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general: update licenses to GPLv3+ and DESCRIPTION to mention non GPL code
author carandraug
date Tue, 13 Mar 2012 22:39:03 +0000
parents 49d1c30a4040
children
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1 ## Copyright (c) 2010 Andrew V. Knyazev <andrew.knyazev@ucdenver.edu>
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2 ## Copyright (c) 2010 Merico .E. Argentati <Merico.Argentati@ucdenver.edu>
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3 ## All rights reserved.
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4 ##
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5 ## Redistribution and use in source and binary forms, with or without
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6 ## modification, are permitted provided that the following conditions are met:
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7 ##
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8 ## 1 Redistributions of source code must retain the above copyright notice,
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9 ## this list of conditions and the following disclaimer.
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10 ## 2 Redistributions in binary form must reproduce the above copyright
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11 ## notice, this list of conditions and the following disclaimer in the
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12 ## documentation and/or other materials provided with the distribution.
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13 ## 3 Neither the name of the author nor the names of its contributors may be
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14 ## used to endorse or promote products derived from this software without
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15 ## specific prior written permission.
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16 ##
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17 ## THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ''AS IS''
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18 ## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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19 ## IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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20 ## ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR
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21 ## ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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22 ## DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
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23 ## SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
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24 ## CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
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25 ## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
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26 ## OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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27
49d1c30a4040 majle: new function for the general package. Submitted by Andrew Knyazev on the mailing list
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28 %MAJLE (Weak) Majorization check
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29 % S = MAJLE(X,Y) checks if the real part of X is (weakly) majorized by
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30 % the real part of Y, where X and Y must be numeric (full or sparse)
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31 % arrays. It returns S=0, if there is no weak majorization of X by Y,
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32 % S=1, if there is a weak majorization of X by Y, or S=2, if there is a
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33 % strong majorization of X by Y. The shapes of X and Y are ignored.
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34 % NUMEL(X) and NUMEL(Y) may be different, in which case one of them is
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35 % appended with zeros to match the sizes with the other and, in case of
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36 % any negative components, a special warning is issued.
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37 %
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38 % S = MAJLE(X,Y,MAJLETOL) allows in addition to specify the tolerance in
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39 % all inequalities [S,Z] = MAJLE(X,Y,MAJLETOL) also outputs a row vector
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40 % Z, which appears in the definition of the (weak) majorization. In the
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41 % traditional case, where the real vectors X and Y are of the same size,
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42 % Z = CUMSUM(SORT(Y,'descend')-SORT(X,'descend')). Here, X is weakly
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43 % majorized by Y, if MIN(Z)>0, and strongly majorized if MIN(Z)=0, see
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44 % http://en.wikipedia.org/wiki/Majorization
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45 %
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46 % The value of MAJLETOL depends on how X and Y have been computed, i.e.,
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47 % on what the level of error in X or Y is. A good minimal starting point
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48 % should be MAJLETOL=eps*MAX(NUMEL(X),NUMEL(Y)). The default is 0.
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49 %
49d1c30a4040 majle: new function for the general package. Submitted by Andrew Knyazev on the mailing list
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50 % % Examples:
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51 % x = [2 2 2]; y = [1 2 3]; s = majle(x,y)
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52 % % returns the value 2.
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53 % x = [2 2 2]; y = [1 2 4]; s = majle(x,y)
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54 % % returns the value 1.
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55 % x = [2 2 2]; y = [1 2 2]; s = majle(x,y)
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56 % % returns the value 0.
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57 % x = [2 2 2]; y = [1 2 2]; [s,z] = majle(x,y)
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58 % % also returns the vector z = [ 0 0 -1].
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59 % x = [2 2 2]; y = [1 2 2]; s = majle(x,y,1)
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60 % % returns the value 2.
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61 % x = [2 2]; y = [1 2 2]; s = majle(x,y)
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62 % % returns the value 1 and warns on tailing with zeros
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63 % x = [2 2]; y = [-1 2 2]; s = majle(x,y)
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64 % % returns the value 0 and gives two warnings on tailing with zeros
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65 % x = [2 -inf]; y = [4 inf]; [s,z] = majle(x,y)
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66 % % returns s = 1 and z = [Inf Inf].
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67 % x = [2 inf]; y = [4 inf]; [s,z] = majle(x,y)
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68 % % returns s = 1 and z = [NaN NaN] and a warning on NaNs in z.
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69 % x=speye(2); y=sparse([0 2; -1 1]); s = majle(x,y)
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70 % % returns the value 2.
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71 % x = [2 2; 2 2]; y = [1 3 4]; [s,z] = majle(x,y) %and
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72 % x = [2 2; 2 2]+i; y = [1 3 4]-2*i; [s,z] = majle(x,y)
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73 % % both return s = 2 and z = [2 3 2 0].
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74 % x = [1 1 1 1 0]; y = [1 1 1 1 1 0 0]'; s = majle(x,y)
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75 % % returns the value 1 and warns on tailing with zeros
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76 %
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77 % % One can use this function to check numerically the validity of the
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78 % Schur-Horn,Lidskii-Mirsky-Wielandt, and Gelfand-Naimark theorems:
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79 % clear all; n=100; majleTol=n*n*eps;
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80 % A = randn(n,n); A = A'+A; eA = -sort(-eig(A)); dA = diag(A);
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81 % majle(dA,eA,majleTol) % returns the value 2
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82 % % which is the Schur-Horn theorem; and
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83 % B=randn(n,n); B=B'+B; eB=-sort(-eig(B));
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84 % eAmB=-sort(-eig(A-B));
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85 % majle(eA-eB,eAmB,majleTol) % returns the value 2
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86 % % which is the Lidskii-Mirsky-Wielandt theorem; finally
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87 % A = randn(n,n); sA = -sort(-svd(A));
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88 % B = randn(n,n); sB = -sort(-svd(B));
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89 % sAB = -sort(-svd(A*B));
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90 % majle(log2(sAB)-log2(sA), log2(sB), majleTol) % retuns the value 2
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91 % majle(log2(sAB)-log2(sB), log2(sA), majleTol) % retuns the value 2
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92 % % which are the log versions of the Gelfand-Naimark theorems
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93
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94 % Tested in MATLAB 7.9.0.529 (R2009b) and Octave 3.2.3.
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95 function [s,z]=majle(x,y,majleTol)
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96
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97 if (nargin < 2)
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98 error('MAJORIZATION:majle:NotEnoughInputs',...
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99 'Not enough input arguments.');
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100 end
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101 if (nargin > 3)
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102 error('MAJORIZATION:majle:TooManyInputs',...
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103 'Too many input arguments.');
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104 end
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105 if (nargout > 2)
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106 error('MAJORIZATION:majle:TooManyOutputs',...
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107 'Too many output arguments.');
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108 end
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109
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110 % Assign default values to unspecified parameters
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111 if (nargin == 2)
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112 majleTol = 0;
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113 end
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114
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115 % transform into real (row) vectors
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116 x=real(x); xc=reshape(x,1,numel(x)); clear x;
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117 y=real(y); yc=reshape(y,1,numel(y)); clear y;
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118
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119 % sort both vectors in descending order
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120 xc=-sort(-xc); yc=-sort(-yc);
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121
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122 % tail with zeros the shorter vector to make vectors of the same length
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123 if size(xc,2)~=size(yc,2)
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124 checkForNegative = (xc(end) < -majleTol) || (yc(end) < -majleTol);
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125 warning('MAJORIZATION:majle:ResizeVectors', ...
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126 'The input vectors have different sizes. Tailing with zeros.');
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127 yc=[yc zeros(size(xc,2)-size(yc,2),1)'];
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128 xc=[xc zeros(size(yc,2)-size(xc,2),1)'];
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129 % but warn if negative
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130 if checkForNegative
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131 warning('MAJORIZATION:majle:ResizeNegativeVectors', ...
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132 sprintf('%s%s\n%s\n%s', ...
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133 'At least one of the input vectors ',...
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134 'has negative components.',...
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135 ' Tailing with zeros is most likely senseless.',...
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136 ' Make sure that you know what you are doing.'));
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137 % sort again both vectors in descending order
49d1c30a4040 majle: new function for the general package. Submitted by Andrew Knyazev on the mailing list
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138 xc=-sort(-xc); yc=-sort(-yc);
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139 end
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140 end
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141 z=cumsum(yc-xc);
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142
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143 %check for NaNs in z
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144 if any(isnan(z))
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145 warning('MAJORIZATION:majle:NaNsInComparisons', ...
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146 sprintf('%s%s\n%s\n%s', ...
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147 'At least one of the input vectors ',...
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148 'includes -Inf, Inf, or NaN components.',...
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149 ' Some comparisons could not be made. ',...
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150 ' Make sure that you know what you are doing.'));
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151 end
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152
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153 if min(z) < -majleTol
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154 s=0; % no majorization
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155 elseif abs(z(end)) <= majleTol
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156 s=2; % strong majorization
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157 else
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158 s=1; % weak majorization
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159 end
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160 endfunction