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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      82127 37127  0      78233  53606  0      -32674 0      -42446 |
     | 19183  94560 0      -15505 0      0      0      0      97442  -50409 |
     | 56302  0     -30156 0      35189  0      36565  0      0      55786  |
     | 0      0     0      65582  30881  26543  -13274 0      23400  0      |
     | -66965 0     -92879 0      0      0      -52114 95511  91745  0      |
     | 0      0     -21833 0      -19816 0      0      -33050 0      -20667 |
     | 0      74968 -66624 0      0      -36039 0      0      17977  0      |
     | 0      68790 41272  75084  -89530 0      0      0      -10898 0      |
     | 0      0     0      0      0      0      0      0      0      84253  |
     | 87266  86320 0      0      0      51473  0      0      0      -54338 |

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

o2 = (| 0      53606  -53606 -32674 8148   8148   -4453  53606  -9772  49453 
      | 0      0      19183  0      19183  3678   0      78259  -31226 -22065
      | 36565  0      19737  36565  -48360 -11795 30156  53393  2393   -19737
      | -13274 26543  -13269 -13274 17617  69925  0      10121  39822  -36674
      | -52114 0      -14851 43397  -3502  -55616 -2632  2368   -6134  18617 
      | 0      0      0      -33050 -41649 -41649 54883  0      12383  -33050
      | 0      -36039 36039  0      5454   5454   66624  -18062 0      56991 
      | 0      0      0      0      -48258 26826  -41272 -10898 0      79688 
      | 0      0      0      0      0      0      0      0      84253  0     
      | 0      51473  35793  0      -15680 -15680 0      -35793 32928  -946  
     ------------------------------------------------------------------------
     |, | 0 0 1  0 1  1  0  -1 1  -1 |)
     |  | 0 0 0  0 0  0  0  0  0  1  |
     |  | 0 0 0  0 1  1  -1 0  0  0  |
     |  | 0 0 0  0 0  1  0  0  0  0  |
     |  | 0 0 0  0 1  1  0  0  0  0  |
     |  | 0 1 -1 0 -2 -2 0  1  0  0  |
     |  | 1 0 -1 1 -3 -2 0  3  -3 1  |
     |  | 0 0 0  1 0  0  -1 0  -1 1  |
     |  | 0 0 0  0 0  0  0  1  0  -1 |
     |  | 0 0 0  0 0  0  0  0  1  0  |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also