# Function: refpkg[ref] - compute the row echelon form of a matrix
# Calling Sequence:
# refpkg[ref](A)
# refpkg[ref](A, m)
# Parameters:
# A  - a matrix
# m  - a non-negative integer
# Description:
# The call ref(A) returns the row echelon form of the matrix A, where
# elimination stops after the last row/column, whichever comes first
# The call ref(A,m) returns the row echelon form of the matrix A, where
# elimination stops in column m, unless it reaches the bottom row first
# if infolevel[refpkg] > 2 a trace of the Gaussian elimination process
# is printed
# Examples:
> read("/afs/eos.ncsu.edu/users/k/kaltofen/www/courses/LinAlgebra/Maple/
> initlib.mpl");

  libname := /afs/eos.ncsu.edu/users/k/kaltofen/www/courses/LinAlg\

        ebra/Maple, "/afs/bp.ncsu.edu/dist/maple551/update",

        "/afs/bp.ncsu.edu/dist/maple551/lib"

> with(refpkg, ref);

                                [ref]

> eval(refpkg);

  table([

        E_III = readlib(refpkg/E_III)

        E_II = readlib(refpkg/E_II)

        mydet = readlib(refpkg/mydet)

        ref = readlib(refpkg/ref)

        E_I = readlib(refpkg/E_I)

        myinverse = readlib(refpkg/myinverse)

        mysolve = readlib(refpkg/mysolve)

        xref = readlib(refpkg/xref)

        ])

> infolevel['refpkg']:=3:
> B := matrix([ [1, 1, 1, 1, 1, 1], [-1, -1, 0, 0, 1, -1],
>      [-2, -2, 0, 0, 3, 1], [0,0,1,1,3,-1], [1, 1, 2, 2,4,1] ]);

                      [ 1     1    1    1    1     1]
                      [                             ]
                      [-1    -1    0    0    1    -1]
                      [                             ]
                 B := [-2    -2    0    0    3     1]
                      [                             ]
                      [ 0     0    1    1    3    -1]
                      [                             ]
                      [ 1     1    2    2    4     1]

> ref(B);

                          Searching column 1


                   Added row 1 times , 1,  to row 2


                   [ 1     1    1    1    1     1]
                   [                             ]
                   [ 0     0    1    1    2     0]
                   [                             ]
                   [-2    -2    0    0    3     1]
                   [                             ]
                   [ 0     0    1    1    3    -1]
                   [                             ]
                   [ 1     1    2    2    4     1]


                   Added row 1 times , 2,  to row 3


                    [1    1    1    1    1     1]
                    [                           ]
                    [0    0    1    1    2     0]
                    [                           ]
                    [0    0    2    2    5     3]
                    [                           ]
                    [0    0    1    1    3    -1]
                    [                           ]
                    [1    1    2    2    4     1]


                   Added row 1 times , 0,  to row 4


                    [1    1    1    1    1     1]
                    [                           ]
                    [0    0    1    1    2     0]
                    [                           ]
                    [0    0    2    2    5     3]
                    [                           ]
                    [0    0    1    1    3    -1]
                    [                           ]
                    [1    1    2    2    4     1]


                  Added row 1 times , -1,  to row 5


                    [1    1    1    1    1     1]
                    [                           ]
                    [0    0    1    1    2     0]
                    [                           ]
                    [0    0    2    2    5     3]
                    [                           ]
                    [0    0    1    1    3    -1]
                    [                           ]
                    [0    0    1    1    3     0]


                          Searching column 2


                          Searching column 3


                  Added row 2 times , -2,  to row 3


                    [1    1    1    1    1     1]
                    [                           ]
                    [0    0    1    1    2     0]
                    [                           ]
                    [0    0    0    0    1     3]
                    [                           ]
                    [0    0    1    1    3    -1]
                    [                           ]
                    [0    0    1    1    3     0]


                  Added row 2 times , -1,  to row 4


                    [1    1    1    1    1     1]
                    [                           ]
                    [0    0    1    1    2     0]
                    [                           ]
                    [0    0    0    0    1     3]
                    [                           ]
                    [0    0    0    0    1    -1]
                    [                           ]
                    [0    0    1    1    3     0]


                  Added row 2 times , -1,  to row 5


                    [1    1    1    1    1     1]
                    [                           ]
                    [0    0    1    1    2     0]
                    [                           ]
                    [0    0    0    0    1     3]
                    [                           ]
                    [0    0    0    0    1    -1]
                    [                           ]
                    [0    0    0    0    1     0]


                          Searching column 4


                          Searching column 5


                  Added row 3 times , -1,  to row 4


                    [1    1    1    1    1     1]
                    [                           ]
                    [0    0    1    1    2     0]
                    [                           ]
                    [0    0    0    0    1     3]
                    [                           ]
                    [0    0    0    0    0    -4]
                    [                           ]
                    [0    0    0    0    1     0]


                  Added row 3 times , -1,  to row 5


                    [1    1    1    1    1     1]
                    [                           ]
                    [0    0    1    1    2     0]
                    [                           ]
                    [0    0    0    0    1     3]
                    [                           ]
                    [0    0    0    0    0    -4]
                    [                           ]
                    [0    0    0    0    0    -3]


                          Searching column 6


                 Added row 4 times , -3/4,  to row 5


                    [1    1    1    1    1     1]
                    [                           ]
                    [0    0    1    1    2     0]
                    [                           ]
                    [0    0    0    0    1     3]
                    [                           ]
                    [0    0    0    0    0    -4]
                    [                           ]
                    [0    0    0    0    0     0]


                    [1    1    1    1    1     1]
                    [                           ]
                    [0    0    1    1    2     0]
                    [                           ]
                    [0    0    0    0    1     3]
                    [                           ]
                    [0    0    0    0    0    -4]
                    [                           ]
                    [0    0    0    0    0     0]

> with(linalg);
Warning, new definition for norm
Warning, new definition for trace

  [BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp, QRdecomp,

        Wronskian, addcol, addrow, adj, adjoint, angle, augment,

        backsub, band, basis, bezout, blockmatrix, charmat, charpoly,

        cholesky, col, coldim, colspace, colspan, companion, concat,

        cond, copyinto, crossprod, curl, definite, delcols, delrows,

        det, diag, diverge, dotprod, eigenvals, eigenvalues,

        eigenvectors, eigenvects, entermatrix, equal, exponential,

        extend, ffgausselim, fibonacci, forwardsub, frobenius,

        gausselim, gaussjord, geneqns, genmatrix, grad, hadamard,

        hermite, hessian, hilbert, htranspose, ihermite, indexfunc,

        innerprod, intbasis, inverse, ismith, issimilar, iszero,

        jacobian, jordan, kernel, laplacian, leastsqrs, linsolve,

        matadd, matrix, minor, minpoly, mulcol, mulrow, multiply,

        norm, normalize, nullspace, orthog, permanent, pivot,

        potential, randmatrix, randvector, rank, ratform, row, rowdim,

        rowspace, rowspan, rref, scalarmul, singularvals, smith,

        stackmatrix, submatrix, subvector, sumbasis, swapcol, swaprow,

        sylvester, toeplitz, trace, transpose, vandermonde, vecpotent,

        vectdim, vector, wronskian]

> rref(B);

                     [1    1    0    0    0    0]
                     [                          ]
                     [0    0    1    1    0    0]
                     [                          ]
                     [0    0    0    0    1    0]
                     [                          ]
                     [0    0    0    0    0    1]
                     [                          ]
                     [0    0    0    0    0    0]

> infolevel['refpkg']:=1: ref(B,2);

                    [1    1    1    1    1     1]
                    [                           ]
                    [0    0    1    1    2     0]
                    [                           ]
                    [0    0    2    2    5     3]
                    [                           ]
                    [0    0    1    1    3    -1]
                    [                           ]
                    [0    0    1    1    3     0]

# See Also: refpkg[xref], refpkg[mysolve]