Mercurial > octave
view scripts/linear-algebra/rank.m @ 31221:f5755dbacd8d
maint: merge stable to default
author | Pantxo Diribarne <pantxo.diribarne@gmail.com> |
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date | Wed, 31 Aug 2022 22:04:02 +0200 |
parents | 5d3faba0342e |
children | 597f3ee61a48 |
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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 {} {@var{k} =} rank (@var{A}) ## @deftypefnx {} {@var{k} =} rank (@var{A}, @var{tol}) ## Compute the rank of matrix @var{A}, using the singular value decomposition. ## ## The rank is taken to be the number of singular values of @var{A} that are ## greater than the specified tolerance @var{tol}. If the second argument is ## omitted, it is taken to be ## ## @example ## tol = max (size (@var{A})) * sigma(1) * eps; ## @end example ## ## @noindent ## where @code{eps} is machine precision and @code{sigma(1)} is the largest ## singular value of @var{A}. ## ## The rank of a matrix is the number of linearly independent rows or columns ## and equals the dimension of the row and column space. The function ## @code{orth} may be used to compute an orthonormal basis of the column space. ## ## For testing if a system @code{@var{A}*@var{x} = @var{b}} of linear equations ## is solvable, one can use ## ## @example ## rank (@var{A}) == rank ([@var{A} @var{b}]) ## @end example ## ## In this case, @code{@var{x} = @var{A} \ @var{b}} finds a particular solution ## @var{x}. The general solution is @var{x} plus the null space of matrix ## @var{A}. The function @code{null} may be used to compute a basis of the ## null space. ## ## Example: ## ## @example ## @group ## A = [1 2 3 ## 4 5 6 ## 7 8 9]; ## rank (A) ## @result{} 2 ## @end group ## @end example ## ## @noindent ## In this example, the number of linearly independent rows is only 2 because ## the final row is a linear combination of the first two rows: ## ## @example ## A(3,:) == -A(1,:) + 2 * A(2,:) ## @end example ## ## @seealso{null, orth, sprank, svd, eps} ## @end deftypefn function k = rank (A, tol) if (nargin < 1) print_usage (); endif if (nargin == 1) sigma = svd (A); if (isempty (sigma)) tolerance = 0; else if (isa (A, "single")) tolerance = max (size (A)) * sigma (1) * eps ("single"); else tolerance = max (size (A)) * sigma (1) * eps; endif endif else sigma = svd (A); tolerance = tol; endif k = sum (sigma > tolerance); endfunction %!test %! A = [1 2 3 4 5 6 7; %! 4 5 6 7 8 9 12; %! 1 2 3.1 4 5 6 7; %! 2 3 4 5 6 7 8; %! 3 4 5 6 7 8 9; %! 4 5 6 7 8 9 10; %! 5 6 7 8 9 10 11]; %! assert (rank (A), 4); %!test %! A = [1 2 3 4 5 6 7; %! 4 5 6 7 8 9 12; %! 1 2 3.0000001 4 5 6 7; %! 4 5 6 7 8 9 12.00001; %! 3 4 5 6 7 8 9; %! 4 5 6 7 8 9 10; %! 5 6 7 8 9 10 11]; %! assert (rank (A), 4); %!test %! A = [1 2 3 4 5 6 7; %! 4 5 6 7 8 9 12; %! 1 2 3 4 5 6 7; %! 4 5 6 7 8 9 12.00001; %! 3 4 5 6 7 8 9; %! 4 5 6 7 8 9 10; %! 5 6 7 8 9 10 11]; %! assert (rank (A), 3); %!test %! A = [1 2 3 4 5 6 7; %! 4 5 6 7 8 9 12; %! 1 2 3 4 5 6 7; %! 4 5 6 7 8 9 12; %! 3 4 5 6 7 8 9; %! 4 5 6 7 8 9 10; %! 5 6 7 8 9 10 11]; %! assert (rank (A), 3); %!test %! A = eye (100); %! assert (rank (A), 100); %!assert (rank ([]), 0) %!assert (rank ([1:9]), 1) %!assert (rank ([1:9]'), 1) %!test %! A = [1, 2, 3; 1, 2.001, 3; 1, 2, 3.0000001]; %! assert (rank (A), 3); %! assert (rank (A,0.0009), 1); %! assert (rank (A,0.0006), 2); %! assert (rank (A,0.00000002), 3);