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########################################################################
##
## Copyright (C) 1996-2024 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{tf} =} issymmetric (@var{A})
## @deftypefnx {} {@var{tf} =} issymmetric (@var{A}, @var{tol})
## @deftypefnx {} {@var{tf} =} issymmetric (@var{A}, @qcode{"skew"})
## @deftypefnx {} {@var{tf} =} issymmetric (@var{A}, @qcode{"skew"}, @var{tol})
## Return true if @var{A} is a symmetric or skew-symmetric numeric matrix
## within the tolerance specified by @var{tol}.
##
## The default tolerance is zero (uses faster code).
##
## The type of symmetry to check may be specified with the additional input
## @qcode{"nonskew"} (default) for regular symmetry or @qcode{"skew"} for
## skew-symmetry.
##
## Background: A matrix is symmetric if the transpose of the matrix is equal
## to the original matrix: @w{@tcode{@var{A} == @var{A}.'}}. If a tolerance
## is given then symmetry is determined by
## @code{norm (@var{A} - @var{A}.', Inf) / norm (@var{A}, Inf) < @var{tol}}.
##
## A matrix is skew-symmetric if the transpose of the matrix is equal to the
## negative of the original matrix: @w{@tcode{@var{A} == -@var{A}.'}}. If a
## tolerance is given then skew-symmetry is determined by
## @code{norm (@var{A} + @var{A}.', Inf) / norm (@var{A}, Inf) < @var{tol}}.
## @seealso{ishermitian, isdefinite}
## @end deftypefn
function tf = issymmetric (A, skewopt = "nonskew", tol = 0)
if (nargin < 1)
print_usage ();
endif
if (nargin == 2)
## Decode whether second argument is skewopt or tol
if (isnumeric (skewopt))
tol = skewopt;
skewopt = "nonskew";
elseif (! ischar (skewopt))
error ("issymmetric: second argument must be a non-negative scalar TOL, or one of the strings: 'skew' / 'nonskew'");
endif
endif
## Validate inputs
if (! (isnumeric (A) || islogical (A) || ischar (A)))
error ("issymmetric: A must be a numeric, logical, or character matrix");
endif
if (! (strcmp (skewopt, "skew") || strcmp (skewopt, "nonskew")))
error ("issymmetric: SKEWOPT must be 'skew' or 'nonskew'");
endif
if (! (isnumeric (tol) && isscalar (tol) && tol >= 0))
error ("issymmetric: TOL must be a scalar >= 0");
endif
if (! issquare (A))
tf = false;
return;
endif
## Calculate symmetry
if (strcmp (skewopt, "nonskew"))
if (tol == 0)
## check for exact symmetry
tf = full (! any ((A != A.')(:)));
else
if (! isnumeric (A))
## Hack to allow norm to work. Choose single to minimize memory.
A = single (A);
endif
norm_x = norm (A, Inf);
tf = norm_x == 0 || norm (A - A.', Inf) / norm_x <= tol;
endif
else
## skew symmetry
if (tol == 0)
tf = full (! any ((A != -A.')(:)));
else
if (! isnumeric (A))
## Hack to allow norm to work. Choose single to minimize memory.
A = single (A);
endif
norm_x = norm (A, Inf);
tf = norm_x == 0 || norm (A + A.', Inf) / norm_x <= tol;
endif
endif
endfunction
%!assert (issymmetric (1))
%!assert (! issymmetric ([1, 2]))
%!assert (issymmetric ([]))
%!assert (issymmetric ([1, 2; 2, 1]))
%!assert (issymmetric ([1, 2.1; 2, 1.1], 0.2))
%!assert (issymmetric ([1, 2i; 2i, 1]))
%!assert (issymmetric (speye (100)), true) # Return full logical value.
%!assert (! issymmetric ([0, 2; -2, 0], "nonskew"))
%!assert (issymmetric ([0, 2; -2, 0], "skew"))
%!assert (! issymmetric ([0, 2; -2, eps], "skew"))
%!assert (issymmetric ([0, 2; -2, eps], "skew", eps))
%!assert (issymmetric (logical (eye (2))))
%!assert (! issymmetric (logical ([1 1; 0 1])))
%!assert (issymmetric (logical ([1 1; 0 1]), 0.5))
%!assert (! issymmetric ("test"))
%!assert (issymmetric ("t"))
%!assert (issymmetric (["te"; "et"]))
## Test input validation
%!error <Invalid call> issymmetric ()
%!error <second argument must be> issymmetric (1, {"skew"})
%!error <A must be a numeric,.* matrix> issymmetric ({1})
%!error <SKEWOPT must be 'skew' or 'nonskew'> issymmetric (1, "foobar")
%!error <TOL must be a scalar .= 0> issymmetric (1, "skew", {1})
%!error <TOL must be a scalar .= 0> issymmetric (1, "skew", [1 1])
%!error <TOL must be a scalar .= 0> issymmetric (1, -1)
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