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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     0      0      0      -24751 0      0      0      0      63966  |
     | 14733 64651  -8524  0      -71976 97879  -19610 -33282 -15616 0      |
     | 2986  0      -5432  0      0      36029  -22098 0      0      0      |
     | 0     -48039 0      0      0      87866  -55253 -42175 0      0      |
     | 255   -73933 0      -42299 18192  0      0      0      -52132 0      |
     | 0     0      -26964 0      0      -24452 0      0      24990  -25787 |
     | 4537  0      0      0      0      83157  0      0      0      17585  |
     | 0     35187  -85443 0      0      0      0      0      -51555 0      |
     | 0     0      0      0      0      0      -26019 0      -69181 -44344 |
     | 0     0      0      0      0      6594   -77393 0      0      -77017 |

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

o2 = (| 0     0      -24751 14464  24751  0      0      -24751 -14464 0     
      | 14733 0      -13044 2590   3509   -3816  -2338  5038   2713   -1061 
      | 2986  0      11944  24887  5972   5972   -14193 5775   -30662 -25084
      | 0     0      0      9562   5864   -42175 -8212  2348   -17774 -13078
      | 255   -42299 19212  -2600  13952  510    17505  11097  10527  -255  
      | 0     0      0      -1335  0      0      -51416 24452  1873   0     
      | 4537  0      18148  -6591  9074   9074   15102  -19639 26230  -4537 
      | 0     0      0      0      -35187 0      -15069 -35187 18819  0     
      | 0     0      0      -18325 0      0      0      0      -50856 -26019
      | 0     0      0      -6218  0      0      6594   -6594  12812  -77393
     ------------------------------------------------------------------------
     |, | 1 0 4 13 2  2 -15 14 -27 -1 |)
     |  | 0 0 0 0  -1 0 2   -1 2   0  |
     |  | 0 0 0 0  0  0 1   0  0   0  |
     |  | 0 1 0 1  1  0 -4  2  -6  0  |
     |  | 0 0 1 2  -1 0 0   1  -2  0  |
     |  | 0 0 0 -1 0  0 1   -1 2   0  |
     |  | 0 0 0 -1 0  0 0   0  1   1  |
     |  | 0 0 0 -1 1  1 0   -1 1   -1 |
     |  | 0 0 0 0  0  0 0   0  1   0  |
     |  | 0 0 0 1  0  0 0   0  -1  0  |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also