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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 = | 0      -93318 0      85304  0      0      0      0     0      68749  |
     | 94726  -77647 0      -44549 -64392 0      -42988 0     11680  0      |
     | 0      0      73140  0      -38985 39975  0      0     44084  -3710  |
     | 0      0      0      -7855  0      27195  -78210 0     11248  0      |
     | -57880 0      0      0      0      0      0      0     72626  91000  |
     | -7001  58738  42783  0      0      0      75369  80081 85475  41046  |
     | 40652  0      0      0      0      -82987 0      0     0      -57017 |
     | 0      0      0      -47531 0      0      -54500 92038 0      5892   |
     | -98330 0      0      26032  -41700 0      -48067 0     -41644 0      |
     | 47206  0      -65250 11297  -85066 -63905 0      10070 0      99965  |

              10        10
o1 : Matrix ZZ   <--- ZZ
i2 : (B,U) = LLL(A, ChangeMatrix=>true)

o2 = (| 0      0      -16555 16555  33110  68749  -27625 -76763 -41124 0     
      | 0      0      -31523 -32869 -1346  -11680 59208  -46124 -26124 11680 
      | 73140  -33165 -53614 -25346 -44872 25346  8742   -19526 44872  -29056
      | 0      27195  -30588 3393   33981  -11248 2069   30588  -41836 11248 
      | 0      0      18374  -18374 -36748 18374  2758   -18374 -21132 72626 
      | 42783  -42783 -1646  1646   40590  -1646  65868  17601  11147  42692 
      | 0      -82987 25970  57017  31047  -57017 -35575 -25970 9605   0     
      | 0      0      53423  -53423 -14808 5892   -27669 -53423 -32723 0     
      | 0      0      -26088 -15612 10476  41644  34763  26088  -82774 -41644
      | -65250 1345   2257   -23418 49645  34715  -22412 62993  8858   65250 
     ------------------------------------------------------------------------
     |, | 0 0  0  0  0  0  -1 0  1  0  |)
     |  | 0 0  0  0  0  0  -1 1  1  0  |
     |  | 1 -1 1  -1 -3 1  -4 -2 3  -1 |
     |  | 0 0  -1 1  2  0  1  1  -1 0  |
     |  | 0 0  1  0  -1 0  -2 -1 1  0  |
     |  | 0 1  -1 0  1  0  2  1  -1 0  |
     |  | 0 0  0  0  0  0  1  0  0  0  |
     |  | 0 0  0  0  1  0  1  0  -1 0  |
     |  | 0 0  -1 1  2  -1 3  1  -2 1  |
     |  | 0 0  1  -1 -2 1  -3 -1 2  0  |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also