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D.3.1.23 rowred
Procedure from library matrix.lib (see matrix_lib).
- Usage:
- rowred(A[,e]); A matrix, e any type
- Return:
- - a matrix B, being the row reduced form of A, if rowred is called
with one argument.
(as far as this is possible over the polynomial ring; no division
by polynomials)
- a list L of two matrices, such that L[1] = L[2] * A with L[1]
the row-reduced form of A and L[2] the transformation matrix
(if rowred is called with two arguments).
- Assume:
- The entries of A are in the base field. It is not verified whether
this assumption holds.
- Note:
- * This procedure is designed for teaching purposes mainly.
* The straight forward Gaussian algorithm is implemented in the
library (no standard basis computation).
The transformation matrix is obtained by concatenating a unit
matrix to A. proc gauss_row should be faster.
* It should only be used with exact coefficient field (there is no
pivoting) over the polynomial ring (ordering lp or dp).
* Parameters are allowed. Hence, if the entries of A are parameters
the computation takes place over the field of rational functions.
Example:
| LIB "matrix.lib";
ring r=(0,a,b),(A,B,C),dp;
matrix m[6][8]=
0, 0, b*B, -A,-4C,2A,0, 0,
2C,-4C,-A,B, 0, B, 3B,AB,
0,a*A, 0, 0, B, 0, 0, 0,
0, 0, 0, 0, 2, 0, 0, 2A,
0, 0, 0, 0, 0, 0, 2b, A,
0, 0, 0, 0, 0, 0, 0, 2a;"";
==>
print(rowred(m));"";
==> 0, 0, 0, 0, 1,0, 0,0,
==> 0, 0, 0, 0, 0,0, 1,0,
==> 0, 0, 0, 0, 0,0, 0,1,
==> 0, 0, (b)*B,-A,0,2*A,0,0,
==> 2*C,-4*C, -A, B, 0,B, 0,0,
==> 0, (a)*A,0, 0, 0,0, 0,0
==>
list L=rowred(m,1);
print(L[1]);
==> 0, 0, 0, 0, 1,0, 0,0,
==> 0, 0, 0, 0, 0,0, 1,0,
==> 0, 0, 0, 0, 0,0, 0,1,
==> 0, 0, (b)*B,-A,0,2*A,0,0,
==> 2*C,-4*C, -A, B, 0,B, 0,0,
==> 0, (a)*A,0, 0, 0,0, 0,0
print(L[2]);
==> 0,0,0,1/2, 0, -1/(2a)*A,
==> 0,0,0,0, 1/(2b), -1/(4ab)*A,
==> 0,0,0,0, 0, 1/(2a),
==> 1,0,0,2*C, 0, -2/(a)*AC,
==> 0,1,0,0, -3/(2b)*B,(-2b+3)/(4ab)*AB,
==> 0,0,1,-1/2*B,0, 1/(2a)*AB
| See also:
gauss_row.
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