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solve -- solve a linear equation

Synopsis

Description

(Disambiguation: for division of matrices, which can also be thought of as solving a system of linear equations, see instead Matrix // Matrix. For lifting a map between modules to a map between their free resolutions, see extend.)

There are several restrictions. The first is that there are only a limited number of rings for which this function is implemented. Second, over RR or CC, the matrix A must be a square non-singular matrix. Third, if A and b are mutable matrices over RR or CC, they must be dense matrices.
i1 : kk = ZZ/101;
i2 : A = matrix"1,2,3,4;1,3,6,10;19,7,11,13" ** kk

o2 = | 1  2 3  4  |
     | 1  3 6  10 |
     | 19 7 11 13 |

              3        4
o2 : Matrix kk  <--- kk
i3 : b = matrix"1;1;1" ** kk

o3 = | 1 |
     | 1 |
     | 1 |

              3        1
o3 : Matrix kk  <--- kk
i4 : x = solve(A,b)

o4 = | 2  |
     | -1 |
     | 34 |
     | 0  |

              4        1
o4 : Matrix kk  <--- kk
i5 : A*x-b

o5 = 0

              3        1
o5 : Matrix kk  <--- kk
Over RR or CC, the matrix A must be a non-singular square matrix.
i6 : printingPrecision = 2;
i7 : A = matrix "1,2,3;1,3,6;19,7,11" ** RR

o7 = | 1  2 3  |
     | 1  3 6  |
     | 19 7 11 |

                3          3
o7 : Matrix RR    <--- RR
              53         53
i8 : b = matrix "1;1;1" ** RR

o8 = | 1 |
     | 1 |
     | 1 |

                3          1
o8 : Matrix RR    <--- RR
              53         53
i9 : x = solve(A,b)

o9 = | -.15 |
     | 1.1  |
     | -.38 |

                3          1
o9 : Matrix RR    <--- RR
              53         53
i10 : A*x-b

o10 = | 2.2e-16  |
      | -2.2e-16 |
      | 0        |

                 3          1
o10 : Matrix RR    <--- RR
               53         53
i11 : norm oo

o11 = 2.22044604925031e-16

o11 : RR (of precision 53)
For large dense matrices over RR or CC, this function calls the lapack routines.
i12 : n = 10;
i13 : A = random(CC^n,CC^n)

o13 = | .78+.72i .036+.28i .22+.77i .33+.45i   .63+.55i .36+.4i   .35+.038i  
      | .7+.39i  .38+.39i  .4+.41i  .04+.076i  .34+.81i .54+.47i  .22+.23i   
      | .26+.35i .36+.85i  .91+.97i .037+.25i  .65+.24i .73+.51i  .33+.54i   
      | .11+.41i .53+.95i  .23+.21i .069+.053i .6+.65i  .57+.41i  .72+.95i   
      | .06+.97i .02+.95i  .63+.24i .67+.38i   .74+.33i .93+.1i   .0085+.038i
      | .96+.42i .84+.17i  .75+.45i .88+.93i   .77+.66i .15+.97i  .23+.85i   
      | .58+.56i .58+.14i  .58+.59i .72+.36i   .87+.34i .091+.19i .71+.33i   
      | .22+.67i .66+.88i  .23+.79i .92+.54i   .65+.3i  .86+.85i  .92+.92i   
      | .19+.82i .5+.44i   .71+.97i 1+.31i     .93+.89i .52+.18i  .12+.25i   
      | .88+.8i  .36+.43i  .77+.01i .56+.54i   .7+.28i  .79+.78i  .71+.15i   
      -----------------------------------------------------------------------
      .71+.96i    .34+.47i .46+.82i  |
      .71+.39i    .21+.53i .072+.16i |
      .63+.36i    .52+.72i .45+.91i  |
      .22+.44i    .23+.87i .53+.93i  |
      .059+.0085i .89+.93i .72+.26i  |
      .21+.42i    .41+.24i .33+.59i  |
      .67+.18i    .29+.17i .2+.6i    |
      .86+.05i    .51+.47i .65+.43i  |
      .85+.84i    .5+.53i  .27+.41i  |
      .14+.68i    .6+.13i  .82+.85i  |

                 10          10
o13 : Matrix CC     <--- CC
               53          53
i14 : b = random(CC^n,CC^2)

o14 = | .69+.98i  .33+.68i  |
      | .94+.34i  .83+.25i  |
      | .14+.44i  .64+.98i  |
      | .39+.013i .75+.81i  |
      | .054+.4i  .22+.44i  |
      | .49+.25i  .58+.48i  |
      | .1+.41i   .1+.67i   |
      | .26+.32i  .15+.058i |
      | .17+.14i  .16+.3i   |
      | .96+.28i  .4+.46i   |

                 10          2
o14 : Matrix CC     <--- CC
               53          53
i15 : x = solve(A,b)

o15 = | .45+.68i  .2+.81i   |
      | -.57-.15i .71-.19i  |
      | -.23+.11i .31+.85i  |
      | .14+.77i  .25+.027i |
      | -1-2.1i   -1.1+.21i |
      | -.31-.41i -.63-.34i |
      | .15+.33i  .24+.46i  |
      | .3+.67i   .66-.52i  |
      | 1.7+1.1i  1.1-1.4i  |
      | .7-.86i   -.41-.25i |

                 10          2
o15 : Matrix CC     <--- CC
               53          53
i16 : norm ( matrix A * matrix x - matrix b )

o16 = 9.55049957678547e-16

o16 : RR (of precision 53)
This may be used to invert a matrix over ZZ/p, RR or QQ.
i17 : A = random(RR^5, RR^5)

o17 = | .67 .88   .45  .86 .17 |
      | .96 .78   .97  .64 .27 |
      | .99 .0062 .88  .22 .6  |
      | .11 .41   .89  .63 .13 |
      | .65 .99   .035 .75 .89 |

                 5          5
o17 : Matrix RR    <--- RR
               53         53
i18 : I = id_(target A)

o18 = | 1 0 0 0 0 |
      | 0 1 0 0 0 |
      | 0 0 1 0 0 |
      | 0 0 0 1 0 |
      | 0 0 0 0 1 |

                 5          5
o18 : Matrix RR    <--- RR
               53         53
i19 : A' = solve(A,I)

o19 = | 1.3  -.097 .75  -1.3 -.54  |
      | -2.1 2.8   -1.8 -.27 .8    |
      | -1.3 1.1   -.26 .83  -.039 |
      | 3.3  -3.3  1.3  .71  -.6   |
      | -1.3 -.3   .35  .59  1.1   |

                 5          5
o19 : Matrix RR    <--- RR
               53         53
i20 : norm(A*A' - I)

o20 = 6.66133814775094e-16

o20 : RR (of precision 53)
i21 : norm(A'*A - I)

o21 = 8.88178419700125e-16

o21 : RR (of precision 53)
Another method, which isn't generally as fast, and isn't as stable over RR or CC, is to lift the matrix b along the matrix A (see Matrix // Matrix).
i22 : A'' = I // A

o22 = | 1.3  -.097 .75  -1.3 -.54  |
      | -2.1 2.8   -1.8 -.27 .8    |
      | -1.3 1.1   -.26 .83  -.039 |
      | 3.3  -3.3  1.3  .71  -.6   |
      | -1.3 -.3   .35  .59  1.1   |

                 5          5
o22 : Matrix RR    <--- RR
               53         53
i23 : norm(A' - A'')

o23 = 0

o23 : RR (of precision 53)

Caveat

This function is limited in scope, but is sometimes useful for very large matrices

See also

Ways to use solve :