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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     -20491 10702  55716  0      -23738 -58960 0      67105  19619  |
     | 92315 0      0      -53530 0      -44620 0      0      -35635 -58102 |
     | 0     0      0      0      0      0      71298  64957  81842  -71453 |
     | 0     0      0      -47935 0      -84591 0      -53192 0      60339  |
     | 0     0      0      0      0      -2572  36800  0      -89158 96457  |
     | 0     0      -69510 0      1265   0      18066  -83316 0      -66789 |
     | 0     0      0      -79230 0      0      0      24403  0      0      |
     | 49140 0      0      3962   -61126 0      0      34759  0      0      |
     | 0     -82508 44880  0      0      0      0      0      0      0      |
     | 11396 0      39632  35719  0      -95764 0      0      4954   0      |

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

o2 = (| 0      -20491 -9789  0      55716  -58960 13036  9789   -14680 -5763 
      | 0      0      0      92315  38785  0      -47695 0      12060  46273 
      | 0      0      0      0      0      71298  64957  64957  23226  16730 
      | 0      0      0      0      -47935 0      31399  -53192 21793  28940 
      | 0      0      0      0      0      36800  2572   0      -54930 41527 
      | 1265   0      -69510 1265   1265   18066  -13806 -12541 45678  -33652
      | 0      0      0      0      -79230 0      24403  24403  -48806 -24403
      | -61126 0      0      -11986 -8024  0      -14381 -26367 -20378 2395  
      | 0      -82508 -37628 0      0      0      -44880 37628  7252   -37628
      | 0      0      39632  11396  47115  0      44736  -39632 -150   -28386
     ------------------------------------------------------------------------
     |, | 0 0 0 1 1 0 -1 0  1  2  |)
     |  | 0 1 1 0 0 0 0  -1 1  1  |
     |  | 0 0 1 0 0 0 -1 -1 2  1  |
     |  | 0 0 0 0 1 0 0  0  0  0  |
     |  | 1 0 0 1 1 0 0  1  0  1  |
     |  | 0 0 0 0 0 0 -1 0  1  1  |
     |  | 0 0 0 0 0 1 0  0  1  1  |
     |  | 0 0 0 0 0 0 1  1  -2 -1 |
     |  | 0 0 0 0 0 0 0  0  1  1  |
     |  | 0 0 0 0 0 0 0  0  0  1  |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also