--- /dev/null
+## 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
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff