Mercurial > octave
view scripts/linear-algebra/isdefinite.m @ 31230:6646f2b5a3d1
lcm.m: Emit warnings when results not exact (Bug #32924)
author | Arun Giridhar <arungiridhar@gmail.com> |
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date | Sat, 17 Sep 2022 04:22:38 -0400 |
parents | 83f9f8bda883 |
children | 597f3ee61a48 |
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######################################################################## ## ## Copyright (C) 2003-2022 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} =} isdefinite (@var{A}) ## @deftypefnx {} {@var{tf} =} isdefinite (@var{A}, @var{tol}) ## Return true if @var{A} is symmetric positive definite matrix within the ## tolerance specified by @var{tol}. ## ## If @var{tol} is omitted, use a tolerance of ## @code{100 * eps * norm (@var{A}, "fro")}. ## ## Background: A positive definite matrix has eigenvalues which are all ## greater than zero. A positive semi-definite matrix has eigenvalues which ## are all greater than or equal to zero. The matrix @var{A} is very likely to ## be positive semi-definite if the following two conditions hold for a ## suitably small tolerance @var{tol}. ## ## @example ## @group ## isdefinite (@var{A}) @result{} 0 ## isdefinite (@var{A} + 5*@var{tol}, @var{tol}) @result{} 1 ## @end group ## @end example ## @seealso{issymmetric, ishermitian} ## @end deftypefn function tf = isdefinite (A, tol) if (nargin < 1) print_usage (); endif ## Validate inputs tf = false; if (! isnumeric (A)) return; endif if (! isfloat (A)) A = double (A); endif if (nargin == 1) tol = 100 * eps (class (A)) * norm (A, "fro"); elseif (! (isnumeric (tol) && isscalar (tol) && tol >= 0)) error ("isdefinite: TOL must be a scalar >= 0"); endif if (! ishermitian (A, tol)) return; endif e = tol * eye (rows (A)); [~, p] = chol (A - e); if (p == 0) tf = true; endif endfunction %!test %! A = [-1, 0; 0, -1]; %! assert (isdefinite (A), false); %!test %! A = [1, 0; 0, 1]; %! assert (isdefinite (A), true); %!test %! A = [2, -1, 0; -1, 2, -1; 0, -1, 2]; %! assert (isdefinite (A), true); ## Test for positive semi-definite matrix %!test %! A = [1, 0; 0, 0]; %! assert (isdefinite (A), false); %! tol = 100*eps; %! assert (isdefinite (A+5*tol, tol), true); %!assert (! isdefinite (magic (3))) %!error <Invalid call> isdefinite () %!error <TOL must be a scalar .= 0> isdefinite (1, {1}) %!error <TOL must be a scalar .= 0> isdefinite (1, [1 1]) %!error <TOL must be a scalar .= 0> isdefinite (1, -1)