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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 = | 62893  85976  -84232 -13400 -2685  -83662 0      0      -37920 0     
     | -28030 0      0      0      0      67699  577    0      0      92407 
     | 0      0      0      0      0      0      0      33255  50092  0     
     | 0      0      78955  44564  -39504 46834  9752   -23804 0      72682 
     | 26114  -25365 43681  16843  -25204 49865  -50837 -63205 0      0     
     | 0      68359  -11154 0      98361  0      62275  0      -75858 5227  
     | 0      -36405 41577  0      -60255 0      -64436 0      0      -59902
     | 0      19179  0      -37993 -55241 91927  -70231 5840   0      0     
     | 82077  0      0      0      -98261 -21462 71744  0      0      0     
     | 0      0      10663  0      -8827  55349  0      -97590 0      0     
     ------------------------------------------------------------------------
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              10        10
o1 : Matrix ZZ   <--- ZZ
i2 : (B,U) = LLL(A, ChangeMatrix=>true)

o2 = (| -13400 15144 -50549 22027  48056  -10433 18683  -27663 31027  -13711
      | 0      0     6425   6425   0      -14961 -2307  22401  -23912 -47109
      | 0      0     -16418 -16418 50092  -33255 15999  -17256 -419   17256 
      | 44564  34391 -22224 22340  0      -17528 -6403  -11488 -28627 -38578
      | 16843  1473  -38284 -46806 -25365 -42649 526    32383  -11644 6014  
      | 0      57205 -28740 39619  -7499  -17081 38980  -35306 10240  -15972
      | 0      5172  9501   -26904 -36405 23252  8893   26973  18394  -13820
      | -37993 57172 21287  2473   19179  -37722 -24646 -27613 -3359  -13618
      | 0      0     -12413 -12413 0      -39153 -17525 3937   52139  -25399
      | 0      10663 28935  28935  0      8218   -25074 27830  3861   6193  
     ------------------------------------------------------------------------
     |, | 0 0  0   0   0 -1 -3  -3  -2  3   |)
     |  | 0 1  0   1   1 0  0   -1  0   0   |
     |  | 0 1  6   6   0 2  -17 -16 -11 14  |
     |  | 1 -1 -12 -11 0 -4 32  29  20  -26 |
     |  | 0 0  1   1   0 0  -4  -4  -3  4   |
     |  | 0 0  -4  -4  0 -2 11  10  7   -9  |
     |  | 0 0  0   0   0 0  1   1   1   -1  |
     |  | 0 0  -2  -2  0 -1 5   4   3   -4  |
     |  | 0 0  1   1   1 0  -3  -3  -2  3   |
     |  | 0 0  3   3   0 1  -9  -8  -6  7   |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also