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D.3.2.11 gaussred
Procedure from library linalg.lib (see linalg_lib).
- Usage:
- gaussred(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:
- This procedure is designed for teaching purposes mainly.
The straight forward implementation in the interpreted library
is not very efficient (no standard basis computation).
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;
print(A);
==> 1, 3,-1,4,
==> 2, 5,-1,3,
==> 1, 3,-1,4,
==> 0, 4,-3,1,
==> -3,1,-5,-2
list Z=gaussred(A); //construct P,U,S s.t. P*A=U*S
print(Z[1]); //P
==> 1,0,0,0,0,
==> 0,1,0,0,0,
==> 0,0,0,0,1,
==> 0,0,0,1,0,
==> 0,0,1,0,0
print(Z[2]); //U
==> 1, 0, 0, 0,0,
==> 2, 1, 0, 0,0,
==> -3,-10,1, 0,0,
==> 0, -4, 1/2,1,0,
==> 1, 0, 0, 0,1
print(Z[3]); //S
==> 1,3, -1,4,
==> 0,-1,1, -5,
==> 0,0, 2, -40,
==> 0,0, 0, 1,
==> 0,0, 0, 0
print(Z[4]); //rank
==> 4
print(Z[1]*A); //P*A
==> 1, 3,-1,4,
==> 2, 5,-1,3,
==> -3,1,-5,-2,
==> 0, 4,-3,1,
==> 1, 3,-1,4
print(Z[2]*Z[3]); //U*S
==> 1, 3,-1,4,
==> 2, 5,-1,3,
==> -3,1,-5,-2,
==> 0, 4,-3,1,
==> 1, 3,-1,4
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