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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      61401  0      0      0      -13579 0      0      80004 
     | 0      -48059 41977  53940  36955  0      0      39640  92816  64823 
     | 0      -84826 42809  40433  -85996 0      0      0      32638  0     
     | 0      0      -30200 -42920 0      0      97389  0      0      56531 
     | 15526  0      -71198 0      0      82378  87592  0      0      -25708
     | 0      93006  -47025 64679  35967  -10618 69017  74876  35040  0     
     | -91017 -53146 85299  -86562 0      20524  0      0      0      0     
     | -10675 0      0      -91200 0      0      0      -91159 43730  0     
     | 0      -67856 0      70407  67309  -61933 0      64539  -30017 -40170
     | -43500 -3686  0      0      0      0      99390  -42445 0      0     
     ------------------------------------------------------------------------
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              10        10
o1 : Matrix ZZ   <--- ZZ
i2 : (B,U) = LLL(A, ChangeMatrix=>true)

o2 = (| 0      0      61401  0      0      -29219 29219  -58438 18603  -21566
      | 0      0      41977  36955  -14300 -4831  -28928 26782  22846  1211  
      | 0      0      42809  -85996 -40433 -45185 792    40019  -42809 -7381 
      | 0      0      -30200 0      42920  -23378 -19542 -3836  86731  -52695
      | 15526  82378  -55672 0      15526  -17482 -13570 -3912  29964  29620 
      | 0      -10618 -47025 35967  10197  14836  67973  -14228 47025  49268 
      | -91017 20524  -5718  0      -4455  15891  26435  -10544 5718   10544 
      | -10675 0      -10675 0      -10634 31984  -10675 -48500 10675  92230 
      | 0      -61933 0      67309  -5868  -34302 -27686 -9386  -40170 19539 
      | -43500 0      -43500 0      -85945 73555  52204  -21094 43500  21094 
     ------------------------------------------------------------------------
     |, | 1 0 1 0 1  -3 1  -4 -1 4  |)
     |  | 0 0 0 0 0  0  1  -1 0  1  |
     |  | 0 0 1 0 0  -2 2  -4 -1 4  |
     |  | 0 0 0 0 -1 1  0  1  0  -1 |
     |  | 0 0 0 1 0  0  0  -1 0  1  |
     |  | 0 1 0 0 0  0  0  0  0  0  |
     |  | 0 0 0 0 0  -1 1  -2 0  2  |
     |  | 0 0 0 0 1  -1 0  0  0  0  |
     |  | 0 0 0 0 0  0  0  0  0  1  |
     |  | 0 0 0 0 0  1  -1 2  1  -3 |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also