If flag = 0
, assuming that the a_k
are the moments of a positive
measure on [0,1]
, the relative error is O(3+
sqrt 8)^{-n}
after using
a_k
for k <= n
. If realprecision
is p
, we thus set
n =
log (10)p/
log (3+
sqrt 8) ~ 1.3 p
; besides the time needed to
compute the a_k
, k <= n
, the algorithm overhead is negligible: time
O(p^2)
and space O(p)
.
If flag = 1
, use a variant with more complicated polynomials, see
polzagier
. If the a_k
are the moments of w(x)dx
where w
(or only xw(x^2)
) is a smooth function extending analytically to the whole
complex plane, convergence is in O(14.4^{-n})
. If xw(x^2)
extends
analytically to a smaller region, we still have exponential convergence,
with worse constants. Usually faster when the computation of a_k
is
expensive. If realprecision
is p
, we thus set
n =
log (10)p/
log (14.4) ~ 0.86 p
; besides the time needed to
compute the a_k
, k <= n
, the algorithm overhead is not
negligible: time O(p^3)
and space O(p^2)
. Thus, even if the analytic
conditions for rigorous use are met, this variant is only worthwile if the
a_k
are hard to compute, at least O(p^2)
individually on average:
otherwise we gain a small constant factor (1.5, say) in the number of
needed a_k
at the expense of a large overhead.
The conditions for rigorous use are hard to check but the routine is best used heuristically: even divergent alternating series can sometimes be summed by this method, as well as series which are not exactly alternating (see for example Label se:user_defined). It should be used to try and guess the value of an infinite sum. (However, see the example at the end of Label se:userfundef.)
If the series already converges geometrically,
suminf
is often a better choice:
? \p28 ? sumalt(i = 1, -(-1)^i / i) - log(2) time = 0 ms. %1 = -2.524354897 E-29 ? suminf(i = 1, -(-1)^i / i) \\ Had to hit C<C-C> *** at top-level: suminf(i=1,-(-1)^i/i) *** ^------ *** suminf: user interrupt after 10min, 20,100 ms. ? \p1000 ? sumalt(i = 1, -(-1)^i / i) - log(2) time = 90 ms. %2 = 4.459597722 E-1002
? sumalt(i = 0, (-1)^i / i!) - exp(-1) time = 670 ms. %3 = -4.03698781490633483156497361352190615794353338591897830587 E-944 ? suminf(i = 0, (-1)^i / i!) - exp(-1) time = 110 ms. %4 = -8.39147638 E-1000 \\ faster and more accurate
The library syntax is sumalt(void *E, GEN (*eval)(void*,GEN),GEN a,long prec)
. Also
available is sumalt2
with the same arguments (flag = 1
).
(n,X,
expr)
Sum of expression expr over the positive divisors of n
.
This function is a trivial wrapper essentially equivalent to
D = divisors(n); for (i = 1, #D, X = D[i]; eval(expr))
@3(except that X
is lexically scoped to the sumdiv
loop). If expr is a multiplicative function, use sumdivmult
.
(n,d,
expr)
Sum of multiplicative expression expr over the positive
divisors d
of n
. Assume that expr evaluates to f(d)
where f
is multiplicative: f(1) = 1
and f(ab) = f(a)f(b)
for coprime
a
and b
.
(X = a,
expr)
infinite sum of expression
expr, the formal parameter X
starting at a
. The evaluation stops
when the relative error of the expression is less than the default precision
for 3 consecutive evaluations. The expressions must always evaluate to a
complex number.
If the series converges slowly, make sure realprecision
is low (even 28
digits may be too much). In this case, if the series is alternating or the
terms have a constant sign, sumalt
and sumpos
should be used
instead.
? \p28 ? suminf(i = 1, -(-1)^i / i) \\ Had to hit C<C-C> *** at top-level: suminf(i=1,-(-1)^i/i) *** ^------ *** suminf: user interrupt after 10min, 20,100 ms. ? sumalt(i = 1, -(-1)^i / i) - log(2) time = 0 ms. %1 = -2.524354897 E-29
The library syntax is suminf(void *E, GEN (*eval)(void*,GEN), GEN a, long prec)
.
sumnum(n = a,f,{
tab})
Numerical summation of f(n)
at high accuracy using Euler-MacLaurin,
the variable n
taking values from a
to + oo
, where f
is assumed to
have positive values and is a C^ oo
function; a
must be an integer
and tab
, if given, is the output of sumnuminit
. The latter
precomputes abcissas and weights, speeding up the computation; it also allows
to specify the behaviour at infinity via sumnuminit([+oo, asymp])
.
? \p500 ? z3 = zeta(3); ? sumpos(n = 1, n^-3) - z3 time = 2,332 ms. %2 = 2.438468843 E-501 ? sumnum(n = 1, n^-3) - z3 \\ here slower than sumpos time = 2,752 ms. %3 = 0.E-500
@3Complexity.
The function f
will be evaluated at O(D
log D)
real arguments,
where D ~ realprecision.
log (10)
. The routine is geared
towards slowly decreasing functions: if f
decreases exponentially fast,
then one of suminf
or sumpos
should be preferred.
If f
satisfies the stronger hypotheses required for Monien summation,
i.e. if f(1/z)
is holomorphic in a complex neighbourhood of [0,1]
,
then sumnummonien
will be faster since it only requires O(D/
log D)
evaluations:
? sumnummonien(n = 1, 1/n^3) - z3 time = 1,985 ms. %3 = 0.E-500
@3The tab
argument precomputes technical data
not depending on the expression being summed and valid for a given accuracy,
speeding up immensely later calls:
? tab = sumnuminit(); time = 2,709 ms. ? sumnum(n = 1, 1/n^3, tab) - z3 \\ now much faster than sumpos time = 40 ms. %5 = 0.E-500
? tabmon = sumnummonieninit(); \\ Monien summation allows precomputations too time = 1,781 ms. ? sumnummonien(n = 1, 1/n^3, tabmon) - z3 time = 2 ms. %7 = 0.E-500
@3The speedup due to precomputations becomes less impressive
when the function f
is expensive to evaluate, though:
? sumnum(n = 1, lngamma(1+1/n)/n, tab); time = 14,180 ms.
? sumnummonien(n = 1, lngamma(1+1/n)/n, tabmon); \\ fewer evaluations time = 717 ms.
@3Behaviour at infinity.
By default, sumnum
assumes that expr decreases slowly at infinity,
but at least like O(n^{-2})
. If the function decreases like n^{
alpha}
for some -2 <
alpha < -1
, then it must be indicated via
tab = sumnuminit([+oo, alpha]); /* alpha < 0 slow decrease */
@3otherwise loss of accuracy is expected.
If the functions decreases quickly, like exp (-
alpha n)
for some
alpha > 0
, then it must be indicated via
tab = sumnuminit([+oo, alpha]); /* alpha > 0 exponential decrease */
@3otherwise exponent overflow will occur.
? sumnum(n=1,2^-n) *** at top-level: sumnum(n=1,2^-n) *** ^---- *** _^_: overflow in expo(). ? tab = sumnuminit([+oo,log(2)]); sumnum(n=1,2^-n, tab) %1 = 1.000[...]
As a shortcut, one can also input
sumnum(n = [a, asymp], f)
@3instead of
tab = sumnuminit(asymp); sumnum(n = a, f, tab)
@3Further examples.
? \p200 ? sumnum(n = 1, n^(-2)) - zeta(2) \\ accurate, fast time = 200 ms. %1 = -2.376364457868949779 E-212 ? sumpos(n = 1, n^(-2)) - zeta(2) \\ even faster time = 96 ms. %2 = 0.E-211 ? sumpos(n=1,n^(-4/3)) - zeta(4/3) \\ now much slower time = 13,045 ms. %3 = -9.980730723049589073 E-210 ? sumnum(n=1,n^(-4/3)) - zeta(4/3) \\ fast but inaccurate time = 365 ms. %4 = -9.85[...]E-85 ? sumnum(n=[1,-4/3],n^(-4/3)) - zeta(4/3) \\ with decrease rate, now accurate time = 416 ms. %5 = -4.134874156691972616 E-210
? tab = sumnuminit([+oo,-4/3]); time = 196 ms. ? sumnum(n=1, n^(-4/3), tab) - zeta(4/3) \\ faster with precomputations time = 216 ms. %5 = -4.134874156691972616 E-210 ? sumnum(n=1,-log(n)*n^(-4/3), tab) - zeta'(4/3) time = 321 ms. %7 = 7.224147951921607329 E-210
Note that in the case of slow decrease (alpha < 0
), the exact
decrease rate must be indicated, while in the case of exponential decrease,
a rough value will do. In fact, for exponentially decreasing functions,
sumnum
is given for completeness and comparison purposes only: one
of suminf
or sumpos
should always be preferred.
? sumnum(n=[1, 1], 2^-n) \\ pretend we decrease as exp(-n) time = 240 ms. %8 = 1.000[...] \\ perfect ? sumpos(n=1, 2^-n) %9 = 1.000[...] \\ perfect and instantaneous
The library syntax is sumnum((void *E, GEN (*eval)(void*, GEN), GEN a, GEN tab, long prec))
where an omitted tab is coded as NULL
.
sumnuminit({
asymp})
Initialize tables for Euler--MacLaurin delta summation of a series with
positive terms. If given, asymp
is of the form [+oo,
alpha]
,
as in intnum
and indicates the decrease rate at infinity of functions
to be summed. A positive
alpha > 0
encodes an exponential decrease of type exp (-
alpha n)
and
a negative -2 <
alpha < -1
encodes a slow polynomial decrease of type
n^{
alpha}
.
? \p200 ? sumnum(n=1, n^-2); time = 200 ms. ? tab = sumnuminit(); time = 188 ms. ? sumnum(n=1, n^-2, tab); \\ faster time = 8 ms.
? tab = sumnuminit([+oo, log(2)]); \\ decrease like 2^-n time = 200 ms. ? sumnum(n=1, 2^-n, tab) time = 44 ms.
? tab = sumnuminit([+oo, -4/3]); \\ decrease like n^(-4/3) time = 200 ms. ? sumnum(n=1, n^(-4/3), tab); time = 221 ms.
The library syntax is GEN
sumnuminit(GEN asymp = NULL, long prec)
.
sumnummonien(n = a,f,{
tab})
Numerical summation sum_{n >= a} f(n)
at high accuracy, the variable
n
taking values from the integer a
to + oo
using Monien summation,
which assumes that f(1/z)
has a complex analytic continuation in a (complex)
neighbourhood of the segment [0,1]
.
The function f
is evaluated at O(D /
log D)
real arguments,
where D ~ realprecision.
log (10)
.
By default, assume that f(n) = O(n^{-2})
and has a non-zero asymptotic
expansion
f(n) =
sum_{i >= 2} a_i n^{-i}
at infinity. To handle more complicated behaviours and allow time-saving
precomputations (for a given realprecision
), see sumnummonieninit
.
The library syntax is GEN
sumnummonien0(GEN n, GEN f, GEN tab = NULL, long prec)
.
sumnummonieninit({
asymp},{w},{
n0 = 1})
Initialize tables for Monien summation of a series sum_{n >= n_0}
f(n)
where f(1/z)
has a complex analytic continuation in a (complex)
neighbourhood of the segment [0,1]
.
By default, assume that f(n) = O(n^{-2})
and has a non-zero asymptotic
expansion
f(n) =
sum_{i >= 2} a_i / n^i
at infinity. Note that the sum starts at i = 2
! The argument asymp
allows to specify different expansions:
@3* a real number alpha > 1
means
f(n) =
sum_{i >= 1} a_i / n^{
alpha i}
(Now the summation starts at 1
.)
@3* a vector [
alpha,
beta]
of reals, where we must have alpha > 0
and alpha +
beta > 1
to ensure convergence, means that
f(n) =
sum_{i >= 1} a_i / n^{
alpha i +
beta}
Note that asymp = [
alpha,
alpha]
is equivalent to
asymp =
alpha.
? \p38 ? s = sumnum(n = 1, sin(1/sqrt(n)) / n) %1 = 2.3979771206715998375659850036324914714
? sumnummonien(n = 1, sin(1/sqrt(n)) / n) - s %2 = -0.001[...] \\ completely wrong !
? t = sumnummonieninit([1/2,1]); \\ f(n) = sum_i 1 / n^(i/2+1) ? sumnummonien(n = 1, sin(1/sqrt(n)) / n, t) - s %3 = 0.E-37 \\ now correct
The argument w
is used to sum expressions of the form
sum_{n >= n_0} f(n) w(n),
for varying f
as above, and fixed weight function w
, where we
further assume that the auxiliary sums
g_w(m) =
sum_{n >= n_0} w(n) / n^{
alpha m +
beta}
converge for all m >= 1
. Note that for non-negative integers k
,
and weight w(n) = (
log n)^k
, the function g_w(m) =
zeta^{(k)}(
alpha m +
beta)
has a simple expression; for general weights, g_w
is
computed using sumnum
. The following variants are available
@3* an integer k >= 0
, to code w(n) = (
log n)^k
;
only the cases k = 0,1
are presently implemented; due to a poor
implementation of zeta derivatives, it is not currently worth it
to exploit the special shape of g_w
when k > 0
;
@3* a t_CLOSURE
computing the values w(n)
, where we
assume that w(n) = O(n^
epsilon)
for all epsilon > 0
;
@3* a vector [w, fast]
, where w
is a closure as above
and fast
is a scalar;
we assume that w(n) = O(n^{fast+
epsilon})
; note that
w = [w, 0]
is equivalent to w = w
.
@3* a vector [w, oo]
, where w
is a closure as above;
we assume that w(n)
decreases exponentially. Note that in this case,
sumnummonien
is provided for completeness and comparison purposes only:
one of suminf
or sumpos
should be preferred in practice.
The cases where w
is a closure or w(n) =
log n
are the only ones where
n_0
is taken into account and stored in the result. The subsequent call to
sumnummonien
must use the same value.
? \p300 ? sumnummonien(n = 1, n^-2*log(n)) + zeta'(2) time = 536 ms. %1 = -1.323[...]E-6 \\ completely wrong, f does not satisfy hypotheses ! ? tab = sumnummonieninit(, 1); \\ codes w(n) = log(n) time = 18,316 ms. ? sumnummonien(n = 1, n^-2, tab) + zeta'(2) time = 44 ms. %3 = -5.562684646268003458 E-309 \\ now perfect
? tab = sumnummonieninit(, n->log(n)); \\ generic, about as fast time = 18,693 ms. ? sumnummonien(n = 1, n^-2, tab) + zeta'(2) time = 40 ms. %5 = -5.562684646268003458 E-309 \\ identical result
The library syntax is GEN
sumnummonieninit(GEN asymp = NULL, GEN w = NULL, GEN n0 = NULL, long prec)
.
(X = a,
expr,{
flag = 0})
Numerical summation of the series expr, which must be a series of
terms having the same sign, the formal variable X
starting at a
. The
algorithm used is Van Wijngaarden's trick for converting such a series into
an alternating one, then we use sumalt
. For regular functions, the
function sumnum
is in general much faster once the initializations
have been made using sumnuminit
.
The routine is heuristic and assumes that expr is more or less a
decreasing function of X
. In particular, the result will be completely
wrong if expr is 0 too often. We do not check either that all terms
have the same sign. As sumalt
, this function should be used to
try and guess the value of an infinite sum.
If flag = 1
, use sumalt
(,1)
instead of sumalt
(,0)
, see
Label se:sumalt. Requiring more stringent analytic properties for
rigorous use, but allowing to compute fewer series terms.
To reach accuracy 10^{-p}
, both algorithms require O(p^2)
space;
furthermore, assuming the terms decrease polynomially (in O(n^{-C})
), both
need to compute O(p^2)
terms. The sumpos
(,1)
variant has a smaller
implied constant (roughly 1.5 times smaller). Since the sumalt
(,1)
overhead is now small compared to the time needed to compute series terms,
this last variant should be about 1.5 faster. On the other hand, the
achieved accuracy may be much worse: as for sumalt
, since
conditions for rigorous use are hard to check, the routine is best used
heuristically.
The library syntax is sumpos(void *E, GEN (*eval)(void*,GEN),GEN a,long prec)
. Also
available is sumpos2
with the same arguments (flag = 1
).
Although plotting is not even a side purpose of PARI, a number of plotting functions are provided. Moreover, a lot of people suggested ideas or submitted patches for this section of the code. There are three types of graphic functions.
(all the functions starting with
ploth
) in which the user has little to do but explain what type of plot
he wants, and whose syntax is similar to the one used in the preceding
section.
(called rectplot functions,
sharing the prefix plot
), where every drawing primitive (point, line,
box, etc.) is specified by the user. These low-level functions work as
follows. You have at your disposal 16 virtual windows which are filled
independently, and can then be physically ORed on a single window at
user-defined positions. These windows are numbered from 0 to 15, and must be
initialized before being used by the function plotinit
, which specifies
the height and width of the virtual window (called a rectwindow in the
sequel). At all times, a virtual cursor (initialized at [0,0]
) is attached
to the window, and its current value can be obtained using the function
plotcursor
.
A number of primitive graphic objects (called rect objects) can then
be drawn in these windows, using a default color attached to that window
(which can be changed using the plotcolor
function) and only the part
of the object which is inside the window will be drawn, with the exception of
polygons and strings which are drawn entirely. The ones sharing the prefix
plotr
draw relatively to the current position of the virtual cursor,
the others use absolute coordinates. Those having the prefix plotrecth
put in the rectwindow a large batch of rect objects corresponding to the
output of the related ploth
function.
Finally, the actual physical drawing is done using plotdraw
. The
rectwindows are preserved so that further drawings using the same windows at
different positions or different windows can be done without extra work. To
erase a window, use plotkill
. It is not possible to partially erase a
window: erase it completely, initialize it again, then fill it with the
graphic objects that you want to keep.
In addition to initializing the window, you may use a scaled window to
avoid unnecessary conversions. For this, use plotscale
. As long as this
function is not called, the scaling is simply the number of pixels, the
origin being at the upper left and the y
-coordinates going downwards.
Plotting functions are platform independent, but a number of graphical
drivers are available for screen output: X11-windows (hence also for GUI's
based on X11 such as Openwindows and Motif), and the Qt and FLTK graphical
libraries. The physical window opened by plotdraw
or any of the
ploth*
functions is completely separated from gp
(technically, a
fork
is done, and the non-graphical memory is immediately freed in the
child process), which means you can go on working in the current gp
session, without having to kill the window first. This window can be closed,
enlarged or reduced using the standard window manager functions. No zooming
procedure is implemented though (yet).
in the same way that printtex
allows you to have a TeX output
corresponding to printed results, the functions starting with ps
allow
you to have PostScript
output of the plots. This will not be identical
with the screen output, but sufficiently close. Note that you can use
PostScript output even if you do not have the plotting routines enabled. The
PostScript output is written in a file whose name is derived from the
psfile
default (./pari.ps
if you did not tamper with it). Each
time a new PostScript output is asked for, the PostScript output is appended
to that file. Hence you probably want to remove this file, or change the
value of psfile
, in between plots. On the other hand, in this manner,
as many plots as desired can be kept in a single file.
None of the graphic functions are available
within the PARI library, you must be under gp
to use them. The reason
for that is that you really should not use PARI for heavy-duty graphical work,
there are better specialized alternatives around. This whole set of routines
was only meant as a convenient, but simple-minded, visual aid. If you really
insist on using these in your program (we warned you), the source
(plot*.c
) should be readable enough for you to achieve something.
plot(X = a,b,
expr,{
Ymin},{
Ymax})
Crude ASCII plot of the function represented by expression expr
from a
to b
, with Y ranging from Ymin to Ymax. If
Ymin (resp. Ymax) is not given, the minimum (resp. the maximum)
of the computed values of the expression is used instead.
The library syntax is void
pariplot(GEN X, GEN b, GEN expr, GEN Ymin = NULL, GEN Ymax = NULL, long prec)
.
(w,
x2,
y2)
Let (x1,y1)
be the current position of the virtual cursor. Draw in the
rectwindow w
the outline of the rectangle which is such that the points
(x1,y1)
and (x2,y2)
are opposite corners. Only the part of the rectangle
which is in w
is drawn. The virtual cursor does not move.
(w)
`clips' the content of rectwindow w
, i.e remove all parts of the
drawing that would not be visible on the screen. Together with
plotcopy
this function enables you to draw on a scratchpad before
committing the part you're interested in to the final picture.
(w,c)
Set default color to c
in rectwindow w
.
This is only implemented for the X-windows, fltk and Qt graphing engines.
Possible values for c
are given by the graphcolormap
default,
factory setting are
1 = black, 2 = blue, 3 = violetred, 4 = red, 5 = green, 6 = grey, 7 = gainsborough.
but this can be considerably extended.
(
sourcew,
destw,
dx,
dy,{
flag = 0})
Copy the contents of rectwindow sourcew to rectwindow destw with offset (dx,dy). If flag's bit 1 is set, dx and dy express fractions of the size of the current output device, otherwise dx and dy are in pixels. dx and dy are relative positions of northwest corners if other bits of flag vanish, otherwise of: 2: southwest, 4: southeast, 6: northeast corners
(w)
Give as a 2-component vector the current
(scaled) position of the virtual cursor corresponding to the rectwindow w
.
(
list, {
flag = 0})
Physically draw the rectwindows given in list
which must be a vector whose number of components is divisible by 3. If
list = [w1,x1,y1,w2,x2,y2,...]
, the windows w1
, w2
, etc. are
physically placed with their upper left corner at physical position
(x1,y1)
, (x2,y2)
,...respectively, and are then drawn together.
Overlapping regions will thus be drawn twice, and the windows are considered
transparent. Then display the whole drawing in a special window on your
screen. If flag != 0
, x1, y1 etc. express fractions of the size of the
current output device
ploth(X = a,b,
expr,{
flags = 0},{n = 0})
High precision plot of the function y = f(x)
represented by the expression
expr, x
going from a
to b
. This opens a specific window (which is
killed whenever you click on it), and returns a four-component vector giving
the coordinates of the bounding box in the form
[
xmin,
xmax,
ymin,
ymax]
.
@3Important note. ploth
may evaluate expr
thousands of
times; given the relatively low resolution of plotting devices, few
significant digits of the result will be meaningful. Hence you should keep
the current precision to a minimum (e.g. 9) before calling this function.
n
specifies the number of reference point on the graph, where a value of 0
means we use the hardwired default values (1000 for general plot, 1500 for
parametric plot, and 8 for recursive plot).
If no flag is given, expr is either a scalar expression f(X)
, in which
case the plane curve y = f(X)
will be drawn, or a vector
[f_1(X),...,f_k(X)]
, and then all the curves y = f_i(X)
will be drawn in
the same window.
@3The binary digits of flag mean:
@3* 1 = Parametric
: parametric plot. Here expr must
be a vector with an even number of components. Successive pairs are then
understood as the parametric coordinates of a plane curve. Each of these are
then drawn.
For instance:
ploth(X=0,2*Pi,[sin(X),cos(X)], "Parametric") ploth(X=0,2*Pi,[sin(X),cos(X)]) ploth(X=0,2*Pi,[X,X,sin(X),cos(X)], "Parametric")
@3draw successively a circle, two entwined sinusoidal curves
and a circle cut by the line y = x
.
@3* 2 = Recursive
: recursive plot. If this flag is set,
only one curve can be drawn at a time, i.e. expr must be either a
two-component vector (for a single parametric curve, and the parametric flag
has to be set), or a scalar function. The idea is to choose pairs of
successive reference points, and if their middle point is not too far away
from the segment joining them, draw this as a local approximation to the
curve. Otherwise, add the middle point to the reference points. This is
fast, and usually more precise than usual plot. Compare the results of
ploth(X=-1,1, sin(1/X), "Recursive") ploth(X=-1,1, sin(1/X))
for instance. But beware that if you are extremely unlucky, or choose too few reference points, you may draw some nice polygon bearing little resemblance to the original curve. For instance you should never plot recursively an odd function in a symmetric interval around 0. Try
ploth(x = -20, 20, sin(x), "Recursive")
to see why. Hence, it's usually a good idea to try and plot the same curve with slightly different parameters.
The other values toggle various display options:
@3* 4 = no_Rescale
: do not rescale plot according to the
computed extrema. This is used in conjunction with plotscale
when
graphing multiple functions on a rectwindow (as a plotrecth
call):
s = plothsizes(); plotinit(0, s[2]-1, s[2]-1); plotscale(0, -1,1, -1,1); plotrecth(0, t=0,2*Pi, [cos(t),sin(t)], "Parametric|no_Rescale") plotdraw([0, -1,1]);
This way we get a proper circle instead of the distorted ellipse produced by
ploth(t=0,2*Pi, [cos(t),sin(t)], "Parametric")
@3* 8 = no_X_axis
: do not print the x
-axis.
@3* 16 = no_Y_axis
: do not print the y
-axis.
@3* 32 = no_Frame
: do not print frame.
@3* 64 = no_Lines
: only plot reference points, do not join them.
@3* 128 = Points_too
: plot both lines and points.
@3* 256 = Splines
: use splines to interpolate the points.
@3* 512 = no_X_ticks
: plot no x
-ticks.
@3* 1024 = no_Y_ticks
: plot no y
-ticks.
@3* 2048 = Same_ticks
: plot all ticks with the same length.
@3* 4096 = Complex
: is a parametric plot but where each member of
expr
is considered a complex number encoding the two coordinates of a
point. For instance:
ploth(X=0,2*Pi,exp(I*X), "Complex") ploth(X=0,2*Pi,[(1+I)*X,exp(I*X)], "Complex")
@3will draw respectively a circle and a circle cut by the line
y = x
.
(
listx,
listy,{
flag = 0})
Given listx and listy two vectors of equal length, plots (in
high precision) the points whose (x,y)
-coordinates are given in
listx and listy. Automatic positioning and scaling is done, but
with the same scaling factor on x
and y
. If flag is 1, join points,
other non-0 flags toggle display options and should be combinations of bits
2^k
, k >= 3
as in ploth
.
({
flag = 0})
Return data corresponding to the output window
in the form of a 6-component vector: window width and height, sizes for ticks
in horizontal and vertical directions (this is intended for the gnuplot
interface and is currently not significant), width and height of characters.
If flag = 0
, sizes of ticks and characters are in
pixels, otherwise are fractions of the screen size
(w,{x},{y},{
flag = 0})
Initialize the rectwindow w
,
destroying any rect objects you may have already drawn in w
. The virtual
cursor is set to (0,0)
. The rectwindow size is set to width x
and height
y
; omitting either x
or y
means we use the full size of the device
in that direction.
If flag = 0
, x
and y
represent pixel units. Otherwise, x
and y
are understood as fractions of the size of the current output device (hence
must be between 0
and 1
) and internally converted to pixels.
The plotting device imposes an upper bound for x
and y
, for instance the
number of pixels for screen output. These bounds are available through the
plothsizes
function. The following sequence initializes in a portable
way (i.e independent of the output device) a window of maximal size, accessed
through coordinates in the [0,1000] x [0,1000]
range:
s = plothsizes(); plotinit(0, s[1]-1, s[2]-1); plotscale(0, 0,1000, 0,1000);
(w)
Erase rectwindow w
and free the corresponding memory. Note that if you
want to use the rectwindow w
again, you have to use plotinit
first
to specify the new size. So it's better in this case to use plotinit
directly as this throws away any previous work in the given rectwindow.
(w,X,Y,{
flag = 0})
Draw on the rectwindow w
the polygon such that the (x,y)-coordinates of the vertices are in the
vectors of equal length X
and Y
. For simplicity, the whole
polygon is drawn, not only the part of the polygon which is inside the
rectwindow. If flag is non-zero, close the polygon. In any case, the
virtual cursor does not move.
X
and Y
are allowed to be scalars (in this case, both have to).
There, a single segment will be drawn, between the virtual cursor current
position and the point (X,Y)
. And only the part thereof which
actually lies within the boundary of w
. Then move the virtual cursor
to (X,Y)
, even if it is outside the window. If you want to draw a
line from (x1,y1)
to (x2,y2)
where (x1,y1)
is not necessarily the
position of the virtual cursor, use plotmove(w,x1,y1)
before using this
function.
(w,
type)
This function is obsolete and currently a no-op.
Change the type of lines subsequently plotted in rectwindow w
.
type -2
corresponds to frames, -1
to axes, larger values may
correspond to something else. w = -1
changes highlevel plotting.
(w,x,y)
Move the virtual cursor of the rectwindow w
to position (x,y)
.
(w,X,Y)
Draw on the rectwindow w
the
points whose (x,y)
-coordinates are in the vectors of equal length X
and
Y
and which are inside w
. The virtual cursor does not move. This
is basically the same function as plothraw
, but either with no scaling
factor or with a scale chosen using the function plotscale
.
As was the case with the plotlines
function, X
and Y
are allowed to
be (simultaneously) scalar. In this case, draw the single point (X,Y)
on
the rectwindow w
(if it is actually inside w
), and in any case
move the virtual cursor to position (x,y)
.
(w,
size)
This function is obsolete. It is currently a no-op.
Changes the ``size'' of following points in rectwindow w
. If w = -1
,
change it in all rectwindows.
(w,
type)
This function is obsolete and currently a no-op.
change the type of points subsequently plotted in rectwindow w
.
type = -1
corresponds to a dot, larger values may correspond to
something else. w = -1
changes highlevel plotting.
(w,
dx,
dy)
Draw in the rectwindow w
the outline of the rectangle which is such
that the points (x1,y1)
and (x1+dx,y1+dy)
are opposite corners, where
(x1,y1)
is the current position of the cursor. Only the part of the
rectangle which is in w
is drawn. The virtual cursor does not move.
(w,X = a,b,
expr,{
flag = 0},{n = 0})
Writes to rectwindow w
the curve output of
ploth
(w,X = a,b,
expr,
flag,n)
. Returns a vector for the bounding box.
plotrecthraw(w,
data,{
flags = 0})
Plot graph(s) for
data in rectwindow w
. flag has the same significance here as in
ploth
, though recursive plot is no more significant.
data is a vector of vectors, each corresponding to a list a coordinates.
If parametric plot is set, there must be an even number of vectors, each
successive pair corresponding to a curve. Otherwise, the first one contains
the x
coordinates, and the other ones contain the y
-coordinates
of curves to plot.
(w,
dx,
dy)
Draw in the rectwindow w
the part of the segment
(x1,y1)-(x1+dx,y1+dy)
which is inside w
, where (x1,y1)
is the current
position of the virtual cursor, and move the virtual cursor to
(x1+dx,y1+dy)
(even if it is outside the window).
(w,
dx,
dy)
Move the virtual cursor of the rectwindow w
to position
(x1+dx,y1+dy)
, where (x1,y1)
is the initial position of the cursor
(i.e. to position (dx,dy)
relative to the initial cursor).
(w,
dx,
dy)
Draw the point (x1+dx,y1+dy)
on the rectwindow w
(if it is inside
w
), where (x1,y1)
is the current position of the cursor, and in any case
move the virtual cursor to position (x1+dx,y1+dy)
.
(w,
x1,
x2,
y1,
y2)
Scale the local coordinates of the rectwindow w
so that x
goes from
x1
to x2
and y
goes from y1
to y2
(x2 < x1
and y2 < y1
being
allowed). Initially, after the initialization of the rectwindow w
using
the function plotinit
, the default scaling is the graphic pixel count,
and in particular the y
axis is oriented downwards since the origin is at
the upper left. The function plotscale
allows to change all these
defaults and should be used whenever functions are graphed.
plotstring(w,x,{
flags = 0})
Draw on the rectwindow w
the String x
(see Label se:strings), at
the current position of the cursor.
flag is used for justification: bits 1 and 2 regulate horizontal alignment: left if 0, right if 2, center if 1. Bits 4 and 8 regulate vertical alignment: bottom if 0, top if 8, v-center if 4. Can insert additional small gap between point and string: horizontal if bit 16 is set, vertical if bit 32 is set (see the tutorial for an example).
(
list, {
flag = 0})
Same as plotdraw
, except that the output is a PostScript program
appended to the psfile
, and flag != 0 scales the plot from size of the
current output device to the standard PostScript plotting size
psploth(X = a,b,
expr,{
flags = 0},{n = 0})
Same as ploth
, except that the output is a PostScript program
appended to the psfile
.
(
listx,
listy,{
flag = 0})
Same as plothraw
, except that the output is a PostScript program
appended to the psfile
.
A number of control statements are available in GP. They are simpler and
have a syntax slightly different from their C counterparts, but are quite
powerful enough to write any kind of program. Some of them are specific to
GP, since they are made for number theorists. As usual, X
will denote any
simple variable name, and seq will always denote a sequence of
expressions, including the empty sequence.
@3Caveat. In constructs like
for (X = a,b, seq)
the variable X
is lexically scoped to the loop, leading to possibly
unexpected behavior:
n = 5; for (n = 1, 10, if (something_nice(), break); ); \\ at this point C<n> is 5 !
If the sequence seq
modifies the loop index, then the loop
is modified accordingly:
? for (n = 1, 10, n += 2; print(n)) 3 6 9 12
({n = 1})
Interrupts execution of current seq, and
immediately exits from the n
innermost enclosing loops, within the
current function call (or the top level loop); the integer n
must be
positive. If n
is greater than the number of enclosing loops, all
enclosing loops are exited.
()
Interrupt the program and enter the breakloop. The program continues when the breakloop is exited.
? f(N,x)=my(z=x^2+1);breakpoint();gcd(N,z^2+1-z); ? f(221,3) *** at top-level: f(221,3) *** ^-------- *** in function f: my(z=x^2+1);breakpoint();gcd(N,z *** ^-------------------- *** Break loop: type <Return> to continue; 'break' to go back to GP break> z 10 break> %2 = 13
({n = 1})
(In the break loop) go down n frames. This allows to cancel a previous call to
dbg_up
.
()
In the break loop, return the error data of the current error, if any.
See iferr
for details about error data. Compare:
? iferr(1/(Mod(2,12019)^(6!)-1),E,Vec(E)) %1 = ["e_INV", "Fp_inv", Mod(119, 12019)] ? 1/(Mod(2,12019)^(6!)-1) *** at top-level: 1/(Mod(2,12019)^(6!)- *** ^-------------------- *** _/_: impossible inverse in Fp_inv: Mod(119, 12019). *** Break loop: type 'break' to go back to GP prompt break> Vec(dbg_err()) ["e_INV", "Fp_inv", Mod(119, 12019)]
({n = 1})
(In the break loop) go up n frames. This allows to inspect data of the
parent function. To cancel a dbg_up
call, use dbg_down
(A{,n})
Print the inner structure of A
, complete if n
is omitted, up
to level n
otherwise. This is useful for debugging. This is similar to
\x
but does not require A
to be an history entry. In particular,
it can be used in the break loop.
(X = a,b,
seq)
Evaluates seq, where
the formal variable X
goes from a
to b
. Nothing is done if a > b
.
a
and b
must be in R. If b
is set to +oo
, the loop will not
stop; it is expected that the caller will break out of the loop itself at some
point, using break
or return
.
(n = a,{b},
seq)
Evaluates seq,
where the formal variable n
ranges over the composite numbers between the
non-negative real numbers a
to b
, including a
and b
if they are
composite. Nothing is done if a > b
.
? forcomposite(n = 0, 10, print(n)) 4 6 8 9 10
@3Omitting b
means we will run through all composites >= a
,
starting an infinite loop; it is expected that the user will break out of
the loop himself at some point, using break
or return
.
Note that the value of n
cannot be modified within seq:
? forcomposite(n = 2, 10, n = []) *** at top-level: forcomposite(n=2,10,n=[]) *** ^--- *** index read-only: was changed to [].
(n,X,
seq)
Evaluates seq, where
the formal variable X
ranges through the divisors of n
(see divisors
, which is used as a subroutine). It is assumed that
factor
can handle n
, without negative exponents. Instead of n
,
it is possible to input a factorization matrix, i.e. the output of
factor(n)
.
This routine uses divisors
as a subroutine, then loops over the
divisors. In particular, if n
is an integer, divisors are sorted by
increasing size.
To avoid storing all divisors, possibly using a lot of memory, the following (much slower) routine loops over the divisors using essentially constant space:
FORDIV(N)= { my(P, E);
P = factor(N); E = P[,2]; P = P[,1]; forvec( v = vector(#E, i, [0,E[i]]), X = factorback(P, v) \\ ... ); } ? for(i=1,10^5, FORDIV(i)) time = 3,445 ms. ? for(i=1,10^5, fordiv(i, d, )) time = 490 ms.
(E,a,b,
seq,{
flag = 0})
Evaluates seq, where the formal variable E = [
name, M, G]
ranges through all elliptic curves of conductors from a
to b
. In this
notation name is the curve name in Cremona's elliptic curve database,
M
is the minimal model, G
is a Z-basis of the free part of the
Mordell-Weil group E(
Q)
. If flag is non-zero, select
only the first curve in each isogeny class.
? forell(E, 1, 500, my([name,M,G] = E); \ if (#G > 1, print(name))) 389a1 433a1 446d1 ? c = 0; forell(E, 1, 500, c++); c \\ number of curves %2 = 2214 ? c = 0; forell(E, 1, 500, c++, 1); c \\ number of isogeny classes %3 = 971
The elldata
database must be installed and contain data for the
specified conductors.
The library syntax is forell(void *data, long (*call)(void*,GEN), long a, long b, long flag)
.
(X = k,
seq,{a = k},{n = k})
Evaluate seq over the partitions X = [x_1,...x_n]
of the
integer k
, i.e. increasing sequences x_1 <= x_2... <= x_n
of sum
x_1+...+ x_n = k
. By convention, 0
admits only the empty partition and
negative numbers have no partitions. A partition is given by a
t_VECSMALL
, where parts are sorted in nondecreasing order:
? forpart(X=3, print(X)) Vecsmall([3]) Vecsmall([1, 2]) Vecsmall([1, 1, 1])
@3Optional parameters n
and a
are as follows:
@3* n =
nmax (resp. n = [
nmin,
nmax]
) restricts
partitions to length less than nmax (resp. length between
nmin and nmax
), where the length is the number of nonzero
entries.
@3* a =
amax (resp. a = [
amin,
amax]
) restricts the parts
to integers less than amax (resp. between amin and
amax).
By default, parts are positive and we remove zero entries unless amin <= 0
,
in which case X
is of constant length nmax.
\\ at most 3 non-zero parts, all <= 4 ? forpart(v=5,print(Vec(v)),4,3) [1, 4] [2, 3] [1, 1, 3] [1, 2, 2]
\\ between 2 and 4 parts less than 5, fill with zeros ? forpart(v=5,print(Vec(v)),[0,5],[2,4]) [0, 0, 1, 4] [0, 0, 2, 3] [0, 1, 1, 3] [0, 1, 2, 2] [1, 1, 1, 2]
@3
The behavior is unspecified if X
is modified inside the loop.
The library syntax is forpart(void *data, long (*call)(void*,GEN), long k, GEN a, GEN n)
.
(p = a,{b},
seq)
Evaluates seq,
where the formal variable p
ranges over the prime numbers between the real
numbers a
to b
, including a
and b
if they are prime. More precisely,
the value of
p
is incremented to nextprime(p + 1)
, the smallest prime strictly
larger than p
, at the end of each iteration. Nothing is done if a > b
.
? forprime(p = 4, 10, print(p)) 5 7
@3Setting b
to +oo
means we will run through all primes
>= a
, starting an infinite loop; it is expected that the caller will break
out of the loop itself at some point, using break
or return
.
Note that the value of p
cannot be modified within seq:
? forprime(p = 2, 10, p = []) *** at top-level: forprime(p=2,10,p=[]) *** ^--- *** prime index read-only: was changed to [].
(X = a,b,s,
seq)
Evaluates seq,
where the formal variable X
goes from a
to b
, in increments of s
.
Nothing is done if s > 0
and a > b
or if s < 0
and a < b
. s
must be in
R^*
or a vector of steps [s_1,...,s_n]
. In the latter case, the
successive steps are used in the order they appear in s
.
? forstep(x=5, 20, [2,4], print(x)) 5 7 11 13 17 19
@3Setting b
to +oo
will start an infinite loop; it is
expected that the caller will break out of the loop itself at some point,
using break
or return
.
forsubgroup(H = G,{
bound},
seq)
Evaluates seq for
each subgroup H
of the abelian group G
(given in
SNF form or as a vector of elementary divisors).
If bound is present, and is a positive integer, restrict the output to
subgroups of index less than bound. If bound is a vector
containing a single positive integer B
, then only subgroups of index
exactly equal to B
are computed
The subgroups are not ordered in any
obvious way, unless G
is a p
-group in which case Birkhoff's algorithm
produces them by decreasing index. A subgroup is given as a matrix
whose columns give its generators on the implicit generators of G
. For
example, the following prints all subgroups of index less than 2 in G =
Z/2
Z g_1 x
Z/2
Z g_2
:
? G = [2,2]; forsubgroup(H=G, 2, print(H)) [1; 1] [1; 2] [2; 1] [1, 0; 1, 1]
The last one, for instance is generated by (g_1, g_1 + g_2)
. This
routine is intended to treat huge groups, when subgrouplist
is not an
option due to the sheer size of the output.
For maximal speed the subgroups have been left as produced by the algorithm.
To print them in canonical form (as left divisors of G
in HNF form), one
can for instance use
? G = matdiagonal([2,2]); forsubgroup(H=G, 2, print(mathnf(concat(G,H)))) [2, 1; 0, 1] [1, 0; 0, 2] [2, 0; 0, 1] [1, 0; 0, 1]
Note that in this last representation, the index [G:H]
is given by the
determinant. See galoissubcyclo
and galoisfixedfield
for
applications to Galois theory.
The library syntax is forsubgroup(void *data, long (*call)(void*,GEN), GEN G, GEN bound)
.
(X = v,
seq,{
flag = 0})
Let v
be an n
-component
vector (where n
is arbitrary) of two-component vectors [a_i,b_i]
for 1 <= i <= n
, where all entries a_i
, b_i
are real numbers.
This routine lets X
vary over the n
-dimensional hyperrectangle
given by v
, that is, X
is an n
-dimensional vector taking
successively its entries X[i]
in the range [a_i,b_i]
with lexicographic
ordering. (The component with the highest index moves the fastest.)
The type of X
is the same as the type of v
: t_VEC
or t_COL
.
The expression seq is evaluated with the successive values of X
.
If flag = 1
, generate only nondecreasing vectors X
, and
if flag = 2
, generate only strictly increasing vectors X
.
? forvec (X=[[0,1],[-1,1]], print(X)); [0, -1] [0, 0] [0, 1] [1, -1] [1, 0] [1, 1] ? forvec (X=[[0,1],[-1,1]], print(X), 1); [0, 0] [0, 1] [1, 1] ? forvec (X=[[0,1],[-1,1]], print(X), 2) [0, 1]
if(a,{
seq1},{
seq2})
Evaluates the expression sequence seq1 if a
is non-zero, otherwise
the expression seq2. Of course, seq1 or seq2 may be empty:
if (a,
seq)
evaluates seq if a
is not equal to zero
(you don't have to write the second comma), and does nothing otherwise,
if (a,,
seq)
evaluates seq if a
is equal to zero, and
does nothing otherwise. You could get the same result using the !
(not
) operator: if (!a,
seq)
.
The value of an if
statement is the value of the branch that gets
evaluated: for instance
x = if(n % 4 == 1, y, z);
@3sets x
to y
if n
is 1
modulo 4
, and to z
otherwise.
Successive 'else' blocks can be abbreviated in a single compound if
as follows:
if (test1, seq1, test2, seq2, ... testn, seqn, seqdefault);
@3is equivalent to
if (test1, seq1 , if (test2, seq2 , ... if (testn, seqn, seqdefault)...));
For instance, this allows to write traditional switch / case constructions:
if (x == 0, do0(), x == 1, do1(), x == 2, do2(), dodefault());
@3Remark.
The boolean operators &&
and ||
are evaluated
according to operator precedence as explained in Label se:operators, but,
contrary to other operators, the evaluation of the arguments is stopped
as soon as the final truth value has been determined. For instance
if (x != 0 && f(1/x), ...)
@3is a perfectly safe statement.
@3Remark. Functions such as break
and next
operate on
loops, such as forxxx
, while
, until
. The if
statement is not a loop. (Obviously!)
iferr(
seq1,E,
seq2{,
pred})
Evaluates the expression sequence seq1. If an error occurs,
set the formal parameter E set to the error data.
If pred is not present or evaluates to true, catch the error
and evaluate seq2. Both pred and seq2 can reference E.
The error type is given by errname(E)
, and other data can be
accessed using the component
function. The code seq2 should check
whether the error is the one expected. In the negative the error can be
rethrown using error(E)
(and possibly caught by an higher iferr
instance). The following uses iferr
to implement Lenstra's ECM factoring
method
? ecm(N, B = 1000!, nb = 100)= { for(a = 1, nb, iferr(ellmul(ellinit([a,1]*Mod(1,N)), [0,1]*Mod(1,N), B), E, return(gcd(lift(component(E,2)),N)), errname(E)=="e_INV" && type(component(E,2)) == "t_INTMOD")) } ? ecm(2^101-1) %2 = 7432339208719
The return value of iferr
itself is the value of seq2 if an
error occurs, and the value of seq1 otherwise. We now describe the
list of valid error types, and the attached error data E; in each
case, we list in order the components of E, accessed via
component(E,1)
, component(E,2)
, etc.
@3Internal errors, ``system'' errors.
@3* "e_ARCH"
. A requested feature s
is not available on this
architecture or operating system.
E has one component (t_STR
): the missing feature name s
.
@3* "e_BUG"
. A bug in the PARI library, in function s
.
E has one component (t_STR
): the function name s
.
@3* "e_FILE"
. Error while trying to open a file.
E has two components, 1 (t_STR
): the file type (input, output,
etc.), 2 (t_STR
): the file name.
@3* "e_IMPL"
. A requested feature s
is not implemented.
E has one component, 1 (t_STR
): the feature name s
.
@3* "e_PACKAGE"
. Missing optional package s
.
E has one component, 1 (t_STR
): the package name s
.
@3Syntax errors, type errors.
@3* "e_DIM"
. The dimensions of arguments x
and y
submitted
to function s
does not match up.
E.g., multiplying matrices of inconsistent dimension, adding vectors of
different lengths,...
E has three component, 1 (t_STR
): the function name s
, 2: the
argument x
, 3: the argument y
.
@3* "e_FLAG"
. A flag argument is out of bounds in function s
.
E has one component, 1 (t_STR
): the function name s
.
@3* "e_NOTFUNC"
. Generated by the PARI evaluator; tried to use a
GEN
x
which is not a t_CLOSURE
in a function call syntax (as in
f = 1; f(2);
).
E has one component, 1: the offending GEN
x
.
@3* "e_OP"
. Impossible operation between two objects than cannot
be typecast to a sensible common domain for deeper reasons than a type
mismatch, usually for arithmetic reasons. As in O(2) + O(3)
: it is
valid to add two t_PADIC
s, provided the underlying prime is the same; so
the addition is not forbidden a priori for type reasons, it only becomes so
when inspecting the objects and trying to perform the operation.
E has three components, 1 (t_STR
): the operator name op,
2: first argument, 3: second argument.
@3* "e_TYPE"
. An argument x
of function s
had an unexpected type.
(As in factor("blah")
.)
E has two components, 1 (t_STR
): the function name s
,
2: the offending argument x
.
@3* "e_TYPE2"
. Forbidden operation between two objects than cannot be
typecast to a sensible common domain, because their types do not match up.
(As in Mod(1,2) + Pi
.)
E has three components, 1 (t_STR
): the operator name op,
2: first argument, 3: second argument.
@3* "e_PRIORITY"
. Object o
in function s
contains
variables whose priority is incompatible with the expected operation.
E.g. Pol([x,1], 'y)
: this raises an error because it's not possible to
create a polynomial whose coefficients involve variables with higher priority
than the main variable. E
has four components: 1 (t_STR
): the function
name s
, 2: the offending argument o
, 3 (t_STR
): an operator
op describing the priority error, 4 (t_POL
):
the variable v
describing the priority error. The argument
satisfies variable(x)
op variable(v)
.
@3* "e_VAR"
. The variables of arguments x
and y
submitted
to function s
does not match up. E.g., considering the algebraic number
Mod(t,t^2+1)
in nfinit(x^2+1)
.
E has three component, 1 (t_STR
): the function name s
, 2
(t_POL
): the argument x
, 3 (t_POL
): the argument y
.
@3Overflows.
@3* "e_COMPONENT"
. Trying to access an inexistent component in a
vector/matrix/list in a function: the index is less than 1
or greater
than the allowed length.
E has four components,
1 (t_STR
): the function name
2 (t_STR
): an operator op ( <
or >
),
2 (t_GEN
): a numerical limit l
bounding the allowed range,
3 (GEN
): the index x
. It satisfies x
op l
.
@3* "e_DOMAIN"
. An argument is not in the function's domain.
E has five components, 1 (t_STR
): the function name,
2 (t_STR
): the mathematical name of the out-of-domain argument
3 (t_STR
): an operator op describing the domain error,
4 (t_GEN
): the numerical limit l
describing the domain error,
5 (GEN
): the out-of-domain argument x
. The argument satisfies x
op l
, which prevents it from belonging to the function's domain.
@3* "e_MAXPRIME"
. A function using the precomputed list of prime
numbers ran out of primes.
E has one component, 1 (t_INT
): the requested prime bound, which
overflowed primelimit
or 0
(bound is unknown).
@3* "e_MEM"
. A call to pari_malloc
or pari_realloc
failed. E has no component.
@3* "e_OVERFLOW"
. An object in function s
becomes too large to be
represented within PARI's hardcoded limits. (As in 2^2^2^10
or
exp(1e100)
, which overflow in lg
and expo
.)
E has one component, 1 (t_STR
): the function name s
.
@3* "e_PREC"
. Function s
fails because input accuracy is too low.
(As in floor(1e100)
at default accuracy.)
E has one component, 1 (t_STR
): the function name s
.
@3* "e_STACK"
. The PARI stack overflows.
E has no component.
@3Errors triggered intentionally.
@3* "e_ALARM"
. A timeout, generated by the alarm
function.
E has one component (t_STR
): the error message to print.
@3* "e_USER"
. A user error, as triggered by
error
(g_1,...,g_n)
.
E has one component, 1 (t_VEC
): the vector of n
arguments given
to error
.
@3Mathematical errors.
@3* "e_CONSTPOL"
. An argument of function s
is a constant
polynomial, which does not make sense. (As in galoisinit(Pol(1))
.)
E has one component, 1 (t_STR
): the function name s
.
@3* "e_COPRIME"
. Function s
expected coprime arguments,
and did receive x,y
, which were not.
E has three component, 1 (t_STR
): the function name s
,
2: the argument x
, 3: the argument y
.
@3* "e_INV"
. Tried to invert a non-invertible object x
in
function s
.
E has two components, 1 (t_STR
): the function name s
,
2: the non-invertible x
. If x = Mod(a,b)
is a t_INTMOD
and a
is not 0
mod b
, this allows to factor
the modulus, as gcd
(a,b)
is a non-trivial divisor of b
.
@3* "e_IRREDPOL"
. Function s
expected an irreducible polynomial,
and did receive T
, which was not. (As in nfinit(x^2-1)
.)
E has two component, 1 (t_STR
): the function name s
,
2 (t_POL
): the polynomial x
.
@3* "e_MISC"
. Generic uncategorized error.
E has one component (t_STR
): the error message to print.
@3* "e_MODULUS"
. moduli x
and y
submitted to function s
are
inconsistent. As in
nfalgtobasis(nfinit(t^3-2), Mod(t,t^2+1)
E has three component, 1 (t_STR
): the function s
,
2: the argument x
, 3: the argument x
.
@3* "e_PRIME"
. Function s
expected a prime number,
and did receive p
, which was not. (As in idealprimedec(nf, 4)
.)
E has two component, 1 (t_STR
): the function name s
,
2: the argument p
.
@3* "e_ROOTS0"
. An argument of function s
is a zero polynomial,
and we need to consider its roots. (As in polroots(0)
.) E has
one component, 1 (t_STR
): the function name s
.
@3* "e_SQRTN"
. Trying to compute an n
-th root of x
, which does
not exist, in function s
. (As in sqrt(Mod(-1,3))
.)
E has two components, 1 (t_STR
): the function name s
,
2: the argument x
.
({n = 1})
Interrupts execution of current seq
,
resume the next iteration of the innermost enclosing loop, within the
current function call (or top level loop). If n
is specified, resume at
the n
-th enclosing loop. If n
is bigger than the number of enclosing
loops, all enclosing loops are exited.
({x = 0})
Returns from current subroutine, with
result x
. If x
is omitted, return the (void)
value (return no
result, like print
).
(a,
seq)
Evaluates seq until a
is not
equal to 0 (i.e. until a
is true). If a
is initially not equal to 0,
seq is evaluated once (more generally, the condition on a
is tested
after execution of the seq, not before as in while
).
(a,
seq)
While a
is non-zero, evaluates the expression sequence seq. The
test is made before evaluating the seq
, hence in particular if a
is initially equal to zero the seq will not be evaluated at all.
In addition to the general PARI functions, it is necessary to have some
functions which will be of use specifically for gp
, though a few of these can
be accessed under library mode. Before we start describing these, we recall
the difference between strings and keywords (see
Label se:strings): the latter don't get expanded at all, and you can type
them without any enclosing quotes. The former are dynamic objects, where
everything outside quotes gets immediately expanded.
(
fmt,{x}*)
Returns a string built from the remaining arguments according to the
format fmt. The format consists of ordinary characters (not %), printed
unchanged, and conversions specifications. See printf
.
(
sym,
str)
Changes the help message for the symbol sym
. The string str
is expanded on the spot and stored as the online help for sym
. It is
recommended to document global variables and user functions in this way,
although gp
will not protest if you don't.
You can attach a help text to an alias, but it will never be
shown: aliases are expanded by the ?
help operator and we get the help
of the symbol the alias points to. Nothing prevents you from modifying the
help of built-in PARI functions. But if you do, we would like to hear why you
needed it!
Without addhelp
, the standard help for user functions consists of its
name and definition.
gp> f(x) = x^2; gp> ?f f = (x)->x^2
@3Once addhelp is applied to f
, the function code is no
longer included. It can still be consulted by typing the function name:
gp> addhelp(f, "Square") gp> ?f Square
gp> f %2 = (x)->x^2
The library syntax is void
addhelp(const char *sym, const char *str)
.
alarm({s = 0},{
code})
If code is omitted, trigger an e_ALARM exception after s
seconds, cancelling any previously set alarm; stop a pending alarm if s =
0
or is omitted.
Otherwise, if s
is positive, the function evaluates code,
aborting after s
seconds. The return value is the value of code if
it ran to completion before the alarm timeout, and a t_ERROR
object
otherwise.
? p = nextprime(10^25); q = nextprime(10^26); N = p*q; ? E = alarm(1, factor(N)); ? type(E) %3 = "t_ERROR" ? print(E) %4 = error("alarm interrupt after 964 ms.") ? alarm(10, factor(N)); \\ enough time %5 = [ 10000000000000000000000013 1]
[100000000000000000000000067 1]
@3Here is a more involved example: the function
timefact(N,sec)
below tries to factor N
and gives up after sec
seconds, returning a partial factorisation.
\\ Time-bounded partial factorization default(factor_add_primes,1); timefact(N,sec)= { F = alarm(sec, factor(N)); if (type(F) == "t_ERROR", factor(N, 2^24), F); }
@3We either return the factorization directly, or replace the
t_ERROR
result by a simple bounded factorization factor(N, 2^24)
.
Note the factor_add_primes
trick: any prime larger than 2^{24}
discovered while attempting the initial factorization is stored and
remembered. When the alarm rings, the subsequent bounded factorization finds
it right away.
@3Caveat. It is not possible to set a new alarm within
another alarm
code: the new timer erases the parent one.
The library syntax is GEN
gp_alarm(long s, GEN code = NULL)
.
(
newsym,
sym)
Defines the symbol newsym as an alias for the symbol sym:
? alias("det", "matdet"); ? det([1,2;3,4]) %1 = -2
You are not restricted to ordinary functions, as in the above example:
to alias (from/to) member functions, prefix them with `_.
';
to alias operators, use their internal name, obtained by writing
_
in lieu of the operators argument: for instance, _!
and
!_
are the internal names of the factorial and the
logical negation, respectively.
? alias("mod", "_.mod"); ? alias("add", "_+_"); ? alias("_.sin", "sin"); ? mod(Mod(x,x^4+1)) %2 = x^4 + 1 ? add(4,6) %3 = 10 ? Pi.sin %4 = 0.E-37
Alias expansion is performed directly by the internal GP compiler. Note that since alias is performed at compilation-time, it does not require any run-time processing, however it only affects GP code compiled after the alias command is evaluated. A slower but more flexible alternative is to use variables. Compare
? fun = sin; ? g(a,b) = intnum(t=a,b,fun(t)); ? g(0, Pi) %3 = 2.0000000000000000000000000000000000000 ? fun = cos; ? g(0, Pi) %5 = 1.8830410776607851098 E-39
with
? alias(fun, sin); ? g(a,b) = intnum(t=a,b,fun(t)); ? g(0,Pi) %2 = 2.0000000000000000000000000000000000000 ? alias(fun, cos); \\ Oops. Does not affect *previous* definition! ? g(0,Pi) %3 = 2.0000000000000000000000000000000000000 ? g(a,b) = intnum(t=a,b,fun(t)); \\ Redefine, taking new alias into account ? g(0,Pi) %5 = 1.8830410776607851098 E-39
A sample alias file misc/gpalias
is provided with
the standard distribution.
The library syntax is void
alias0(const char *newsym, const char *sym)
.
({s = 0})
This special operation changes the stack size after
initialization. x
must be a non-negative integer. If x > 0
, a new stack
of at least x
bytes is allocated. We may allocate more than x
bytes if
x
is way too small, or for alignment reasons: the current formula is
max (16*ceil{x/16}, 500032)
bytes.
If x = 0
, the size of the new stack is twice the size of the old one.
This command is much more useful if parisizemax
is non-zero, and we
describe this case first. With parisizemax
enabled, there are three
sizes of interest:
@3* a virtual stack size, parisizemax
, which is an absolute upper
limit for the stack size; this is set by default(parisizemax, ...)
.
@3* the desired typical stack size, parisize
, that will grow as
needed, up to parisizemax
; this is set by default(parisize, ...)
.
@3* the current stack size, which is less that parisizemax
,
typically equal to parisize
but possibly larger and increasing
dynamically as needed; allocatemem
allows to change that one
explicitly.
The allocatemem
command forces stack
usage to increase temporarily (up to parisizemax
of course); for
instance if you notice using \gm2
that we seem to collect garbage a
lot, e.g.
? \gm2 debugmem = 2 ? default(parisize,"32M") *** Warning: new stack size = 32000000 (30.518 Mbytes). ? bnfinit('x^2+10^30-1) *** bnfinit: collecting garbage in hnffinal, i = 1. *** bnfinit: collecting garbage in hnffinal, i = 2. *** bnfinit: collecting garbage in hnffinal, i = 3.
@3and so on for hundred of lines. Then, provided the
breakloop
default is set, you can interrupt the computation, type
allocatemem(100*10^6)
at the break loop prompt, then let the
computation go on by typing <Enter>
. Back at the gp
prompt,
the desired stack size of parisize
is restored. Note that changing either
parisize
or parisizemax
at the break loop prompt would interrupt
the computation, contrary to the above.
In most cases, parisize
will increase automatically (up to
parisizemax
) and there is no need to perform the above maneuvers.
But that the garbage collector is sufficiently efficient that
a given computation can still run without increasing the stack size,
albeit very slowly due to the frequent garbage collections.
@3Deprecated: when parisizemax.
is unset
This is currently still the default behavior in order not to break backward
compatibility. The rest of this section documents the
behavior of allocatemem
in that (deprecated) situation: it becomes a
synonym for default(parisize,...)
. In that case, there is no
notion of a virtual stack, and the stack size is always equal to
parisize
. If more memory is needed, the PARI stack overflows, aborting
the computation.
Thus, increasing parisize
via allocatemem
or
default(parisize,...)
before a big computation is important.
Unfortunately, either must be typed at the gp
prompt in
interactive usage, or left by itself at the start of batch files.
They cannot be used meaningfully in loop-like constructs, or as part of a
larger expression sequence, e.g
allocatemem(); x = 1; \\ This will not set C<x>!
In fact, all loops are immediately exited, user functions terminated, and
the rest of the sequence following allocatemem()
is silently
discarded, as well as all pending sequences of instructions. We just go on
reading the next instruction sequence from the file we are in (or from the
user). In particular, we have the following possibly unexpected behavior: in
read("file.gp"); x = 1
@3were file.gp
contains an allocatemem
statement,
the x = 1
is never executed, since all pending instructions in the
current sequence are discarded.
The reason for these unfortunate side-effects is that, with
parisizemax
disabled, increasing the stack size physically
moves the stack, so temporary objects created during the current expression
evaluation are not correct anymore. (In particular byte-compiled expressions,
which are allocated on the stack.) To avoid accessing obsolete pointers to
the old stack, this routine ends by a longjmp
.
The library syntax is void
gp_allocatemem(GEN s = NULL)
.
(f, A)
Apply the t_CLOSURE
f
to the entries of A
. If A
is a scalar, return f(A)
. If A
is a polynomial or power series,
apply f
on all coefficients. If A
is a vector or list, return
the elements f(x)
where x
runs through A
. If A
is a matrix,
return the matrix whose entries are the f(A[i,j])
.
? apply(x->x^2, [1,2,3,4]) %1 = [1, 4, 9, 16] ? apply(x->x^2, [1,2;3,4]) %2 = [1 4]
[9 16] ? apply(x->x^2, 4*x^2 + 3*x+ 2) %3 = 16*x^2 + 9*x + 4
@3Note that many functions already act componentwise on
vectors or matrices, but they almost never act on lists; in this
case, apply
is a good solution:
? L = List([Mod(1,3), Mod(2,4)]); ? lift(L) *** at top-level: lift(L) *** ^------- *** lift: incorrect type in lift. ? apply(lift, L); %2 = List([1, 2])
@3Remark. For v
a t_VEC
, t_COL
, t_LIST
or t_MAT
,
the alternative set-notations
[g(x) | x <- v, f(x)] [x | x <- v, f(x)] [g(x) | x <- v]
@3 are available as shortcuts for
apply(g, select(f, Vec(v))) select(f, Vec(v)) apply(g, Vec(v))
@3respectively:
? L = List([Mod(1,3), Mod(2,4)]); ? [ lift(x) | x<-L ] %2 = [1, 2]
The library syntax is genapply(void *E, GEN (*fun)(void*,GEN), GEN a)
.
(f, A)
A = [a_1,..., a_n]
being a vector and f
being a function, returns the
evaluation of f(a_1,...,a_n)
.
f
can also be the name of a built-in GP function.
If # A = 1
, call
(f,A
) = apply
(f,A
)[1].
If f
is variadic, the variadic arguments must grouped in a vector in
the last component of A
.
This function is useful
@3* when writing a variadic function, to call another one:
fprintf(file,format,args[..]) = write(file,call(Strprintf,[format,args]))
@3* when dealing with function arguments with unspecified arity
The function below implements a global memoization interface:
memo=Map(); memoize(f,A[..])= { my(res); if(!mapisdefined(memo, [f,A], &res), res = call(f,A); mapput(memo,[f,A],res)); res; }
for example:
? memoize(factor,2^128+1) %3 = [59649589127497217,1;5704689200685129054721,1] ? ## *** last result computed in 76 ms. ? memoize(factor,2^128+1) %4 = [59649589127497217,1;5704689200685129054721,1] ? ## *** last result computed in 0 ms. ? memoize(ffinit,3,3) %5 = Mod(1,3)*x^3+Mod(1,3)*x^2+Mod(1,3)*x+Mod(2,3) ? fibo(n)=if(n==0,0,n==1,1,memoize(fibo,n-2)+memoize(fibo,n-1)); ? fibo(100) %7 = 354224848179261915075
@3* to call operators through their internal names without using
alias
matnbelts(M) = call("_*_",matsize(M))
The library syntax is GEN
call0(GEN f, GEN A)
.
default({
key},{
val})
Returns the default corresponding to keyword key. If val is
present, sets the default to val first (which is subject to string
expansion first). Typing default()
(or \d
) yields the complete
default list as well as their current values. See Label se:defaults for an
introduction to GP defaults, Label se:gp_defaults for a
list of available defaults, and Label se:meta for some shortcut
alternatives. Note that the shortcuts are meant for interactive use and
usually display more information than default
.
The library syntax is GEN
default0(const char *key = NULL, const char *val = NULL)
.
(E)
Returns the type of the error message E
as a string.
The library syntax is GEN
errname(GEN E)
.
error({
str}*)
Outputs its argument list (each of
them interpreted as a string), then interrupts the running gp
program,
returning to the input prompt. For instance
error("n = ", n, " is not squarefree!")
(
str)
The string str is the name of an external command (i.e. one you
would type from your UNIX shell prompt). This command is immediately run and
its output fed into gp
, just as if read from a file.
The library syntax is GEN
gpextern(const char *str)
.
(
str)
The string str is the name of an external command (i.e. one you would type from your UNIX shell prompt). This command is immediately run and its output is returned as a vector of GP strings, one component per output line.
The library syntax is GEN
externstr(const char *str)
.
(f, A)
Apply the t_CLOSURE
f
of arity 2
to the entries of A
,
in order to return f(...f(f(A[1],A[2]),A[3])...,A[#A])
.
? fold((x,y)->x*y, [1,2,3,4]) %1 = 24 ? fold((x,y)->[x,y], [1,2,3,4]) %2 = [[[1, 2], 3], 4] ? fold((x,f)->f(x), [2,sqr,sqr,sqr]) %3 = 256 ? fold((x,y)->(x+y)/(1-x*y),[1..5]) %4 = -9/19 ? bestappr(tan(sum(i=1,5,atan(i)))) %5 = -9/19
The library syntax is GEN
fold0(GEN f, GEN A)
.
Also available is
GEN
genfold(void *E, GEN (*fun)(void*,GEN, GEN), GEN A)
.
()
Returns the CPU time (in milliseconds) elapsed since gp
startup.
This provides a reentrant version of gettime
:
my (t = getabstime()); ... print("Time: ", getabstime() - t);
For a version giving wall-clock time, see getwalltime
.
The library syntax is long
getabstime()
.
(s)
Return the value of the environment variable s
if it is defined, otherwise return 0.
The library syntax is GEN
gp_getenv(const char *s)
.
()
Returns a two-component row vector giving the number of objects on the heap and the amount of memory they occupy in long words. Useful mainly for debugging purposes.
The library syntax is GEN
getheap()
.
()
Returns the current value of the seed used by the
pseudo-random number generator random
. Useful mainly for debugging
purposes, to reproduce a specific chain of computations. The returned value
is technical (reproduces an internal state array), and can only be used as an
argument to setrand
.
The library syntax is GEN
getrand()
.
()
Returns the current value of top-avma
, i.e. the number of
bytes used up to now on the stack. Useful mainly for debugging purposes.
The library syntax is long
getstack()
.
()
Returns the CPU time (in milliseconds) used since either the last call to
gettime
, or to the beginning of the containing GP instruction (if
inside gp
), whichever came last.
For a reentrant version, see getabstime
.
For a version giving wall-clock time, see getwalltime
.
The library syntax is long
gettime()
.
()
Returns the time (in milliseconds) elapsed since the UNIX Epoch (1970-01-01 00:00:00 (UTC)).
my (t = getwalltime()); ... print("Time: ", getwalltime() - t);
The library syntax is GEN
getwalltime()
.
(
list of variables)
Obsolete. Scheduled for deletion.
(x,...,z)
(Experimental) declare x,..., z
as inline variables. Such variables
behave like lexically scoped variable (see my()) but with unlimited scope.
It is however possible to exit the scope by using uninline()
.
When used in a GP script, it is recommended to call uninline()
before
the script's end to avoid inline variables leaking outside the script.
()
Reads a string, interpreted as a GP expression,
from the input file, usually standard input (i.e. the keyboard). If a
sequence of expressions is given, the result is the result of the last
expression of the sequence. When using this instruction, it is useful to
prompt for the string by using the print1
function. Note that in the
present version 2.19 of pari.el
, when using gp
under GNU Emacs (see
Label se:emacs) one must prompt for the string, with a string
which ends with the same prompt as any of the previous ones (a "? "
will do for instance).
The library syntax is GEN
gp_input()
.
install(
name,
code,{
gpname},{
lib})
Loads from dynamic library lib the function name. Assigns to it
the name gpname in this gp
session, with prototype
code (see below). If gpname is omitted, uses name.
If lib is omitted, all symbols known to gp
are available: this
includes the whole of libpari.so
and possibly others (such as
libc.so
).
Most importantly, install
gives you access to all non-static functions
defined in the PARI library. For instance, the function
GEN addii(GEN x, GEN y)
@3adds two PARI integers, and is not directly accessible under
gp
(it is eventually called by the +
operator of course):
? install("addii", "GG") ? addii(1, 2) %1 = 3
It also allows to add external functions to the gp
interpreter.
For instance, it makes the function system
obsolete:
? install(system, vs, sys,/*omitted*/) ? sys("ls gp*") gp.c gp.h gp_rl.c
@3This works because system
is part of libc.so
,
which is linked to gp
. It is also possible to compile a shared library
yourself and provide it to gp in this way: use gp2c
, or do it manually
(see the modules_build
variable in pari.cfg
for hints).
Re-installing a function will print a warning and update the prototype code if needed. However, it will not reload a symbol from the library, even if the latter has been recompiled.
@3Prototype. We only give a simplified description here, covering
most functions, but there are many more possibilities. The full documentation
is available in libpari.dvi
, see
??prototype
@3* First character i
, l
, v
: return type int / long /
void. (Default: GEN
)
@3* One letter for each mandatory argument, in the same order as they appear
in the argument list: G
(GEN
), &
(GEN*
), L
(long
), s
(char *
), n
(variable).
@3* p
to supply realprecision
(usually long prec
in the
argument list), P
to supply seriesprecision
(usually long precdl
).
@3We also have special constructs for optional arguments and default values:
@3* DG
(optional GEN
, NULL
if omitted),
@3* D&
(optional GEN*
, NULL
if omitted),
@3* Dn
(optional variable, -1
if omitted),
For instance the prototype corresponding to
long issquareall(GEN x, GEN *n = NULL)
@3is lGD&
.
@3Caution. This function may not work on all systems, especially
when gp
has been compiled statically. In that case, the first use of an
installed function will provoke a Segmentation Fault (this should never
happen with a dynamically linked executable). If you intend to use this
function, please check first on some harmless example such as the one above
that it works properly on your machine.
The library syntax is void
gpinstall(const char *name, const char *code, const char *gpname, const char *lib)
.
(
sym)
Restores the symbol sym
to its ``undefined'' status, and deletes any
help messages attached to sym
using addhelp
. Variable names
remain known to the interpreter and keep their former priority: you cannot
make a variable ``less important" by killing it!
? z = y = 1; y %1 = 1 ? kill(y) ? y \\ restored to ``undefined'' status %2 = y ? variable() %3 = [x, y, z] \\ but the variable name y is still known, with y > z !
For the same reason, killing a user function (which is an ordinary
variable holding a t_CLOSURE
) does not remove its name from the list of
variable names.
If the symbol is attached to a variable --- user functions being an
important special case ---, one may use the quote operator
a = 'a
to reset variables to their starting values. However, this
will not delete a help message attached to a
, and is also slightly
slower than kill(a)
.
? x = 1; addhelp(x, "foo"); x %1 = 1 ? x = 'x; x \\ same as 'kill', except we don't delete help. %2 = x ? ?x foo
On the other hand, kill
is the only way to remove aliases and installed
functions.
? alias(fun, sin); ? kill(fun);
? install(addii, GG); ? kill(addii);
The library syntax is void
kill0(const char *sym)
.
({n})
This function is obsolete, use List
.
Creates an empty list. This routine used to have a mandatory argument, which is now ignored (for backward compatibility).
(L,x,n)
Inserts the object x
at
position n
in L
(which must be of type t_LIST
). This has
complexity O(#L - n + 1)
: all the
remaining elements of list (from position n+1
onwards) are shifted
to the right.
The library syntax is GEN
listinsert(GEN L, GEN x, long n)
.
(L)
Obsolete, retained for backward compatibility. Just use L = List()
instead of listkill(L)
. In most cases, you won't even need that, e.g.
local variables are automatically cleared when a user function returns.
The library syntax is void
listkill(GEN L)
.
(
list,{n})
Removes the n
-th element of the list
list (which must be of type t_LIST
). If n
is omitted,
or greater than the list current length, removes the last element.
If the list is already empty, do nothing. This runs in time O(#L - n + 1)
.
The library syntax is void
listpop0(GEN list, long n)
.
(
list,x,{n})
Sets the n
-th element of the list
list (which must be of type t_LIST
) equal to x
. If n
is omitted,
or greater than the list length, appends x
. The function returns the
inserted element.
? L = List(); ? listput(L, 1) %2 = 1 ? listput(L, 2) %3 = 2 ? L %4 = List([1, 2])
You may put an element into an occupied cell (not changing the
list length), but it is easier to use the standard list[n] = x
construct.
? listput(L, 3, 1) \\ insert at position 1 %5 = 3 ? L %6 = List([3, 2]) ? L[2] = 4 \\ simpler %7 = List([3, 4]) ? L[10] = 1 \\ can't insert beyond the end of the list *** at top-level: L[10]=1 *** ^------ *** non-existent component: index > 2 ? listput(L, 1, 10) \\ but listput can %8 = 1 ? L %9 = List([3, 2, 1])
This function runs in time O(#L)
in the worst case (when the list must
be reallocated), but in time O(1)
on average: any number of successive
listput
s run in time O(#L)
, where #L
denotes the list
final length.
The library syntax is GEN
listput0(GEN list, GEN x, long n)
.
(L,{
flag = 0})
Sorts the t_LIST
list in place, with respect to the (somewhat
arbitrary) universal comparison function cmp
. In particular, the
ordering is the same as for sets and setsearch
can be used on a sorted
list.
? L = List([1,2,4,1,3,-1]); listsort(L); L %1 = List([-1, 1, 1, 2, 3, 4]) ? setsearch(L, 4) %2 = 6 ? setsearch(L, -2) %3 = 0
@3This is faster than the vecsort
command since the list
is sorted in place: no copy is made. No value returned.
If flag is non-zero, suppresses all repeated coefficients.
The library syntax is void
listsort(GEN L, long flag)
.
(p)
Set the real precision to p
bits in the dynamic scope. All computations
are performed as if realbitprecision
was p
:
transcendental constants (e.g. Pi
) and
conversions from exact to floating point inexact data use p
bits, as well as
iterative routines implicitly using a floating point
accuracy as a termination criterion (e.g. solve
or intnum
).
But realbitprecision
itself is unaffected
and is ``unmasked'' when we exit the dynamic (not lexical) scope.
In effect, this is similar to
my(bit = default(realbitprecision)); default(realbitprecision,p); ... default(realbitprecision, bit);
@3but is both less cumbersome, cleaner (no need to manipulate
a global variable, which in fact never changes and is only temporarily masked)
and more robust: if the above computation is interrupted or an exception
occurs, realbitprecision
will not be restored as intended.
Such localbitprec
statements can be nested, the innermost one taking
precedence as expected. Beware that localbitprec
follows the semantic of
local
, not my
: a subroutine called from localbitprec
scope
uses the local accuracy:
? f()=bitprecision(1.0); ? f() %2 = 128 ? localbitprec(1000); f() %3 = 1024
@3Note that the bit precision of data (1.0
in the
above example) increases by steps of 64 (32 on a 32-bit machine) so we get
1024
instead of the expected 1000
; localbitprec
bounds the
relative error exactly as specified in functions that support that
granularity (e.g. lfun
), and rounded to the next multiple of 64
(resp. 32) everywhere else.
@3Warning. Changing realbitprecision
or realprecision
in programs is deprecated in favor of localbitprec
and
localprec
. Think about the realprecision
and
realbitprecision
defaults as interactive commands for the gp
interpreter, best left out of GP programs. Indeed, the above rules imply that
mixing both constructs yields surprising results:
? \p38 ? localprec(19); default(realprecision,1000); Pi %1 = 3.141592653589793239 ? \p realprecision = 1001 significant digits (1000 digits displayed)
@3Indeed, realprecision
itself is ignored within
localprec
scope, so Pi
is computed to a low accuracy. And when
we leave the localprec
scope, realprecision
only regains precedence,
it is not ``restored'' to the original value.
(p)
Set the real precision to p
in the dynamic scope. All computations
are performed as if realprecision
was p
:
transcendental constants (e.g. Pi
) and
conversions from exact to floating point inexact data use p
decimal
digits, as well as iterative routines implicitly using a floating point
accuracy as a termination criterion (e.g. solve
or intnum
).
But realprecision
itself is unaffected
and is ``unmasked'' when we exit the dynamic (not lexical) scope.
In effect, this is similar to
my(prec = default(realprecision)); default(realprecision,p); ... default(realprecision, prec);
@3but is both less cumbersome, cleaner (no need to manipulate
a global variable, which in fact never changes and is only temporarily masked)
and more robust: if the above computation is interrupted or an exception
occurs, realprecision
will not be restored as intended.
Such localprec
statements can be nested, the innermost one taking
precedence as expected. Beware that localprec
follows the semantic of
local
, not my
: a subroutine called from localprec
scope
uses the local accuracy:
? f()=precision(1.); ? f() %2 = 38 ? localprec(19); f() %3 = 19
@3Warning. Changing realprecision
itself in programs is
now deprecated in favor of localprec
. Think about the
realprecision
default as an interactive command for the gp
interpreter, best left out of GP programs. Indeed, the above rules
imply that mixing both constructs yields surprising results:
? \p38 ? localprec(19); default(realprecision,100); Pi %1 = 3.141592653589793239 ? \p realprecision = 115 significant digits (100 digits displayed)
@3Indeed, realprecision
itself is ignored within
localprec
scope, so Pi
is computed to a low accuracy. And when
we leave localprec
scope, realprecision
only regains precedence,
it is not ``restored'' to the original value.
(M,x)
Removes x
from the domain of the map M
.
? M = Map(["a",1; "b",3; "c",7]); ? mapdelete(M,"b"); ? Mat(M) ["a" 1]
["c" 7]
The library syntax is void
mapdelete(GEN M, GEN x)
.
(M,x)
Returns the image of x
by the map M
.
? M=Map(["a",23;"b",43]); ? mapget(M,"a") %2 = 23 ? mapget(M,"b") %3 = 43
@3Raises an exception when the key x
is not present in M
.
? mapget(M,"c") *** at top-level: mapget(M,"c") *** ^------------- *** mapget: non-existent component in mapget: index not in map
The library syntax is GEN
mapget(GEN M, GEN x)
.
(M,x,{&z})
Returns true (1
) if x
has an image by the map M
, false (0
)
otherwise. If z
is present, set z
to the image of x
, if it exists.
? M1 = Map([1, 10; 2, 20]); ? mapisdefined(M1,3) %1 = 0 ? mapisdefined(M1, 1, &z) %2 = 1 ? z %3 = 10
? M2 = Map(); N = 19; ? for (a=0, N-1, mapput(M2, a^3%N, a)); ? {for (a=0, N-1, if (mapisdefined(M2, a, &b), printf("%d is the cube of %d mod %d\n",a,b,N)));} 0 is the cube of 0 mod 19 1 is the cube of 11 mod 19 7 is the cube of 9 mod 19 8 is the cube of 14 mod 19 11 is the cube of 17 mod 19 12 is the cube of 15 mod 19 18 is the cube of 18 mod 19
The library syntax is GEN
mapisdefined(GEN M, GEN x, GEN *z = NULL)
.
(M,x,y)
Associates x
to y
in the map M
. The value y
can be retrieved
with mapget
.
? M = Map(); ? mapput(M, "foo", 23); ? mapput(M, 7718, "bill"); ? mapget(M, "foo") %4 = 23 ? mapget(M, 7718) %5 = "bill" ? Vec(M) \\ keys %6 = [7718, "foo"] ? Mat(M) %7 = [ 7718 "bill"]
["foo" 23]
The library syntax is void
mapput(GEN M, GEN x, GEN y)
.
print({
str}*)
Outputs its (string) arguments in raw format, ending with a newline.
print1({
str}*)
Outputs its (string) arguments in raw
format, without ending with a newline. Note that you can still embed newlines
within your strings, using the \n
notation !
(
fmt,{x}*)
This function is based on the C library command of the same name. It prints its arguments according to the format fmt, which specifies how subsequent arguments are converted for output. The format is a character string composed of zero or more directives:
@3* ordinary characters (not %
), printed unchanged,
@3* conversions specifications (%
followed by some characters)
which fetch one argument from the list and prints it according to the
specification.
More precisely, a conversion specification consists in a %
, one or more
optional flags (among #
, 0
, -
, +
, ` '), an optional
decimal digit string specifying a minimal field width, an optional precision
in the form of a period (`.
') followed by a decimal digit string, and
the conversion specifier (among d
,i
, o
, u
,
x
,X
, p
, e
,E
, f
, g
,G
, s
).
@3The flag characters. The character %
is followed by zero or
more of the following flags:
@3* #
: the value is converted to an ``alternate form''. For
o
conversion (octal), a 0
is prefixed to the string. For x
and X
conversions (hexa), respectively 0x
and 0X
are
prepended. For other conversions, the flag is ignored.
@3* 0
: the value should be zero padded. For
d
,
i
,
o
,
u
,
x
,
X
e
,
E
,
f
,
F
,
g
, and
G
conversions, the value is padded on the left with zeros rather than
blanks. (If the 0
and -
flags both appear, the 0
flag is
ignored.)
@3* -
: the value is left adjusted on the field boundary. (The
default is right justification.) The value is padded on the right with
blanks, rather than on the left with blanks or zeros. A -
overrides a
0
if both are given.
@3* ` '
(a space): a blank is left before a positive number
produced by a signed conversion.
@3* +
: a sign (+ or -) is placed before a number produced by a
signed conversion. A +
overrides a space if both are used.
@3The field width. An optional decimal digit string (whose first
digit is non-zero) specifying a minimum field width. If the value has
fewer characters than the field width, it is padded with spaces on the left
(or right, if the left-adjustment flag has been given). In no case does a
small field width cause truncation of a field; if the value is wider than
the field width, the field is expanded to contain the conversion result.
Instead of a decimal digit string, one may write *
to specify that the
field width is given in the next argument.
@3The precision. An optional precision in the form of a period
(`.
') followed by a decimal digit string. This gives
the number of digits to appear after the radix character for e
,
E
, f
, and F
conversions, the maximum number of significant
digits for g
and G
conversions, and the maximum number of
characters to be printed from an s
conversion.
Instead of a decimal digit string, one may write *
to specify that the
field width is given in the next argument.
@3The length modifier. This is ignored under gp
, but
necessary for libpari
programming. Description given here for
completeness:
@3* l
: argument is a long
integer.
@3* P
: argument is a GEN
.
@3The conversion specifier. A character that specifies the type of conversion to be applied.
@3* d
, i
: a signed integer.
@3* o
, u
, x
, X
: an unsigned integer, converted
to unsigned octal (o
), decimal (u
) or hexadecimal (x
or
X
) notation. The letters abcdef
are used for x
conversions; the letters ABCDEF
are used for X
conversions.
@3* e
, E
: the (real) argument is converted in the style
[ -]d.ddd e[ -]dd
, where there is one digit before the decimal point,
and the number of digits after it is equal to the precision; if the
precision is missing, use the current realprecision
for the total
number of printed digits. If the precision is explicitly 0, no decimal-point
character appears. An E
conversion uses the letter E
rather
than e
to introduce the exponent.
@3* f
, F
: the (real) argument is converted in the style
[ -]ddd.ddd
, where the number of digits after the decimal point
is equal to the precision; if the precision is missing, use the current
realprecision
for the total number of printed digits. If the precision
is explicitly 0, no decimal-point character appears. If a decimal point
appears, at least one digit appears before it.
@3* g
, G
: the (real) argument is converted in style
e
or f
(or E
or F
for G
conversions)
[ -]ddd.ddd
, where the total number of digits printed
is equal to the precision; if the precision is missing, use the current
realprecision
. If the precision is explicitly 0, it is treated as 1.
Style e
is used when
the decimal exponent is < -4
, to print 0.
, or when the integer
part cannot be decided given the known significant digits, and the f
format otherwise.
@3* c
: the integer argument is converted to an unsigned char, and the
resulting character is written.
@3* s
: convert to a character string. If a precision is given, no
more than the specified number of characters are written.
@3* p
: print the address of the argument in hexadecimal (as if by
%#x
).
@3* %
: a %
is written. No argument is converted. The complete
conversion specification is %%
.
@3Examples:
? printf("floor: %d, field width 3: %3d, with sign: %+3d\n", Pi, 1, 2); floor: 3, field width 3: 1, with sign: +2
? printf("%.5g %.5g %.5g\n",123,123/456,123456789); 123.00 0.26974 1.2346 e8
? printf("%-2.5s:%2.5s:%2.5s\n", "P", "PARI", "PARIGP"); P :PARI:PARIG
\\ min field width and precision given by arguments ? x = 23; y=-1/x; printf("x=%+06.2f y=%+0*.*f\n", x, 6, 2, y); x=+23.00 y=-00.04
\\ minimum fields width 5, pad left with zeroes ? for (i = 2, 5, printf("%05d\n", 10^i)) 00100 01000 10000 100000 \\ don't truncate fields whose length is larger than the minimum width ? printf("%.2f |%06.2f|", Pi,Pi) 3.14 | 3.14|
@3All numerical conversions apply recursively to the entries of vectors and matrices:
? printf("%4d", [1,2,3]); [ 1, 2, 3] ? printf("%5.2f", mathilbert(3)); [ 1.00 0.50 0.33]
[ 0.50 0.33 0.25]
[ 0.33 0.25 0.20]
@3Technical note. Our implementation of printf
deviates from the C89 and C99 standards in a few places:
@3* whenever a precision is missing, the current realprecision
is
used to determine the number of printed digits (C89: use 6 decimals after
the radix character).
@3* in conversion style e
, we do not impose that the
exponent has at least two digits; we never write a +
sign in the
exponent; 0 is printed in a special way, always as 0.E
exp.
@3* in conversion style f
, we switch to style e
if the
exponent is greater or equal to the precision.
@3* in conversion g
and G
, we do not remove trailing zeros
from the fractional part of the result; nor a trailing decimal point;
0 is printed in a special way, always as 0.E
exp.
printsep(
sep,{
str}*)
Outputs its (string) arguments in raw format, ending with a newline. Successive entries are separated by sep:
? printsep(":", 1,2,3,4) 1:2:3:4
printsep1(
sep,{
str}*)
Outputs its (string) arguments in raw format, without ending with a newline. Successive entries are separated by sep:
? printsep1(":", 1,2,3,4);print("|") 1:2:3:4
printtex({
str}*)
Outputs its (string) arguments in TeX format. This output can then be
used in a TeX manuscript.
The printing is done on the standard output. If you want to print it to a
file you should use writetex
(see there).
Another possibility is to enable the log
default
(see Label se:defaults).
You could for instance do:
default(logfile, "new.tex"); default(log, 1); printtex(result);
quit({
status = 0})
Exits gp
and return to the system with exit status
status
, a small integer. A non-zero exit status normally indicates
abnormal termination. (Note: the system actually sees only
status
mod 256
, see your man pages for exit(3)
or wait(2)
).
read({
filename})
Reads in the file
filename (subject to string expansion). If filename is
omitted, re-reads the last file that was fed into gp
. The return
value is the result of the last expression evaluated.
If a GP binary file
is read using this command (see
Label se:writebin), the file is loaded and the last object in the file
is returned.
In case the file you read in contains an allocatemem
statement (to be
generally avoided), you should leave read
instructions by themselves,
and not part of larger instruction sequences.
The library syntax is GEN
gp_read_file(const char *filename)
.
readstr({
filename})
Reads in the file filename and return a vector of GP strings,
each component containing one line from the file. If filename is
omitted, re-reads the last file that was fed into gp
.
The library syntax is GEN
readstr(const char *filename)
.
readvec({
filename})
Reads in the file
filename (subject to string expansion). If filename is
omitted, re-reads the last file that was fed into gp
. The return
value is a vector whose components are the evaluation of all sequences
of instructions contained in the file. For instance, if file contains
1 2 3
then we will get:
? \r a %1 = 1 %2 = 2 %3 = 3 ? read(a) %4 = 3 ? readvec(a) %5 = [1, 2, 3]
In general a sequence is just a single line, but as usual braces and
\
may be used to enter multiline sequences.
The library syntax is GEN
gp_readvec_file(const char *filename)
.
The underlying library function
GEN
gp_readvec_stream(FILE *f)
is usually more flexible.
(f, A, {
flag = 0})
We first describe the default behavior, when flag is 0 or omitted.
Given a vector or list A
and a t_CLOSURE
f
, select
returns the elements x
of A
such that f(x)
is non-zero. In other
words, f
is seen as a selection function returning a boolean value.
? select(x->isprime(x), vector(50,i,i^2+1)) %1 = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601] ? select(x->(x<100), %) %2 = [2, 5, 17, 37]
@3returns the primes of the form i^2+1
for some i <= 50
,
then the elements less than 100 in the preceding result. The select
function also applies to a matrix A
, seen as a vector of columns, i.e. it
selects columns instead of entries, and returns the matrix whose columns are
the selected ones.
@3Remark. For v
a t_VEC
, t_COL
, t_LIST
or t_MAT
,
the alternative set-notations
[g(x) | x <- v, f(x)] [x | x <- v, f(x)] [g(x) | x <- v]
are available as shortcuts for
apply(g, select(f, Vec(v))) select(f, Vec(v)) apply(g, Vec(v))
@3respectively:
? [ x | x <- vector(50,i,i^2+1), isprime(x) ] %1 = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601]
@3If flag = 1
, this function returns instead the indices of
the selected elements, and not the elements themselves (indirect selection):
? V = vector(50,i,i^2+1); ? select(x->isprime(x), V, 1) %2 = Vecsmall([1, 2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40]) ? vecextract(V, %) %3 = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601]
The following function lists the elements in (
Z/N
Z)^*
:
? invertibles(N) = select(x->gcd(x,N) == 1, [1..N])
@3Finally
? select(x->x, M)
@3selects the non-0 entries in M
. If the latter is a
t_MAT
, we extract the matrix of non-0 columns. Note that removing
entries instead of selecting them just involves replacing the selection
function f
with its negation:
? select(x->!isprime(x), vector(50,i,i^2+1))
The library syntax is genselect(void *E, long (*fun)(void*,GEN), GEN a)
. Also available
is GEN
genindexselect(void *E, long (*fun)(void*, GEN), GEN a)
,
corresponding to flag = 1
.
()
Return the calling function or closure as a t_CLOSURE
object.
This is useful for defining anonymous recursive functions.
? (n->if(n==0,1,n*self()(n-1)))(5) %1 = 120
The library syntax is GEN
pari_self()
.
(n)
Reseeds the random number generator using the seed n
. No value is
returned. The seed is either a technical array output by getrand
, or a
small positive integer, used to generate deterministically a suitable state
array. For instance, running a randomized computation starting by
setrand(1)
twice will generate the exact same output.
The library syntax is void
setrand(GEN n)
.
(
str)
str is a string representing a system command. This command is
executed, its output written to the standard output (this won't get into your
logfile), and control returns to the PARI system. This simply calls the C
system
command.
The library syntax is void
gpsystem(const char *str)
.
trap({e}, {
rec},
seq)
This function is obsolete, use iferr
, which has a nicer and much
more powerful interface. For compatibility's sake we now describe the
obsolete function trap
.
This function tries to
evaluate seq, trapping runtime error e
, that is effectively preventing
it from aborting computations in the usual way; the recovery sequence
rec is executed if the error occurs and the evaluation of rec
becomes the result of the command. If e
is omitted, all exceptions are
trapped. See Label se:errorrec for an introduction to error recovery
under gp
.
? \\ trap division by 0 ? inv(x) = trap (e_INV, INFINITY, 1/x) ? inv(2) %1 = 1/2 ? inv(0) %2 = INFINITY
Note that seq is effectively evaluated up to the point that produced the error, and the recovery sequence is evaluated starting from that same context, it does not "undo" whatever happened in the other branch (restore the evaluation context):
? x = 1; trap (, /* recover: */ x, /* try: */ x = 0; 1/x) %1 = 0
@3Note. The interface is currently not adequate for trapping
individual exceptions. In the current version 2.9.1, the following keywords
are recognized, but the name list will be expanded and changed in the
future (all library mode errors can be trapped: it's a matter of defining
the keywords to gp
):
e_ALARM
: alarm time-out
e_ARCH
: not available on this architecture or operating system
e_STACK
: the PARI stack overflows
e_INV
: impossible inverse
e_IMPL
: not yet implemented
e_OVERFLOW
: all forms of arithmetic overflow, including length
or exponent overflow (when a larger value is supplied than the
implementation can handle).
e_SYNTAX
: syntax error
e_MISC
: miscellaneous error
e_TYPE
: wrong type
e_USER
: user error (from the error
function)
The library syntax is GEN
trap0(const char *e = NULL, GEN rec = NULL, GEN seq = NULL)
.
(x)
This is useful only under gp
. Returns the internal type name of
the PARI object x
as a string. Check out existing type names with the
metacommand \t
. For example type(1)
will return "t_INT
".
The library syntax is GEN
type0(GEN x)
.
The macro typ
is usually simpler to use since it returns a
long
that can easily be matched with the symbols t_*
. The name
type
was avoided since it is a reserved identifier for some compilers.
()
(Experimental) Exit the scope of all current inline
variables.
()
Returns the current version number as a t_VEC
with three integer
components (major version number, minor version number and patchlevel);
if your sources were obtained through our version control system, this will
be followed by further more precise arguments, including
e.g. a git
commit hash.
This function is present in all versions of PARI following releases 2.3.4 (stable) and 2.4.3 (testing).
Unless you are working with multiple development versions, you probably only
care about the 3 first numeric components. In any case, the lex
function
offers a clever way to check against a particular version number, since it will
compare each successive vector entry, numerically or as strings, and will not
mind if the vectors it compares have different lengths:
if (lex(version(), [2,3,5]) >= 0, \\ code to be executed if we are running 2.3.5 or more recent. , \\ compatibility code );
@3On a number of different machines, version()
could return either of
%1 = [2, 3, 4] \\ released version, stable branch %1 = [2, 4, 3] \\ released version, testing branch %1 = [2, 6, 1, 15174, ""505ab9b"] \\ development
In particular, if you are only working with released versions, the first line of the gp introductory message can be emulated by
[M,m,p] = version(); printf("GP/PARI CALCULATOR Version %s.%s.%s", M,m,p);
@3If you are working with many development versions of PARI/GP, the 4th and/or 5th components can be profitably included in the name of your logfiles, for instance.
@3Technical note. For development versions obtained via git
,
the 4th and 5th components are liable to change eventually, but we document
their current meaning for completeness. The 4th component counts the number
of reachable commits in the branch (analogous to svn
's revision
number), and the 5th is the git
commit hash. In particular, lex
comparison still orders correctly development versions with respect to each
others or to released versions (provided we stay within a given branch,
e.g. master
)!
The library syntax is GEN
pari_version()
.
warning({
str}*)
Outputs the message ``user warning'' and the argument list (each of them interpreted as a string). If colors are enabled, this warning will be in a different color, making it easy to distinguish.
warning(n, " is very large, this might take a while.")
(
key)
If keyword key is the name of a function that was present in GP version 1.39.15, outputs the new function name and syntax, if it changed at all. Functions that where introduced since then, then modified are also recognized.
? whatnow("mu") New syntax: mu(n) ===> moebius(n)
moebius(x): Moebius function of x.
? whatnow("sin") This function did not change
When a function was removed and the underlying functionality is not available under a compatible interface, no equivalent is mentioned:
? whatnow("buchfu") This function no longer exists
@3(The closest equivalent would be to set K = bnfinit(T)
then access K.fu
.)
write(
filename,{
str}*)
Writes (appends) to filename the remaining arguments, and appends a
newline (same output as print
).
write1(
filename,{
str}*)
Writes (appends) to filename the remaining arguments without a
trailing newline (same output as print1
).
(
filename,{x})
Writes (appends) to
filename the object x
in binary format. This format is not human
readable, but contains the exact internal structure of x
, and is much
faster to save/load than a string expression, as would be produced by
write
. The binary file format includes a magic number, so that such a
file can be recognized and correctly input by the regular read
or \r
function. If saved objects refer to polynomial variables that are not
defined in the new session, they will be displayed as tn
for some
integer n
(the attached variable number).
Installed functions and history objects can not be saved via this function.
If x
is omitted, saves all user variables from the session, together with
their names. Reading such a ``named object'' back in a gp
session will set
the corresponding user variable to the saved value. E.g after
x = 1; writebin("log")
reading log
into a clean session will set x
to 1
.
The relative variables priorities (see Label se:priority) of new variables
set in this way remain the same (preset variables retain their former
priority, but are set to the new value). In particular, reading such a
session log into a clean session will restore all variables exactly as they
were in the original one.
Just as a regular input file, a binary file can be compressed
using gzip
, provided the file name has the standard .gz
extension.
In the present implementation, the binary files are architecture dependent
and compatibility with future versions of gp
is not guaranteed. Hence
binary files should not be used for long term storage (also, they are
larger and harder to compress than text files).
The library syntax is void
gpwritebin(const char *filename, GEN x = NULL)
.
writetex(
filename,{
str}*)
As write
, in TeX format.
These function are only available if PARI was configured using
Configure --mt = ...
. Two multithread interfaces are supported:
@3* POSIX threads
@3* Message passing interface (MPI)
As a rule, POSIX threads are well-suited for single systems, while MPI is used by most clusters. However the parallel GP interface does not depend on the chosen multithread interface: a properly written GP program will work identically with both.
(f, x)
Parallel evaluation of f
on the elements of x
.
The function f
must not access global variables or variables
declared with local()
, and must be free of side effects.
parapply(factor,[2^256 + 1, 2^193 - 1])
factors 2^{256} + 1
and 2^{193} - 1
in parallel.
{ my(E = ellinit([1,3]), V = vector(12,i,randomprime(2^200))); parapply(p->ellcard(E,p), V) }
computes the order of E(
F_p)
for 12
random primes of 200
bits.
The library syntax is GEN
parapply(GEN f, GEN x)
.
(x)
Parallel evaluation of the elements of x
, where x
is a
vector of closures. The closures must be of arity 0
, must not access
global variables or variables declared with local
and must be
free of side effects.
The library syntax is GEN
pareval(GEN x)
.
parfor(i = a,{b},
expr1,{r},{
expr2})
Evaluates in parallel the expression expr1
in the formal
argument i
running from a
to b
.
If b
is set to +oo
, the loop runs indefinitely.
If r
and expr2
are present, the expression expr2
in the
formal variables r
and i
is evaluated with r
running through all
the different results obtained for expr1
and i
takes the
corresponding argument.
The computations of expr1
are started in increasing order
of i
; otherwise said, the computation for i = c
is started after those
for i = 1,..., c-1
have been started, but before the computation for
i = c+1
is started. Notice that the order of completion, that is,
the order in which the different r
become available, may be different;
expr2
is evaluated sequentially on each r
as it appears.
The following example computes the sum of the squares of the integers
from 1
to 10
by computing the squares in parallel and is equivalent
to parsum (i = 1, 10, i^2)
:
? s=0; ? parfor (i=1, 10, i^2, r, s=s+r) ? s %3 = 385
More precisely, apart from a potentially different order of evaluation
due to the parallelism, the line containing parfor
is equivalent to
? my (r); for (i=1, 10, r=i^2; s=s+r)
The sequentiality of the evaluation of expr2
ensures that the
variable s
is not modified concurrently by two different additions,
although the order in which the terms are added is non-deterministic.
It is allowed for expr2
to exit the loop using
break
/next
/return
. If that happens for i = c
,
then the evaluation of expr1
and expr2
is continued
for all values i < c
, and the return value is the one obtained for
the smallest i
causing an interruption in expr2
(it may be
undefined if this is a break
/next
).
In that case, using side-effects
in expr2
may lead to undefined behavior, as the exact
number of values of i
for which it is executed is non-deterministic.
The following example computes nextprime(1000)
in parallel:
? parfor (i=1000, , isprime (i), r, if (r, return (i))) %1 = 1009
parforprime(p = a,{b},
expr1,{r},{
expr2})
Behaves exactly as parfor
, but loops only over prime values p
.
Precisely, the functions evaluates in parallel the expression expr1
in the formal
argument p
running through the primes from a
to b
.
If b
is set to +oo
, the loop runs indefinitely.
If r
and expr2
are present, the expression expr2
in the
formal variables r
and p
is evaluated with r
running through all
the different results obtained for expr1
and p
takes the
corresponding argument.
It is allowed fo expr2
to exit the loop using
break
/next
/return
; see the remarks in the documentation
of parfor
for details.
parforvec(X = v,
expr1,{j},{
expr2},{
flag})
Evaluates the sequence expr2
(dependent on X
and j
) for X
as generated by forvec
, in random order, computed in parallel. Substitute
for j
the value of expr1
(dependent on X
).
It is allowed fo expr2
to exit the loop using
break
/next
/return
, however in that case, expr2
will
still be evaluated for all remaining value of p
less than the current one,
unless a subsequent break
/next
/return
happens.
(f, A, {
flag = 0})
Selects elements of A
according to the selection function f
, done in
parallel. If flag is 1
, return the indices of those elements (indirect
selection) The function f
must not access global variables or
variables declared with local()
, and must be free of side effects.
The library syntax is GEN
parselect(GEN f, GEN A, long flag)
.
(i = a,b,
expr,{x})
Sum of expression expr, initialized at x
, the formal parameter
going from a
to b
, evaluated in parallel in random order.
The expression expr
must not access global variables or
variables declared with local()
, and must be free of side effects.
parsum(i=1,1000,ispseudoprime(2^prime(i)-1))
returns the numbers of prime numbers among the first 1000
Mersenne numbers.
(N,i,
expr)
As vector(N,i,expr)
but the evaluations of expr
are done in
parallel. The expression expr
must not access global variables or
variables declared with local()
, and must be free of side effects.
parvector(10,i,quadclassunit(2^(100+i)+1).no)
computes the class numbers in parallel.
This section documents the GP defaults, be sure to
check out parisize
and parisizemax
!
The bits of this default allow
gp
to use less rigid TeX formatting commands in the logfile. This
default is only taken into account when log = 3
. The bits of
TeXstyle
have the following meaning
2: insert \right
/ \left
pairs where appropriate.
4: insert discretionary breaks in polynomials, to enhance the probability of a good line break.
The default value is 0
.
If true, enables the ``break loop'' debugging mode, see Label se:break_loop.
The default value is 1
if we are running an interactive gp
session, and 0
otherwise.
This default is only usable if gp
is running within certain color-capable terminals. For instance rxvt
,
color_xterm
and modern versions of xterm
under X Windows, or
standard Linux/DOS text consoles. It causes gp
to use a small palette of
colors for its output. With xterms, the colormap used corresponds to the
resources Xterm*colorn
where n
ranges from 0
to 15
(see the
file misc/color.dft
for an example). Accepted values for this
default are strings "a_1,...,a_k"
where k <= 7
and each
a_i
is either
@3* the keyword no
(use the default color, usually
black on transparent background)
@3* an integer between 0 and 15 corresponding to the aforementioned colormap
@3* a triple [c_0,c_1,c_2]
where c_0
stands for foreground
color, c_1
for background color, and c_2
for attributes (0 is default, 1
is bold, 4 is underline).
The output objects thus affected are respectively error messages,
history numbers, prompt, input line, output, help messages, timer (that's
seven of them). If k < 7
, the remaining a_i
are assumed to be no
. For
instance
default(colors, "9, 5, no, no, 4")
typesets error messages in color 9
, history numbers in color 5
, output in
color 4
, and does not affect the rest.
A set of default colors for dark (reverse video or PC console) and light
backgrounds respectively is activated when colors
is set to
darkbg
, resp. lightbg
(or any proper prefix: d
is
recognized as an abbreviation for darkbg
). A bold variant of
darkbg
, called boldfg
, is provided if you find the former too
pale.
In the present version, this default is incompatible with PariEmacs. Changing it will just fail silently (the alternative would be to display escape sequences as is, since Emacs will refuse to interpret them). You must customize color highlighting from the PariEmacs side, see its documentation.
The default value is ""
(no colors).
Obsolete. This default is now a no-op.
The name of directory containing the optional data files. For now,
this includes the elldata
, galdata
, galpol
, seadata
packages.
The default value is /usr/local/share/pari
, or the override specified
via Configure --datadir =
.
Debugging level. If it is non-zero, some extra messages may be printed,
according to what is going on (see \g
).
The default value is 0
(no debugging messages).
File usage debugging level. If it is non-zero, gp
will print
information on file descriptors in use, from PARI's point of view
(see \gf
).
The default value is 0
(no debugging messages).
Memory debugging level. If it is non-zero, gp
will regularly print
information on memory usage. If it's greater than 2, it will indicate any
important garbage collecting and the function it is taking place in
(see \gm
).
@3Important Note: As it noticeably slows down the performance, the first functionality (memory usage) is disabled if you're not running a version compiled for debugging (see Appendix A).
The default value is 0
(no debugging messages).
This toggle is either 1 (on) or 0 (off). When echo
mode is on, each command is reprinted before being executed. This can be
useful when reading a file with the \r
or read
commands. For
example, it is turned on at the beginning of the test files used to check
whether gp
has been built correctly (see \e
).
The default value is 0
(no echo).
This toggle is either 1 (on) or 0 (off). If on,
the integer factorization machinery calls addprimes
on prime
factors that were difficult to find (larger than 2^{24}
), so they are
automatically tried first in other factorizations. If a routine is performing
(or has performed) a factorization and is interrupted by an error or via
Control-C, this lets you recover the prime factors already found. The
downside is that a huge addprimes
table unrelated to the current
computations will slow down arithmetic functions relying on integer
factorization; one should then empty the table using removeprimes
.
The default value is 0
.
This toggle is either 1 (on) or 0 (off). By
default, the factors output by the integer factorization machinery are
only pseudo-primes, not proven primes. If this toggle is
set, a primality proof is done for each factor and all results depending on
integer factorization are fully proven. This flag does not affect partial
factorization when it is explicitly requested. It also does not affect the
private table managed by addprimes
: its entries are included as is in
factorizations, without being tested for primality.
The default value is 0
.
Of the form x.n
, where x (conversion style)
is a letter in {e,f,g}
, and n
(precision) is an
integer; this affects the way real numbers are printed:
@3* If the conversion style is e
, real numbers are printed in
scientific format, always with an explicit exponent,
e.g. 3.3 E-5
.
@3* In style f
, real numbers are generally printed in
fixed floating point format without exponent, e.g. 0.000033
. A
large real number, whose integer part is not well defined (not enough
significant digits), is printed in style e
. For instance
10.^100
known to ten significant digits is always printed in style
e
.
@3* In style g
, non-zero real numbers are printed in f
format,
except when their decimal exponent is < -4
, in which case they are printed
in e
format. Real zeroes (of arbitrary exponent) are printed in e
format.
The precision n
is the number of significant digits printed for real
numbers, except if n < 0
where all the significant digits will be printed
(initial default 28, or 38 for 64-bit machines). For more powerful formatting
possibilities, see printf
and Strprintf
.
The default value is "g.28"
and "g.38"
on 32-bit and
64-bit machines, respectively.
A vector of colors, to be
used by hi-res graphing routines. Its length is arbitrary, but it must
contain at least 3 entries: the first 3 colors are used for background,
frame/ticks and axes respectively. All colors in the colormap may be freely
used in plotcolor
calls.
A color is either given as in the default by character strings or by an RGB
code. For valid character strings, see the standard rgb.txt
file in X11
distributions, where we restrict to lowercase letters and remove all
whitespace from color names. An RGB code is a vector with 3 integer entries
between 0 and 255. For instance [250, 235, 215]
and
"antiquewhite"
represent the same color. RGB codes are cryptic but
often easier to generate.
The default value is ["white"
, "black"
, "blue"
,
"violetred"
, "red"
, "green"
, "grey"
,
"gainsboro"
].
Entries in the
graphcolormap
that will be used to plot multi-curves. The successive
curves are drawn in colors
graphcolormap[graphcolors[1]]
, graphcolormap[graphcolors[2]]
,
...
cycling when the graphcolors
list is exhausted.
The default value is [4,5]
.
Name of the external help program to use from within gp
when
extended help is invoked, usually through a ??
or ???
request
(see Label se:exthelp), or M-H
under readline (see
Label se:readline).
The default value is the path to the gphelp
script we install.
Name of a file where
gp
will keep a history of all input commands (results are
omitted). If this file exists when the value of histfile
changes,
it is read in and becomes part of the session history. Thus, setting this
default in your gprc saves your readline history between sessions. Setting
this default to the empty string ""
changes it to
< undefined >
The default value is < undefined >
(no history file).
gp
keeps a history of the last
histsize
results computed so far, which you can recover using the
%
notation (see Label se:history). When this number is exceeded,
the oldest values are erased. Tampering with this default is the only way to
get rid of the ones you do not need anymore.
The default value is 5000
.
If set to a positive value, gp
prints at
most that many lines from each result, terminating the last line shown with
[+++]
if further material has been suppressed. The various print
commands (see Label se:gp_program) are unaffected, so you can always type
print(%)
or \a
to view the full result. If the actual screen width
cannot be determined, a ``line'' is assumed to be 80 characters long.
The default value is 0
.
If set to a positive value, gp
wraps every single line after
printing that many characters.
The default value is 0
(unset).
This can be either 0 (off) or 1, 2, 3
(on, see below for the various modes). When logging mode is turned on, gp
opens a log file, whose exact name is determined by the logfile
default. Subsequently, all the commands and results will be written to that
file (see \l
). In case a file with this precise name already existed, it
will not be erased: your data will be appended at the end.
The specific positive values of log
have the following meaning
1: plain logfile
2: emit color codes to the logfile (if colors
is set).
3: write LaTeX output to the logfile (can be further customized using
TeXstyle
).
The default value is 0
.
Name of the log file to be used when the log
toggle is on.
Environment and time expansion are performed.
The default value is "pari.log"
.
Number of threads to use for parallel computing.
The exact meaning an default depend on the mt
engine used:
@3* single
: not used (always one thread).
@3* pthread
: number of threads (unlimited, default: number of core)
@3* mpi
: number of MPI process to use (limited to the number allocated by mpirun
,
default: use all allocated process).
This toggle is either 1 (on) or 0 (off). If on,
the polgalois
command will use a different, more
consistent, naming scheme for Galois groups. This default is provided to
ensure that scripts can control this behavior and do not break unexpectedly.
The default value is 0
. This value will change to 1
(set) in the next
major version.
There are three possible values: 0
( = raw), 1 ( = prettymatrix), or 3
( = external prettyprint). This
means that, independently of the default format
for reals which we
explained above, you can print results in three ways:
@3* raw format, i.e. a format which is equivalent to what you input, including explicit multiplication signs, and everything typed on a line instead of two dimensional boxes. This can have several advantages, for instance it allows you to pick the result with a mouse or an editor, and to paste it somewhere else.
@3* prettymatrix format: this is identical to raw format, except that matrices are printed as boxes instead of horizontally. This is prettier, but takes more space and cannot be used for input. Column vectors are still printed horizontally.
@3* external prettyprint: pipes all gp
output in TeX format to an external prettyprinter, according to the value of
prettyprinter
. The default script (tex2mail
) converts its input
to readable two-dimensional text.
Independently of the setting of this default, an object can be printed
in any of the three formats at any time using the commands \a
and \m
and \B
respectively.
The default value is 1
(prettymatrix).
gp
, and in fact any program using the PARI
library, needs a stack in which to do its computations; parisize
is the stack size, in bytes. It is recommended to increase this
default using a gprc
, to the value you believe PARI should be happy
with, given your typical computation. We strongly recommend to also
set parisizemax
to a much larger value, about what you believe your
machine can stand: PARI will then try to fit its computations within about
parisize
bytes, but will increase the stack size if needed (up to
parisizemax
). Once the memory intensive computation is over, PARI
will restore the stack size to the originally requested parisize
.
The default value is 4M, resp. 8M on a 32-bit, resp. 64-bit machine.
gp
, and in fact any program using the PARI library, needs a
stack in which to do its computations. If non-zero, parisizemax
is the maximum size the stack can grow to, in bytes. If zero, the stack will
not automatically grow, and will be limited to the value of parisize
.
We strongly recommend to set parisizemax
to a non-zero value, about
what you believe your machine can stand: PARI will then try to fit its
computations within about parisize
bytes, but will increase the stack
size if needed (up to parisizemax
). Once the memory intensive
computation is over, PARI will restore the stack size to the originally
requested parisize
.
The default value is 0
.
This is a list of directories, separated by colons ':'
(semicolons ';'
in the DOS world, since colons are preempted for drive names).
When asked to read a file whose name is not given by an absolute path
(does not start with /
, ./
or ../
), gp
will look for
it in these directories, in the order they were written in path
. Here,
as usual, .
means the current directory, and ..
its immediate
parent. Environment expansion is performed.
The default value is ".:~:~/gp"
on UNIX systems,
".;C:\;C:\GP"
on DOS, OS/2 and Windows, and "."
otherwise.
The name of an external prettyprinter to use when
output
is 3 (alternate prettyprinter). Note that the default
tex2mail
looks much nicer than the built-in ``beautified
format'' (output = 2
).
The default value is "tex2mail -TeX -noindent -ragged -by_par"
.
gp
precomputes a list of
all primes less than primelimit
at initialization time, and can build
fast sieves on demand to quickly iterate over primes up to the square
of primelimit
. These are used by many arithmetic functions, usually for
trial division purposes. The maximal value is 2^{32} - 2049
(resp 2^{64} -
2049
) on a 32-bit (resp. 64-bit) machine, but values beyond 10^8
,
allowing to iterate over primes up to 10^{16}
, do not seem useful.
Since almost all arithmetic functions eventually require some table of prime
numbers, PARI guarantees that the first 6547 primes, up to and
including 65557, are precomputed, even if primelimit
is 1
.
This default is only used on startup: changing it will not recompute a new table.
@3Deprecated feature. primelimit
was used in some
situations by algebraic number theory functions using the
nf_PARTIALFACT
flag (nfbasis
, nfdisc
, nfinit
,...):
this assumes that all primes p > primelimit
have a certain
property (the equation order is p
-maximal). This is never done by default,
and must be explicitly set by the user of such functions. Nevertheless,
these functions now provide a more flexible interface, and their use
of the global default primelimit
is deprecated.
@3Deprecated feature. factor(N, 0)
was used to partially
factor integers by removing all prime factors <=
primelimit
.
Don't use this, supply an explicit bound: factor(N, bound)
,
which avoids relying on an unpredictable global variable.
The default value is 500k
.
A string that will be printed as
prompt. Note that most usual escape sequences are available there: \e
for
Esc, \n
for Newline,..., \\
for \
. Time expansion is
performed.
This string is sent through the library function strftime
(on a
Unix system, you can try man strftime
at your shell prompt). This means
that %
constructs have a special meaning, usually related to the time
and date. For instance, %H
= hour (24-hour clock) and %M
=
minute [00,59] (use %%
to get a real %
).
If you use readline
, escape sequences in your prompt will result in
display bugs. If you have a relatively recent readline
(see the comment
at the end of \secref{se:def,colors}), you can brace them with special sequences
(\[
and \]
), and you will be safe. If these just result in
extra spaces in your prompt, then you'll have to get a more recent
readline
. See the file misc/gprc.dft
for an example.
S< >Caution: PariEmacs needs to know about the prompt pattern to
separate your input from previous gp
results, without ambiguity. It is
not a trivial problem to adapt automatically this regular expression to an
arbitrary prompt (which can be self-modifying!). See PariEmacs's
documentation.
The default value is "? "
.
A string that will be printed
to prompt for continuation lines (e.g. in between braces, or after a
line-terminating backslash). Everything that applies to prompt
applies to prompt_cont
as well.
The default value is ""
.
Name of the default file where
gp
is to dump its PostScript drawings (these are appended, so that no
previous data are lost). Environment and time expansion are performed.
The default value is "pari.ps"
.
Switches readline line-editing
facilities on and off. This may be useful if you are running gp
in a Sun
cmdtool
, which interacts badly with readline. Of course, until readline
is switched on again, advanced editing features like automatic completion
and editing history are not available.
The default value is 1
.
The number of significant bits used to convert exact inputs given to
transcendental functions (see Label se:trans), or to create
absolute floating point constants (input as 1.0
or Pi
for
instance). Unless you tamper with the format
default, this is also
the number of significant bits used to print a t_REAL
number;
format
will override this latter behaviour, and allow you to have a
large internal precision while outputting few digits for instance.
Note that most PARI's functions currently handle precision on a word basis (by
increments of 32 or 64 bits), hence bit precision may be a little larger
than the number of bits you expected. For instance to get 10 bits of
precision, you need one word of precision which, on a 64-bit machine,
correspond to 64 bits. To make things even more confusing, this internal bit
accuracy is converted to decimal digits when printing floating point numbers:
now 64 bits correspond to 19 printed decimal digits
(19 <
log _{10}(2^{64}) < 20
).
The value returned when typing default(realbitprecision)
is the internal
number of significant bits, not the number of printed decimal digits:
? default(realbitprecision, 10) ? \pb realbitprecision = 64 significant bits ? default(realbitprecision) %1 = 64 ? \p realprecision = 3 significant digits ? default(realprecision) %2 = 19
@3Note that realprecision
and \p
allow
to view and manipulate the internal precision in decimal digits.
The default value is 128
, resp. 96
, on a 64-bit, resp .32-bit,
machine.
The number of significant digits used to convert exact inputs given to
transcendental functions (see Label se:trans), or to create
absolute floating point constants (input as 1.0
or Pi
for
instance). Unless you tamper with the format
default, this is also
the number of significant digits used to print a t_REAL
number;
format
will override this latter behaviour, and allow you to have a
large internal precision while outputting few digits for instance.
Note that PARI's internal precision works on a word basis (by increments of
32 or 64 bits), hence may be a little larger than the number of decimal
digits you expected. For instance to get 2 decimal digits you need one word
of precision which, on a 64-bit machine, actually gives you 19 digits (19 <
log _{10}(2^{64}) < 20
). The value returned when typing
default(realprecision)
is the internal number of significant digits,
not the number of printed digits:
? default(realprecision, 2) realprecision = 19 significant digits (2 digits displayed) ? default(realprecision) %1 = 19
The default value is 38
, resp. 28
, on a 64-bit, resp. 32-bit,
machine.
This toggle is either 1 (on) or 0 (off). If you change this to 0
, any
error becomes fatal and causes the gp interpreter to exit immediately. Can be
useful in batch job scripts.
The default value is 1
.
This toggle is either 1 (on) or 0 (off). If on, the system
and
extern
command are disabled. These two commands are potentially
dangerous when you execute foreign scripts since they let gp
execute
arbitrary UNIX commands. gp
will ask for confirmation before letting
you (or a script) unset this toggle.
The default value is 0
.
Number of significant terms
when converting a polynomial or rational function to a power series
(see \ps
).
The default value is 16
.
This toggle is either 1 (on) or 0 (off). When the PARI library computes
something, the type of the
result is not always the simplest possible. The only type conversions which
the PARI library does automatically are rational numbers to integers (when
they are of type t_FRAC
and equal to integers), and similarly rational
functions to polynomials (when they are of type t_RFRAC
and equal to
polynomials). This feature is useful in many cases, and saves time, but can
be annoying at times. Hence you can disable this and, whenever you feel like
it, use the function simplify
(see Chapter 3) which allows you to
simplify objects to the simplest possible types recursively (see \y
).
The default value is 1
.
This is a list of directories, separated by colons ':'
(semicolons ';'
in the DOS world, since colons are preempted for drive names).
When asked to install
an external symbol from a shared library whose
name is not given by an absolute path (does not start with /
, ./
or ../
), gp
will look for it in these directories, in the order
they were written in sopath
. Here, as usual, .
means the current
directory, and ..
its immediate parent. Environment expansion is
performed.
The default value is ""
, corresponding to an empty list of
directories: install
will use the library name as input (and look in
the current directory if the name is not an absolute path).
This toggle is either 1 (on) or 0 (off). If on, all arguments to new
user functions are mandatory unless the function supplies an explicit default
value.
Otherwise arguments have the default value 0
.
In this example,
fun(a,b=2)=a+b
a
is mandatory, while b
is optional. If strictargs
is on:
? fun() *** at top-level: fun() *** ^----- *** in function fun: a,b=2 *** ^----- *** missing mandatory argument 'a' in user function.
This applies to functions defined while strictargs
is on. Changing strictargs
does not affect the behavior of previously defined functions.
The default value is 0
.
Obsolete. This toggle is now a no-op.
In parallel mode, each thread needs its own private stack in which
to do its computations, see parisize
. This value determines the size
in bytes of the stacks of each thread, so the total memory allocated will be
parisize+nbthreads x threadsize
.
If set to 0
, the value used is the same as parisize
.
The default value is 0
.
In parallel mode, each threads needs its own private stack in which
to do its computations, see parisize
. This value determines the maximal
size in bytes of the stacks of each thread, so the total memory allocated will
be between parisize+nbthreads x threadsize
. and
parisize+nbthreads x threadsizemax
.
If set to 0
, the value used is the same as threadsize
.
The default value is 0
.
This toggle is either 1 (on) or 0 (off). Every instruction sequence
in the gp calculator (anything ended by a newline in your input) is timed,
to some accuracy depending on the hardware and operating system. When
timer
is on, each such timing is printed immediately before the
output as follows:
? factor(2^2^7+1) time = 108 ms. \\ this line omitted if 'timer' is 0 %1 = [ 59649589127497217 1]
[5704689200685129054721 1]
@3(See also #
and ##
.)
The time measured is the user CPU time, not including the time
for printing the results. If the time is negligible ( < 1
ms.), nothing is
printed: in particular, no timing should be printed when defining a user
function or an alias, or installing a symbol from the library.
The default value is 0
(off).