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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} =} ishermitian (@var{A})
## @deftypefnx {} {@var{tf} =} ishermitian (@var{A}, @var{tol})
## @deftypefnx {} {@var{tf} =} ishermitian (@var{A}, @qcode{"skew"})
## @deftypefnx {} {@var{tf} =} ishermitian (@var{A}, @qcode{"skew"}, @var{tol})
## Return true if @var{A} is a Hermitian or skew-Hermitian 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 Hermitian or @qcode{"skew"} for
## skew-Hermitian.
##
## Background: A matrix is Hermitian if the complex conjugate transpose of the
## matrix is equal to the original matrix: @w{@tcode{@var{A} == @var{A}'}}. If
## a tolerance is given then the calculation is
## @code{norm (@var{A} - @var{A}', Inf) / norm (@var{A}, Inf) < @var{tol}}.
##
## A matrix is skew-Hermitian if the complex conjugate 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 the calculation is
## @code{norm (@var{A} + @var{A}', Inf) / norm (@var{A}, Inf) < @var{tol}}.
## @seealso{issymmetric, isdefinite}
## @end deftypefn
function tf = ishermitian (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 ("ishermitian: second argument must be a non-negative scalar TOL, or one of the strings: 'skew' / 'nonskew'");
endif
endif
## Validate inputs
tf = (isnumeric (A) || islogical (A)) && issquare (A);
if (! tf)
return;
endif
if (! (strcmp (skewopt, "skew") || strcmp (skewopt, "nonskew")))
error ("ishermitian: SKEWOPT must be 'skew' or 'nonskew'");
endif
if (! (isnumeric (tol) && isscalar (tol) && tol >= 0))
error ("ishermitian: TOL must be a scalar >= 0");
endif
## Calculate Hermitian-ness
if (strcmp (skewopt, "nonskew"))
if (tol == 0)
## check for exact symmetry
tf = full (! any ((A != A')(:)));
else
if (islogical (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-Hermitian
if (tol == 0)
tf = full (! any ((A != -A')(:)));
else
if (islogical (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 (ishermitian (1))
%!assert (! ishermitian ([1, 2]))
%!assert (ishermitian ([]))
%!assert (ishermitian ([1, 2; 2, 1]))
%!assert (ishermitian ([1, 2.1; 2, 1.1], 0.2))
%!assert (ishermitian ([1, -2i; 2i, 1]))
%!assert (ishermitian (speye (100)), true) # Return full logical value.
%!assert (ishermitian (logical (eye (2))))
%!assert (! ishermitian (logical ([1 1; 0 1])))
%!assert (ishermitian (logical ([1 1; 0 1]), 0.5))
%!assert (ishermitian ([0, 2i; 2i, 0], "skew"))
%!assert (! ishermitian ([0, 2; -2, eps], "skew"))
%!assert (ishermitian ([0, 2; -2, eps], "skew", eps))
%!assert (! (ishermitian ("test")))
%!assert (! (ishermitian ("t")))
%!assert (! (ishermitian (["te"; "et"])))
%!assert (! ishermitian ({1}))
%!test
%! s.a = 1;
%! assert (! ishermitian (s));
## Test input validation
%!error <Invalid call> ishermitian ()
%!error <second argument must be> ishermitian (1, {"skew"})
%!error <SKEWOPT must be 'skew' or 'nonskew'> ishermitian (1, "foobar")
%!error <SKEWOPT must be 'skew' or 'nonskew'> ishermitian (1, "foobar")
%!error <TOL must be a scalar .= 0> ishermitian (1, "skew", {1})
%!error <TOL must be a scalar .= 0> ishermitian (1, "skew", [1 1])
%!error <TOL must be a scalar .= 0> ishermitian (1, -1)
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