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LLL(..., ChangeMatrix => ...) -- also find change of basis matrix

Synopsis

Description

Constructs the change of basis matrix U from the basis A to the basis B. This is an invertible matrix U such thatAU = B.
i1 : A = matrix randomMutableMatrix(10,10,.5,100000)

o1 = | 94323  96946  -64982 -89005 53702  0      0      73362  0      54969 
     | -99296 0      0      58671  0      0      701    -89802 0      0     
     | 14100  0      0      -31655 55686  0      0      -88912 88954  -40453
     | 0      -21885 69528  0      0      94804  13608  12347  0      -1911 
     | 0      0      -49179 0      -70732 0      -75270 0      0      0     
     | 42937  96309  0      7230   0      -16410 -16272 0      -88864 -4575 
     | 0      -85941 -79313 0      42193  0      0      0      -27903 0     
     | -79441 -23097 -93761 0      0      -4682  0      0      0      0     
     | 24565  -6111  0      0      0      0      84905  0      24832  0     
     | 0      -54657 8020   0      -21394 0      -4670  97028  0      0     
     ------------------------------------------------------------------------
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              10        10
o1 : Matrix ZZ   <--- ZZ
i2 : (B,U) = LLL(A, ChangeMatrix=>true)

o2 = (| 54969  5318   -6362  27674  -2311  -27295 -28718 0      -36264 -10385
      | 0      -40625 13085  -45586 -27540 -44885 18046  0      -18046 -13754
      | -40453 -17555 -55769 16339  22815  -32162 -709   0      709    -7734 
      | -1911  0      -19560 -17649 -17649 -2130  -1911  94804  -23365 -16388
      | 0      0      21553  21553  -49179 -53717 0      0      -49179 17015 
      | -4575  50167  -64812 -67467 -10070 9700   -36042 -16410 52452  -2105 
      | 0      0      -7662  -7662  34531  20241  -27903 0      -51410 -21952
      | 0      -79441 13459  13459  -65982 13459  -79441 -4682  -9638  13459 
      | 0      24565  -43286 -43286 -18721 16787  49397  0      -49397 16787 
      | 0      0      -12957 -12957 -34351 -17627 0      0      8020   -93261
     ------------------------------------------------------------------------
     |, | 0 1 -1 -1 0  -1 1 0 -1 -1 |)
     |  | 0 0 -1 -1 -1 -1 0 0 0  -1 |
     |  | 0 0 1  1  1  1  0 0 1  1  |
     |  | 0 1 -3 -4 -2 -4 2 0 -2 -5 |
     |  | 0 0 -1 -1 0  -1 0 0 0  -2 |
     |  | 0 0 -1 -1 -1 -1 0 1 -1 -1 |
     |  | 0 0 0  0  0  1  0 0 0  1  |
     |  | 0 0 -1 -1 -1 -1 0 0 0  -2 |
     |  | 0 0 -1 -1 -1 -2 1 0 -1 -2 |
     |  | 1 0 2  1  1  0  1 0 -1 1  |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also