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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 = | 81078  0      0      62615  0      0      -84546 0      0     0      |
     | 34175  94397  7983   27809  0      -1908  0      -15    61241 0      |
     | 0      22148  -21781 -42630 0      0      51883  70756  50135 0      |
     | 0      29825  0      30799  0      1828   -22838 0      0     -52537 |
     | 58041  -22902 42205  30910  81629  -9424  -24919 63576  92444 0      |
     | -55247 0      0      -22952 0      11959  0      0      0     0      |
     | -2807  0      0      0      -71078 0      0      67355  64577 0      |
     | 0      -67860 0      -9834  0      59878  0      0      0     -59198 |
     | -68082 0      0      -43851 0      -60151 0      -12120 0     74141  |
     | 86642  0      -99784 0      0      71579  0      0      41516 -25855 |

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

o2 = (| -21931 -40394 -62325 -40394 0      21931  62615  -18463 3758   0     
      | 27809  13460  10036  19535  1908   35355  -35355 -14349 -30184 6075  
      | 9253   -11596 -45112 -33377 0      -29874 -22009 -20849 -5479  -21781
      | 7961   38760  15068  -11949 50709  42748  -19910 30799  -1060  1828  
      | 5991   -63345 3840   -30564 9424   32301  -7382  12293  37923  32781 
      | -22952 9343   -25568 21302  -11959 10993  -10993 32295  42604  11959 
      | 0      2807   29     2807   0      -2778  2778   -68271 4669   0     
      | -9834  -19668 -21520 -18988 -680   9154   -9154  -9834  -37976 59878 
      | -43851 -19620 8800   -5630  -13990 41981  -41981 24231  -35500 -60151
      | 0      13142  -16921 -40918 -45724 -4208  4208   13142  -23568 -28205
     ------------------------------------------------------------------------
     |, | 0 -1 -1 -1 0  0  0  -1 -2 0 |)
     |  | 0 0  -1 0  0  0  0  0  0  0 |
     |  | 0 -1 -1 0  0  0  0  -1 -1 1 |
     |  | 1 2  3  2  0  -1 1  1  4  0 |
     |  | 0 0  0  0  0  0  0  1  1  0 |
     |  | 0 0  -1 1  -1 -1 1  0  2  1 |
     |  | 1 1  2  1  0  -1 0  0  1  0 |
     |  | 0 0  -1 0  0  -1 1  0  2  0 |
     |  | 0 0  1  0  0  1  -1 0  -1 0 |
     |  | 0 0  0  1  -1 -1 1  0  2  0 |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also