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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 = | 145    0      -13304 -6479  0      18844  0      84672  0      32302 
     | 94040  10579  0      -25994 0      0      0      -58500 -51892 0     
     | 0      -64799 0      30283  0      18142  0      0      0      -44150
     | 0      0      0      0      0      0      83706  0      0      0     
     | 0      -21901 0      81714  77531  0      76231  0      0      -53735
     | -36742 0      -69726 0      -94265 -13514 -6887  0      0      0     
     | 24931  -95476 0      64324  0      0      0      0      0      0     
     | 0      0      0      0      -86712 0      -82221 -71054 0      0     
     | 0      -85988 19698  0      -48537 -84796 -19144 74006  0      0     
     | 0      -79695 0      0      -39529 -10977 12699  0      0      0     
     ------------------------------------------------------------------------
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              10        10
o1 : Matrix ZZ   <--- ZZ
i2 : (B,U) = LLL(A, ChangeMatrix=>true)

o2 = (| -145   0     -13449 12229  17479  33952  -32302 -41437 -34028 4320  
      | 9744   51892 9744   -6506  12982  16887  0      10513  -9714  3238  
      | 0      0     0      -13867 4275   23501  44150  -43066 5317   4275  
      | 0      0     0      0      0      0      0      0      0      83706 
      | 0      0     0      27979  27979  8038   53735  3855   -6406  26679 
      | 36742  0     -32984 3758   -5998  17023  0      11996  20441  -25088
      | -24931 0     -24931 14462  -35400 4248   0      39648  31407  -10469
      | 0      0     0      0      0      15658  0      0      -86712 4491  
      | 0      0     19698  19698  -45400 16857  0      4812   2867   3691  
      | 0      0     0      0      -10977 -40166 0      -57741 -17575 41251 
     ------------------------------------------------------------------------
     |, | -1 0  -1 -2 -4 4  0  8  9  -3 |)
     |  | 0  0  0  0  0  1  0  1  0  0  |
     |  | 0  0  1  1  2  -1 0  -4 -6 3  |
     |  | 0  0  0  1  1  0  0  -1 -3 1  |
     |  | 0  0  0  0  0  -1 0  0  1  -1 |
     |  | 0  0  0  0  1  0  0  -2 -2 1  |
     |  | 0  0  0  0  0  0  0  0  0  1  |
     |  | 0  0  0  0  0  1  0  0  0  0  |
     |  | -2 -1 -2 -4 -8 6  0  15 18 -6 |
     |  | 0  0  0  1  1  -2 -1 -2 -3 1  |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also