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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      -96910 99876  0      0      0      83951  -94624 0      0     
     | 0      0      -75233 0      0      -60281 0      36561  65194  0     
     | 0      0      -81314 -98716 0      0      0      0      0      96740 
     | -12244 -54883 0      -34984 0      -17982 0      0      0      0     
     | -26729 0      -4526  0      -31046 0      -51317 0      30335  0     
     | 2539   -19135 -91774 0      0      0      48069  0      50529  83182 
     | 0      -91493 19382  0      0      72019  0      78024  51471  -77728
     | 0      0      0      -57682 -69541 42826  0      -30108 0      79780 
     | 98313  0      -57029 69373  0      3349   0      0      0      0     
     | -36911 51201  -45152 -78176 0      5917   0      0      -51868 -82475
     ------------------------------------------------------------------------
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              10        10
o1 : Matrix ZZ   <--- ZZ
i2 : (B,U) = LLL(A, ChangeMatrix=>true)

o2 = (| 0      0      0      10673  2966   -10673 83951  -2286  7707   -8387 
      | 0      0      -60281 -31648 -10039 -28633 0      -31648 -26522 3015  
      | 0      0      0      0      17402  -98716 0      0      -17402 -1976 
      | 0      -12244 -17982 -17982 -32143 -22740 0      -72865 32143  50125 
      | -31046 -26729 -31046 50606  -31966 -23877 -51317 -711   52237  -54212
      | 0      2539   0      2460   -57841 -4999  48069  31394  9772   46789 
      | 0      0      72019  45466  -20640 26553  0      -46027 -57384 -5148 
      | -69541 0      -26715 3393   -11859 -18249 0      3393   -27574 -11403
      | 0      98313  3349   3349   -28089 -28940 0      3349   28089  -32289
      | 0      -36911 5917   -45951 -4554  10603  0      5250   4554   -77122
     ------------------------------------------------------------------------
     |, | 0 1 0 0  1  -1 0 0  -1 -1 |)
     |  | 0 0 0 0  1  0  0 1  -1 -1 |
     |  | 0 0 0 0  1  0  0 0  -1 0  |
     |  | 0 0 0 0  -1 1  0 0  1  1  |
     |  | 1 0 1 1  1  -1 0 1  0  -1 |
     |  | 0 0 1 1  0  0  0 1  0  -1 |
     |  | 0 0 0 -1 0  1  1 0  -1 1  |
     |  | 0 0 0 -1 0  1  0 -1 -1 2  |
     |  | 0 0 0 1  1  -1 0 1  -1 -2 |
     |  | 0 0 0 0  0  0  0 0  0  1  |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also