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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 = | 47913  0      -53571 82675  -2100  0      0      -552   78755  0     
     | 0      0      85075  0      7399   0      0      0      0      -67025
     | 0      0      11186  -16249 12709  0      72463  53206  -98927 0     
     | -49796 0      0      0      0      0      0      -59308 0      0     
     | 0      0      0      -11304 0      -78251 -37256 -80273 37144  43497 
     | 0      -62630 -90086 -59927 0      32486  99129  0      0      36314 
     | 0      7542   -80742 0      0      0      0      -54392 0      -8598 
     | 387    0      23214  -45826 92817  0      0      0      0      0     
     | -90105 -57566 -69128 -52383 23856  0      0      -7886  80292  49246 
     | 0      99510  0      0      -62345 -8939  64632  0      37713  -28015
     ------------------------------------------------------------------------
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              10        10
o1 : Matrix ZZ   <--- ZZ
i2 : (B,U) = LLL(A, ChangeMatrix=>true)

o2 = (| 0      47913  34762  0      0      -2100  34762  29656  -552   25184 
      | 0      0      0      -67025 -67025 -59626 67025  18050  0      18050 
      | 0      0      -16249 0      0      12709  56214  14194  53206  -87741
      | 0      -49796 49796  0      0      0      49796  59308  -59308 0     
      | -78251 0      -11304 43497  -34754 -34754 -13806 -3041  -2022  2390  
      | 32486  0      -59927 36314  6170   6170   33032  17916  -32486 -21286
      | 0      0      0      -8598  -1056  -1056  1056   -34948 -54392 -89340
      | 0      387    -46213 0      0      92817  -46213 -22612 0      23214 
      | 0      -90105 37722  49246  -8320  15536  46042  -64379 -7886  60410 
      | -8939  0      0      -28015 62556  211    2076   27678  8939   759   
     ------------------------------------------------------------------------
     |, | 0 1 -1 0 0 0 -1 0  0  0 |)
     |  | 0 0 0  0 1 1 -1 0  0  0 |
     |  | 0 0 0  0 0 0 0  1  0  1 |
     |  | 0 0 1  0 0 0 1  1  0  0 |
     |  | 0 0 0  0 0 1 0  0  0  0 |
     |  | 1 0 0  0 1 1 -1 1  -1 1 |
     |  | 0 0 0  0 0 0 1  1  0  0 |
     |  | 0 0 0  0 0 0 0  -1 1  0 |
     |  | 0 0 0  0 0 0 0  0  0  1 |
     |  | 0 0 0  1 1 1 -1 1  0  1 |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also