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author | carandraug |
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date | Tue, 13 Mar 2012 22:39:03 +0000 |
parents | 49d1c30a4040 |
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## Copyright (c) 2010 Andrew V. Knyazev <andrew.knyazev@ucdenver.edu> ## Copyright (c) 2010 Merico .E. Argentati <Merico.Argentati@ucdenver.edu> ## All rights reserved. ## ## Redistribution and use in source and binary forms, with or without ## modification, are permitted provided that the following conditions are met: ## ## 1 Redistributions of source code must retain the above copyright notice, ## this list of conditions and the following disclaimer. ## 2 Redistributions in binary form must reproduce the above copyright ## notice, this list of conditions and the following disclaimer in the ## documentation and/or other materials provided with the distribution. ## 3 Neither the name of the author nor the names of its contributors may be ## used to endorse or promote products derived from this software without ## specific prior written permission. ## ## THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ''AS IS'' ## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE ## IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ## ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR ## ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL ## DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR ## SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER ## CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, ## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE ## OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. %MAJLE (Weak) Majorization check % S = MAJLE(X,Y) checks if the real part of X is (weakly) majorized by % the real part of Y, where X and Y must be numeric (full or sparse) % arrays. It returns S=0, if there is no weak majorization of X by Y, % S=1, if there is a weak majorization of X by Y, or S=2, if there is a % strong majorization of X by Y. The shapes of X and Y are ignored. % NUMEL(X) and NUMEL(Y) may be different, in which case one of them is % appended with zeros to match the sizes with the other and, in case of % any negative components, a special warning is issued. % % S = MAJLE(X,Y,MAJLETOL) allows in addition to specify the tolerance in % all inequalities [S,Z] = MAJLE(X,Y,MAJLETOL) also outputs a row vector % Z, which appears in the definition of the (weak) majorization. In the % traditional case, where the real vectors X and Y are of the same size, % Z = CUMSUM(SORT(Y,'descend')-SORT(X,'descend')). Here, X is weakly % majorized by Y, if MIN(Z)>0, and strongly majorized if MIN(Z)=0, see % http://en.wikipedia.org/wiki/Majorization % % The value of MAJLETOL depends on how X and Y have been computed, i.e., % on what the level of error in X or Y is. A good minimal starting point % should be MAJLETOL=eps*MAX(NUMEL(X),NUMEL(Y)). The default is 0. % % % Examples: % x = [2 2 2]; y = [1 2 3]; s = majle(x,y) % % returns the value 2. % x = [2 2 2]; y = [1 2 4]; s = majle(x,y) % % returns the value 1. % x = [2 2 2]; y = [1 2 2]; s = majle(x,y) % % returns the value 0. % x = [2 2 2]; y = [1 2 2]; [s,z] = majle(x,y) % % also returns the vector z = [ 0 0 -1]. % x = [2 2 2]; y = [1 2 2]; s = majle(x,y,1) % % returns the value 2. % x = [2 2]; y = [1 2 2]; s = majle(x,y) % % returns the value 1 and warns on tailing with zeros % x = [2 2]; y = [-1 2 2]; s = majle(x,y) % % returns the value 0 and gives two warnings on tailing with zeros % x = [2 -inf]; y = [4 inf]; [s,z] = majle(x,y) % % returns s = 1 and z = [Inf Inf]. % x = [2 inf]; y = [4 inf]; [s,z] = majle(x,y) % % returns s = 1 and z = [NaN NaN] and a warning on NaNs in z. % x=speye(2); y=sparse([0 2; -1 1]); s = majle(x,y) % % returns the value 2. % x = [2 2; 2 2]; y = [1 3 4]; [s,z] = majle(x,y) %and % x = [2 2; 2 2]+i; y = [1 3 4]-2*i; [s,z] = majle(x,y) % % both return s = 2 and z = [2 3 2 0]. % x = [1 1 1 1 0]; y = [1 1 1 1 1 0 0]'; s = majle(x,y) % % returns the value 1 and warns on tailing with zeros % % % One can use this function to check numerically the validity of the % Schur-Horn,Lidskii-Mirsky-Wielandt, and Gelfand-Naimark theorems: % clear all; n=100; majleTol=n*n*eps; % A = randn(n,n); A = A'+A; eA = -sort(-eig(A)); dA = diag(A); % majle(dA,eA,majleTol) % returns the value 2 % % which is the Schur-Horn theorem; and % B=randn(n,n); B=B'+B; eB=-sort(-eig(B)); % eAmB=-sort(-eig(A-B)); % majle(eA-eB,eAmB,majleTol) % returns the value 2 % % which is the Lidskii-Mirsky-Wielandt theorem; finally % A = randn(n,n); sA = -sort(-svd(A)); % B = randn(n,n); sB = -sort(-svd(B)); % sAB = -sort(-svd(A*B)); % majle(log2(sAB)-log2(sA), log2(sB), majleTol) % retuns the value 2 % majle(log2(sAB)-log2(sB), log2(sA), majleTol) % retuns the value 2 % % which are the log versions of the Gelfand-Naimark theorems % Tested in MATLAB 7.9.0.529 (R2009b) and Octave 3.2.3. function [s,z]=majle(x,y,majleTol) if (nargin < 2) error('MAJORIZATION:majle:NotEnoughInputs',... 'Not enough input arguments.'); end if (nargin > 3) error('MAJORIZATION:majle:TooManyInputs',... 'Too many input arguments.'); end if (nargout > 2) error('MAJORIZATION:majle:TooManyOutputs',... 'Too many output arguments.'); end % Assign default values to unspecified parameters if (nargin == 2) majleTol = 0; end % transform into real (row) vectors x=real(x); xc=reshape(x,1,numel(x)); clear x; y=real(y); yc=reshape(y,1,numel(y)); clear y; % sort both vectors in descending order xc=-sort(-xc); yc=-sort(-yc); % tail with zeros the shorter vector to make vectors of the same length if size(xc,2)~=size(yc,2) checkForNegative = (xc(end) < -majleTol) || (yc(end) < -majleTol); warning('MAJORIZATION:majle:ResizeVectors', ... 'The input vectors have different sizes. Tailing with zeros.'); yc=[yc zeros(size(xc,2)-size(yc,2),1)']; xc=[xc zeros(size(yc,2)-size(xc,2),1)']; % but warn if negative if checkForNegative warning('MAJORIZATION:majle:ResizeNegativeVectors', ... sprintf('%s%s\n%s\n%s', ... 'At least one of the input vectors ',... 'has negative components.',... ' Tailing with zeros is most likely senseless.',... ' Make sure that you know what you are doing.')); % sort again both vectors in descending order xc=-sort(-xc); yc=-sort(-yc); end end z=cumsum(yc-xc); %check for NaNs in z if any(isnan(z)) warning('MAJORIZATION:majle:NaNsInComparisons', ... sprintf('%s%s\n%s\n%s', ... 'At least one of the input vectors ',... 'includes -Inf, Inf, or NaN components.',... ' Some comparisons could not be made. ',... ' Make sure that you know what you are doing.')); end if min(z) < -majleTol s=0; % no majorization elseif abs(z(end)) <= majleTol s=2; % strong majorization else s=1; % weak majorization end endfunction