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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      0      0      75798  0      -80082 41755 
     | -63234 0      0      90772  0      73605  -62249 -18356 0      78182 
     | 49915  0      -91374 38794  -25200 83955  -66144 17357  0      -51232
     | 0      0      77185  0      55845  0      0      0      0      50488 
     | -46113 0      72883  -55274 -26376 0      48786  -23382 0      0     
     | -66425 0      -39050 0      0      -47116 -85056 22062  -78369 0     
     | 79459  -86555 -51350 0      -80122 -33659 0      -34948 0      50321 
     | 0      0      -56340 51107  0      95016  -88100 -45487 70341  88866 
     | 0      -4140  -1837  -65732 6863   73081  0      38126  14170  0     
     | 0      -41308 0      51207  -14404 0      0      0      26510  29983 
     ------------------------------------------------------------------------
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              10        10
o1 : Matrix ZZ   <--- ZZ
i2 : (B,U) = LLL(A, ChangeMatrix=>true)

o2 = (| 0      0      0      41755 38327  3428   72370  3428   -7712  -30615
      | 0      -18356 0      59826 32135  -35543 -8350  36873  -26706 4942  
      | -25200 17357  0      -8675 50073  -34033 -8494  -18856 34063  -41640
      | 55845  0      0      -5357 61202  -10714 45219  -45219 -10626 26609 
      | -26376 -23382 0      2994  -5988  -63507 10040  -16528 13034  19724 
      | 0      22062  0      22062 -12871 -31492 -36576 -48480 -92883 -46837
      | 6433   -34948 -86555 8940  -1218  9495   3114   -3114  -38267 33935 
      | 0      -45487 0      43379 26783  16596  -2869  -34124 21985  -10092
      | 11003  38126  -4140  27123 4665   29321  -47744 -17988 -6451  73030 
      | 26904  0      -41308 3079  -2685  -8640  21140  30067  20746  -18061
     ------------------------------------------------------------------------
     |, | 0  0 0 0  0  1  -1 1  -1 2  |)
     |  | -1 0 1 1  -2 3  -4 4  -3 5  |
     |  | 0  0 0 0  0  0  -1 1  -1 2  |
     |  | 0  0 0 0  0  0  0  1  0  0  |
     |  | 1  0 0 -1 2  -2 4  -4 3  -5 |
     |  | 0  0 0 0  1  -1 1  -1 1  -1 |
     |  | 0  0 0 0  0  0  1  0  1  -1 |
     |  | 0  1 0 1  -2 3  -4 4  -3 5  |
     |  | 0  0 0 0  -1 1  -1 1  0  1  |
     |  | 0  0 0 1  -1 2  -2 2  -2 3  |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also