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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     0      -6757 82021  29504  -27770 80164  61876 67646  0      |
     | 0     -64736 85095 -87061 88004  86455  71521  0     -40732 0      |
     | 0     0      0     0      0      -83602 -72128 0     -63992 0      |
     | 0     -66298 0     0      -67803 0      87899  0     -71375 51822  |
     | 0     51582  -270  14200  0      -86960 27706  45338 0      0      |
     | 0     0      0     0      -67012 97005  0      0     -66433 0      |
     | 25045 66162  0     0      0      47856  97894  0     68520  0      |
     | 70153 -29129 3245  37172  86189  0      75354  0     0      -75586 |
     | 0     -5090  0     82495  -23739 72759  0      0     49840  0      |
     | 0     7083   0     70234  0      55609  0      53688 32649  -81991 |

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

o2 = (| -7048  -7048  291    -7339  -35263 13828  -24572 -27216 6039   42009 
      | 36635  36635  48460  -11825 -2734  -557   28094  2523   23054  -22497
      | 8136   8136   -8136  16272  12620  19924  23262  -19924 -14030 -5894 
      | 26649  26649  -26649 1476   -384   -11140 14269  11140  52058  10904 
      | -3082  -3082  2812   -5894  29424  10100  5224   21308  18890  16348 
      | 579    579    -579   1158   -3975  6291   -24281 -6291  30071  -36362
      | -20401 4644   -4644  -40802 -6759  41407  11666  -16362 14542  -5859 
      | -31468 38685  -35440 9405   30623  27023  -34877 2713   -5267  42964 
      | -3826  -3826  3826   -7652  13663  -34057 3244   -48438 30811  3246  
      | -19283 -19283 19283  43425  16240  -4298  32528  -12248 -2361  -21644
     ------------------------------------------------------------------------
     |, | 3  4  -4 6  -61 75  68  -74 -38 -35 |)
     |  | -1 -1 1  -2 16  -19 -18 19  10  9   |
     |  | 1  1  0  1  -8  11  13  -12 -5  -6  |
     |  | -1 -1 1  -2 19  -23 -21 22  12  11  |
     |  | -1 -1 1  -2 16  -20 -20 20  10  10  |
     |  | 0  0  0  0  -2  2   1   -2  -1  -1  |
     |  | -1 -1 1  -2 19  -23 -21 23  12  11  |
     |  | 2  2  -2 4  -39 47  42  -46 -24 -22 |
     |  | 1  1  -1 2  -19 23  22  -23 -12 -11 |
     |  | 1  1  -1 1  -17 20  17  -20 -10 -9  |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also