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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 = | -26604 0     0      0      6885   0      0      -13837 0      0      |
     | 10784  0     0      38479  -75944 39791  13725  90662  1014   0      |
     | 0      22897 0      70529  0      0      0      -68128 0      0      |
     | 0      0     82914  99039  0      -29914 0      -46923 -53298 -30883 |
     | -41636 0     0      -30182 0      55297  0      4945   0      -38198 |
     | 0      17504 0      0      -30391 41117  0      92222  -39635 -88382 |
     | 90701  48072 75737  -36101 -80343 0      22381  0      36806  -57303 |
     | 0      72480 -94098 20952  23492  0      0      0      -39744 94691  |
     | 0      0     34592  0      -23236 0      -80663 0      0      0      |
     | 0      0     79127  0      0      0      -57204 11698  45303  60095  |

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

o2 = (| -19719 19719  -46323 0      46323  -19719 -39371 46323  -13837 19719 
      | -25369 25369  -14585 -39791 15599  14124  49265  29324  50871  39392 
      | 0      22897  -22897 0      22897  70529  22334  22897  -68128 -70529
      | -29914 29914  -29914 29914  -23384 15827  -20404 -23384 -17009 6290  
      | 13661  -13661 -27975 -55297 27975  -16521 39476  27975  -50352 33620 
      | 10726  6778   -6778  -41117 -32857 -28909 -52626 -32857 51105  -18356
      | 10358  37714  52987  0      -16181 11063  -8851  6200   0      29752 
      | 23492  48988  -48988 0      9244   4700   -10525 9244   0      -4107 
      | -23236 23236  -23236 0      23236  -23236 0      -57427 0      -22835
      | 0      0      0      0      45303  45303  3094   -11901 11698  36715 
     ------------------------------------------------------------------------
     |, | 1 -1 2  0  -2 1 2  -2 0  -1 |)
     |  | 0 1  -1 0  1  0 -2 1  0  0  |
     |  | 0 0  0  0  0  0 0  0  0  1  |
     |  | 0 0  0  0  0  1 0  0  0  -1 |
     |  | 1 -1 1  0  -1 1 0  -1 0  -1 |
     |  | 1 -1 1  -1 -1 1 3  -1 -1 0  |
     |  | 0 0  0  0  0  0 0  1  0  1  |
     |  | 0 0  0  0  0  0 -1 0  1  0  |
     |  | 0 0  0  0  1  1 -1 1  0  -1 |
     |  | 0 0  0  0  0  0 1  0  0  1  |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also