| #### D.3.1.22 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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