next | previous | forward | backward | up | top | index | toc | Macaulay2 web site

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      -60625 0      0      0      -45620 -51158 -17117 0      13007 
     | 81081  0      0      0      0      0      -1007  0      -62769 0     
     | 0      0      73470  35579  -67832 0      0      0      -80372 27631 
     | 25638  -42029 27407  0      0      20058  -18765 -67475 -7481  97383 
     | 88366  0      -63771 0      -99420 67519  0      0      0      0     
     | -19642 21257  0      -95658 0      -78389 -6329  -86326 18135  4863  
     | 986    -22956 0      0      63681  82424  0      68469  0      -26053
     | 0      0      -39274 0      0      0      0      0      40576  -47651
     | -37521 0      0      4149   50940  0      53395  0      0      53375 
     | 0      -6780  -62974 67478  69816  -36124 0      0      -41170 30193 
     ------------------------------------------------------------------------
     |
     |
     |
     |
     |
     |
     |
     |
     |
     |

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

o2 = (| 9467   -51158 -13577 -13577 13577  -37581 -13577 37970  -41510 4110  
      | -1007  -1007  1007   1007   61762  16298  1007   17305  -18312 18312 
      | 0      0      -7948  27631  52741  1046   -81418 -6902  52741  -66786
      | 23264  -18765 6644   6644   837    20155  -20763 -18705 -28007 15656 
      | 0      0      0      0      0      24595  63771  -42924 -20847 -74825
      | -27586 -6329  41781  -53877 35742  -49617 41781  -37030 4581   -15702
      | 22956  0      19460  19460  -19460 -18474 19460  9987   39022  22251 
      | 0      0      -47651 -47651 7075   48953  -8377  1302   7075   48953 
      | 53395  53395  -4169  -20    20     20043  -4169  15874  -15854 -35807
      | 6780   0      -44065 23413  17757  -60079 18909  -61240 -25147 2957  
     ------------------------------------------------------------------------
     |, | 0  0 0  0  0  1  0  1  -1 1  |)
     |  | -1 0 1  1  -1 -1 1  -1 0  0  |
     |  | 0  0 0  0  0  1  -1 1  0  1  |
     |  | 0  0 -1 0  0  1  -1 0  0  1  |
     |  | 0  0 0  0  0  0  0  0  0  1  |
     |  | 0  0 0  0  0  0  0  -1 1  0  |
     |  | 1  1 -1 -1 1  2  -1 1  0  0  |
     |  | 0  0 1  1  -1 -1 1  1  -1 -1 |
     |  | 0  0 0  0  -1 1  0  1  -1 1  |
     |  | 0  0 1  1  -1 -1 1  0  -1 -1 |

o2 : Sequence
i3 : B == A*U

o3 = true

Further information

See also