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D.3.2.12 gaussred_pivot

Procedure from library linalg.lib (see linalg_lib).

Usage:
gaussred_pivot(A); A any constant matrix

Return:
list Z: Z[1]=P , Z[2]=U , Z[3]=S , Z[4]=rank(A)
gives a row reduced matrix S, a permutation matrix P and a normalized lower triangular matrix U, with P*A=U*S

Note:
with row pivoting

Example:
 
LIB "linalg.lib";
ring r=0,(x),dp;
matrix A[5][4] = 1, 3,-1,4,
2, 5,-1,3,
1, 3,-1,4,
0, 4,-3,1,
-3,1,-5,-2;
list Z=gaussred_pivot(A);  //construct P,U,S s.t. P*A=U*S
print(Z[1]);               //P
==> 0,0,0,0,1,
==> 0,1,0,0,0,
==> 0,0,1,0,0,
==> 0,0,0,1,0,
==> 1,0,0,0,0 
print(Z[2]);               //U
==> 1,   0,    0,   0,0,
==> -2/3,1,    0,   0,0,
==> -1/3,10/17,1,   0,0,
==> 0,   12/17,-1/2,1,0,
==> -1/3,10/17,1,   0,1 
print(Z[3]);               //S
==> -3,1,   -5,   -2,   
==> 0, 17/3,-13/3,5/3,  
==> 0, 0,   -2/17,40/17,
==> 0, 0,   0,    1,    
==> 0, 0,   0,    0     
print(Z[4]);               //rank
==> 4
print(Z[1]*A);             //P*A
==> -3,1,-5,-2,
==> 2, 5,-1,3, 
==> 1, 3,-1,4, 
==> 0, 4,-3,1, 
==> 1, 3,-1,4  
print(Z[2]*Z[3]);          //U*S
==> -3,1,-5,-2,
==> 2, 5,-1,3, 
==> 1, 3,-1,4, 
==> 0, 4,-3,1, 
==> 1, 3,-1,4