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D.15.17.49 smithNormalForm

Procedure from library multigrading.lib (see multigrading_lib).

Usage:
smithNormalForm(A[,opt]); intmat A

Purpose:
Computes the Smith Normal Form of A

Return:
if no optional argument is given: intmat, the Smith Normal Form of A, otherwise: a list of 3 integer matrices P, D Q, such that D == P*A*Q.

Example:
 
LIB "multigrading.lib";
intmat A[5][7] =
1,0,1,0,-2,9,-71,
0,-24,248,-32,-96,448,-3496,
0,4,-42,4,-8,30,-260,
0,0,0,18,-90,408,-3168,
0,0,0,-32,224,-1008,7872;
print( smithNormalForm(A) );
==>      1     0     0     0     0     0     0
==>      0     2     2     6    12     0     0
==>      0     0     2     0     0     0     0
==>      0     0     0     8     0     0     0
==>      0     0     0     0    48     0     0
list l = smithNormalForm(A, 5);
l;
==> [1]:
==>    1,0,0,0,0,
==>    0,0,1,0,0,
==>    0,-1,4,-1,0,
==>    0,3,-12,4,-1,
==>    0,-6,24,-8,3 
==> [2]:
==>    1,0,0,0,0,0,0,
==>    0,2,2,6,12,0,0,
==>    0,0,2,0,0,0,0,
==>    0,0,0,8,0,0,0,
==>    0,0,0,0,48,0,0 
==> [3]:
==>    1,5,45,187,384,8,18591,
==>    0,-52,-461,-1914,-3929,-82,-190555,
==>    0,-5,-53,-218,-445,-10,-23236,
==>    0,0,-28,-109,-215,-7,-16260,
==>    0,0,221,871,1729,53,123084,
==>    0,0,50,197,391,12,27876,
==>    0,0,0,0,0,0,1 
l[1]*A*l[3];
==> 1,0,0,0,0,0,0,
==> 0,2,2,6,12,0,0,
==> 0,0,2,0,0,0,0,
==> 0,0,0,8,0,0,0,
==> 0,0,0,0,48,0,0 
det(l[1]);
==> 1
det(l[3]);
==> 1