Mercurial > octave
view scripts/linear-algebra/null.m @ 29154:24750f40cfec
null: overhaul function with respect to empty input (bug #59630)
* scripts/linear-algebra/null.m: use economy-sized svd if possible, like Matlab
publicly documents. Overhaul docstring with respect to common linear-algebra
matrix conventions (bug #59564). Regard corner cases for empty input matrices
to enable proper computations with the output. Complete test coverage for
double and single input.
author | Kai T. Ohlhus <k.ohlhus@gmail.com> |
---|---|
date | Mon, 07 Dec 2020 18:20:54 +0900 |
parents | 9e43deb9bfc3 |
children | 7854d5752dd2 |
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######################################################################## ## ## Copyright (C) 1994-2020 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{Z} =} null (@var{A}) ## @deftypefnx {} {@var{Z} =} null (@var{A}, @var{tol}) ## Return an orthonormal basis @var{Z} of the null space of @var{A}. ## ## The dimension of the null space @var{Z} is taken as the number of singular ## values of @var{A} not greater than @var{tol}. If the argument @var{tol} ## is missing, it is computed as ## ## @example ## max (size (@var{A})) * max (svd (@var{A}, 0)) * eps ## @end example ## @seealso{orth, svd} ## @end deftypefn function Z = null (A, tol) if (nargin < 1) print_usage (); elseif (nargin == 2 && strcmp (tol, "r")) error ("null: option for rational not yet implemented"); endif [~, S, V] = svd (A, 0); # Use economy-sized svd if possible. ## In case of A = [], zeros (0,X), zeros (X,0) Matlab R2020b seems to ## simply return the nullspace "V" of the svd-decomposition (bug #59630). if (isempty (A)) Z = V; else out_cls = class (V); s = diag (S); if (nargin == 1) tol = max (size (A)) * s (1) * eps (out_cls); endif rank = sum (s > tol); cols = columns (A); if (rank < cols) Z = V(:, rank+1:cols); Z(abs (Z) < eps (out_cls)) = 0; else Z = zeros (cols, 0, out_cls); endif endif endfunction ## Exact tests %!test %! A = { ... %! [], []; ... %! zeros(1,0), []; ... %! zeros(4,0), []; ... %! zeros(0,1), 1; ... %! zeros(0,4), eye(4); ... %! 0, 1; ... %! 1, zeros(1,0); ... %! [1 0; 0 1], zeros(2,0); ... %! [1 0; 1 0], [0 1]'; ... %! }; %! for i = 1:size (A, 1) %! assert (null (A{i,1}), A{i,2}); %! assert (null (single (A{i,1})), single (A{i,2})); %! endfor ## Inexact tests %!test %! A = { ... %! [1 1; 0 0], [-1/sqrt(2) 1/sqrt(2)]'; ... %! }; %! for i = 1:size (A, 1) %! assert (null (A{i,1}), A{i,2}, eps); %! assert (null (single (A{i,1})), single (A{i,2}), eps); %! endfor ## Tests with tolerance input %!test %! tol = 1e-4; %! A = { ... %! @(e) [1 0; 0 tol-e], [0 1]'; ... %! @(e) [1 0; 0 tol+e], zeros(2,0); ... %! }; %! for i = 1:size (A, 1) %! assert (null (A{i,1}(eps ("double")), tol), A{i,2}); %! assert (null (single (A{i,1}(eps ("single"))), tol), single (A{i,2})); %! endfor ## Input tests %!assert (null (uint8 ([])), [])