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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 = | -36150 -80575 -24068 -77005 0      0      0      0     -21906 -47506 |
     | 0      -5292  -96090 0      0      0      -48401 0     -76891 11986  |
     | 0      -70985 61758  58558  33524  0      0      0     -91914 0      |
     | 0      94091  0      0      9306   0      55154  0     0      -83672 |
     | 0      -86432 0      45217  0      -33787 35838  0     0      -67094 |
     | 34519  0      9219   -79618 -36216 0      0      0     -4737  0      |
     | 0      0      0      -55715 -80511 0      56413  0     0      0      |
     | 0      -87473 -22309 -11089 0      14334  0      0     64758  0      |
     | -16646 46541  64563  0      0      0      32929  0     0      66971  |
     | 0      0      -42639 -87413 -68621 -27929 -19274 23119 0      0      |

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

o2 = (| 0     0      -36150 77005  0      -36150 17417  -6651  -21906 -20987 
      | 0     0      0      0      0      -48401 11274  -36415 -76891 -16566 
      | 0     0      0      -25034 33524  33524  -53268 8490   -91914 -42751 
      | 0     0      0      9306   9306   64460  45248  -9906  0      58149  
      | 0     -33787 0      -11430 0      2051   -6848  -8899  33787  -57227 
      | 0     0      34519  43402  -36216 -1697  32486  41705  -4737  10916  
      | 0     0      0      -24796 -80511 -24098 7519   -48894 0      -32315 
      | 0     14334  0      -3245  0      14334  19064  -17579 50424  -124116
      | 0     0      -16646 0      0      16283  51620  83254  0      -5079  
      | 23119 -4810  0      483    736    -229   5310   9874   4810   -17    
     ------------------------------------------------------------------------
     |, | 0 0 1 0  0 1 1  1  0  -1 |)
     |  | 0 0 0 0  0 0 0  0  0  1  |
     |  | 0 0 0 0  0 0 -1 0  0  1  |
     |  | 0 0 0 -1 0 0 -1 -1 0  0  |
     |  | 0 0 0 1  1 1 2  2  0  -1 |
     |  | 0 1 0 -1 0 1 -1 -2 -1 -1 |
     |  | 0 0 0 0  0 1 2  1  0  -2 |
     |  | 1 1 0 -2 3 5 1  1  -1 -4 |
     |  | 0 0 0 0  0 0 0  0  1  0  |
     |  | 0 0 0 0  0 0 1  1  0  -1 |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also