Mercurial > octave
view scripts/linear-algebra/cond.m @ 30564:796f54d4ddbf stable
update Octave Project Developers copyright for the new year
In files that have the "Octave Project Developers" copyright notice,
update for 2021.
In all .txi and .texi files except gpl.txi and gpl.texi in the
doc/liboctave and doc/interpreter directories, change the copyright
to "Octave Project Developers", the same as used for other source
files. Update copyright notices for 2022 (not done since 2019). For
gpl.txi and gpl.texi, change the copyright notice to be "Free Software
Foundation, Inc." and leave the date at 2007 only because this file
only contains the text of the GPL, not anything created by the Octave
Project Developers.
Add Paul Thomas to contributors.in.
author | John W. Eaton <jwe@octave.org> |
---|---|
date | Tue, 28 Dec 2021 18:22:40 -0500 |
parents | 7854d5752dd2 |
children | 5d3faba0342e |
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######################################################################## ## ## Copyright (C) 1993-2022 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 {} {} cond (@var{A}) ## @deftypefnx {} {} cond (@var{A}, @var{p}) ## Compute the @var{p}-norm condition number of a matrix with respect to ## inversion. ## ## @code{cond (@var{A})} is defined as ## @tex ## $ {\parallel A \parallel_p * \parallel A^{-1} \parallel_p .} $ ## @end tex ## @ifnottex ## @code{norm (@var{A}, @var{p}) * norm (inv (@var{A}), @var{p})}. ## @end ifnottex ## ## By default, @code{@var{p} = 2} is used which implies a (relatively slow) ## singular value decomposition. Other possible selections are ## @code{@var{p} = 1, Inf, "fro"} which are generally faster. For a full ## discussion of possible @var{p} values, @pxref{XREFnorm,,@code{norm}}. ## ## The condition number of a matrix quantifies the sensitivity of the matrix ## inversion operation when small changes are made to matrix elements. Ideally ## the condition number will be close to 1. When the number is large this ## indicates small changes (such as underflow or round-off error) will produce ## large changes in the resulting output. In such cases the solution results ## from numerical computing are not likely to be accurate. ## @seealso{condest, rcond, condeig, norm, svd} ## @end deftypefn function retval = cond (A, p = 2) if (nargin < 1) print_usage (); endif if (ndims (A) > 2) error ("cond: A must be a 2-D matrix"); endif if (p == 2) if (isempty (A)) retval = 0.0; elseif (any (! isfinite (A(:)))) error ("cond: A must not contain Inf or NaN values"); else sigma = svd (A); sigma_1 = sigma(1); sigma_n = sigma(end); if (sigma_1 == 0 || sigma_n == 0) retval = Inf; else retval = sigma_1 / sigma_n; endif endif else retval = norm (A, p) * norm (inv (A), p); endif endfunction %!test %! y = [7, 2, 3; 1, 3, 4; 6, 4, 5]; %! tol = 1e-6; %! type = {1, 2, "fro", "inf", inf}; %! for n = 1:numel (type) %! rcondition(n) = 1 / cond (y, type{n}); %! endfor %! assert (rcondition, [0.017460, 0.019597, 0.018714, 0.012022, 0.012022], tol); %!assert (cond ([1, 2; 2, 1]), 3, sqrt (eps)) %!assert (cond ([1, 2, 3; 4, 5, 6; 7, 8, 9]) > 1.0e+16) %!error <Invalid call> cond () %!error <A must be a 2-D matrix> cond (ones (1,3,3)) %!error <A must not contain Inf or NaN value> cond ([1, 2;Inf 4]) %!error <A must not contain Inf or NaN value> cond ([1, 2;NaN 4])