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########################################################################
##
## Copyright (C) 2000-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## This file is part of Octave.
##
## Octave is free software: you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {@var{r} =} rref (@var{A})
## @deftypefnx {} {@var{r} =} rref (@var{A}, @var{tol})
## @deftypefnx {} {[@var{r}, @var{k}] =} rref (@dots{})
## Return the reduced row echelon form of @var{A}.
##
## @var{tol} defaults to
## @code{eps * max (size (@var{A})) * norm (@var{A}, inf)}.
##
## The optional return argument @var{k} contains the vector of
## "bound variables", which are those columns on which elimination has been
## performed.
##
## @end deftypefn
function [A, k] = rref (A, tol)
if (nargin < 1)
print_usage ();
endif
if (ndims (A) > 2)
error ("rref: A must be a 2-dimensional matrix");
endif
[rows, cols] = size (A);
if (nargin < 2)
if (isa (A, "single"))
tol = eps ("single") * max (rows, cols) * norm (A, inf ("single"));
else
tol = eps * max (rows, cols) * norm (A, inf);
endif
endif
used = zeros (1, cols);
r = 1;
for c = 1:cols
## Find the pivot row
[m, pivot] = max (abs (A(r:rows,c)));
pivot = r + pivot - 1;
if (m <= tol)
## Skip column c, making sure the approximately zero terms are
## actually zero.
A(r:rows, c) = zeros (rows-r+1, 1);
else
## keep track of bound variables
used(1, c) = 1;
## Swap current row and pivot row
A([pivot, r], c:cols) = A([r, pivot], c:cols);
## Normalize pivot row
A(r, c:cols) = A(r, c:cols) / A(r, c);
## Eliminate the current column
ridx = [1:r-1, r+1:rows];
A(ridx, c:cols) = A(ridx, c:cols) - A(ridx, c) * A(r, c:cols);
## Check if done
if (r++ == rows)
break;
endif
endif
endfor
if (nargout > 1)
k = find (used);
endif
endfunction
%!test
%! a = [1];
%! [r k] = rref (a);
%! assert (r, [1], 2e-8);
%! assert (k, [1], 2e-8);
%!test
%! a = [1 3; 4 5];
%! [r k] = rref (a);
%! assert (rank (a), rank (r), 2e-8);
%! assert (r, eye (2), 2e-8);
%! assert (k == [1, 2] || k == [2, 1]);
%!test
%! a = [1 3; 4 5; 7 9];
%! [r k] = rref (a);
%! assert (rank (a), rank (r), 2e-8);
%! assert (r, eye (3)(:,1:2), 2e-8);
%! assert (k, [1 2], 2e-8);
%!test
%! a = [1 2 3; 2 4 6; 7 2 0];
%! [r k] = rref (a);
%! assert (rank (a), rank (r), 2e-8);
%! assert (r, [1 0 (3-7/2); 0 1 (7/4); 0 0 0], 2e-8);
%! assert (k, [1 2], 2e-8);
%!test
%! a = [1 2 1; 2 4 2.01; 2 4 2.1];
%! tol = 0.02;
%! [r k] = rref (a, tol);
%! assert (rank (a, tol), rank (r, tol), 2e-8);
%! tol = 0.2;
%! [r k] = rref (a, tol);
%! assert (rank (a, tol), rank (r, tol), 2e-8);
## Test input validation
%!error <Invalid call> rref ()
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