Mercurial > jwe > octave
annotate scripts/sparse/svds.m @ 21116:ba3c66f4e1ca stable
svds.m: Modify BIST test for sigma=0 to reliably pass.
* svds.m: Only compare whatever was returned from svds to the correct values.
This may be fewer values then requested.
author | Rik <rik@octave.org> |
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date | Wed, 20 Jan 2016 06:56:52 -0800 |
parents | df437a52bcaf |
children | c1df36373272 e169fe73b57e |
rev | line source |
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1 ## Copyright (C) 2006-2015 David Bateman |
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2 ## |
11104 | 3 ## This file is part of Octave. |
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4 ## |
11104 | 5 ## Octave is free software; you can redistribute it and/or modify it |
6 ## under the terms of the GNU General Public License as published by | |
7 ## the Free Software Foundation; either version 3 of the License, or (at | |
8 ## your option) any later version. | |
9 ## | |
10 ## Octave is distributed in the hope that it will be useful, but | |
11 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU | |
13 ## General Public License for more details. | |
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14 ## |
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15 ## You should have received a copy of the GNU General Public License |
11104 | 16 ## along with Octave; see the file COPYING. If not, see |
17 ## <http://www.gnu.org/licenses/>. | |
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18 |
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19 ## -*- texinfo -*- |
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20 ## @deftypefn {Function File} {@var{s} =} svds (@var{A}) |
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21 ## @deftypefnx {Function File} {@var{s} =} svds (@var{A}, @var{k}) |
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22 ## @deftypefnx {Function File} {@var{s} =} svds (@var{A}, @var{k}, @var{sigma}) |
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23 ## @deftypefnx {Function File} {@var{s} =} svds (@var{A}, @var{k}, @var{sigma}, @var{opts}) |
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24 ## @deftypefnx {Function File} {[@var{u}, @var{s}, @var{v}] =} svds (@dots{}) |
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25 ## @deftypefnx {Function File} {[@var{u}, @var{s}, @var{v}, @var{flag}] =} svds (@dots{}) |
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26 ## |
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27 ## Find a few singular values of the matrix @var{A}. |
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28 ## |
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29 ## The singular values are calculated using |
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30 ## |
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31 ## @example |
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32 ## @group |
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33 ## [@var{m}, @var{n}] = size (@var{A}); |
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34 ## @var{s} = eigs ([sparse(@var{m}, @var{m}), @var{A}; |
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35 ## @var{A}', sparse(@var{n}, @var{n})]) |
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36 ## @end group |
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37 ## @end example |
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38 ## |
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39 ## The eigenvalues returned by @code{eigs} correspond to the singular values |
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40 ## of @var{A}. The number of singular values to calculate is given by @var{k} |
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41 ## and defaults to 6. |
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42 ## |
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43 ## The argument @var{sigma} specifies which singular values to find. When |
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44 ## @var{sigma} is the string @qcode{'L'}, the default, the largest singular |
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45 ## values of @var{A} are found. Otherwise, @var{sigma} must be a real scalar |
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46 ## and the singular values closest to @var{sigma} are found. As a corollary, |
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47 ## @code{@var{sigma} = 0} finds the smallest singular values. Note that for |
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48 ## relatively small values of @var{sigma}, there is a chance that the |
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49 ## requested number of singular values will not be found. In that case |
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50 ## @var{sigma} should be increased. |
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51 ## |
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52 ## @var{opts} is a structure defining options that @code{svds} will pass |
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53 ## to @code{eigs}. The possible fields of this structure are documented in |
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54 ## @code{eigs}. By default, @code{svds} sets the following three fields: |
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55 ## |
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56 ## @table @code |
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57 ## @item tol |
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58 ## The required convergence tolerance for the singular values. The default |
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59 ## value is 1e-10. @code{eigs} is passed @code{@var{tol} / sqrt(2)}. |
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60 ## |
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61 ## @item maxit |
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62 ## The maximum number of iterations. The default is 300. |
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63 ## |
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64 ## @item disp |
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65 ## The level of diagnostic printout (0|1|2). If @code{disp} is 0 then |
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66 ## diagnostics are disabled. The default value is 0. |
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67 ## @end table |
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68 ## |
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69 ## If more than one output is requested then @code{svds} will return an |
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70 ## approximation of the singular value decomposition of @var{A} |
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71 ## |
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72 ## @example |
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73 ## @var{A}_approx = @var{u}*@var{s}*@var{v}' |
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74 ## @end example |
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75 ## |
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76 ## @noindent |
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77 ## where @var{A}_approx is a matrix of size @var{A} but only rank @var{k}. |
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78 ## |
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79 ## @var{flag} returns 0 if the algorithm has succesfully converged, and 1 |
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80 ## otherwise. The test for convergence is |
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81 ## |
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82 ## @example |
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83 ## @group |
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84 ## norm (@var{A}*@var{v} - @var{u}*@var{s}, 1) <= @var{tol} * norm (@var{A}, 1) |
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85 ## @end group |
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86 ## @end example |
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87 ## |
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88 ## @code{svds} is best for finding only a few singular values from a large |
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89 ## sparse matrix. Otherwise, @code{svd (full (@var{A}))} will likely be more |
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90 ## efficient. |
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91 ## @end deftypefn |
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92 ## @seealso{svd, eigs} |
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93 |
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94 function [u, s, v, flag] = svds (A, k, sigma, opts) |
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95 |
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97 | |
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98 if (nargin < 1 || nargin > 4) |
8507 | 99 print_usage (); |
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100 endif |
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101 |
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102 if (ndims (A) > 2) |
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104 endif |
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105 |
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106 if (nargin < 4) |
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107 opts.tol = 0; # use ARPACK default |
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108 opts.disp = 0; |
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109 opts.maxit = 300; |
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110 else |
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111 if (! isstruct (opts)) |
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112 error ("svds: OPTS must be a structure"); |
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113 endif |
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114 if (! isfield (opts, "tol")) |
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115 opts.tol = 0; # use ARPACK default |
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116 else |
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117 opts.tol = opts.tol / root2; |
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118 endif |
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119 if (isfield (opts, "v0")) |
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120 if (! isvector (opts.v0) || (length (opts.v0) != sum (size (A)))) |
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121 error ("svds: OPTS.v0 must be a vector with rows(A)+columns(A) entries"); |
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122 endif |
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123 endif |
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124 endif |
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125 |
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127 if (isreal (A)) |
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128 sigma = "LA"; |
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129 else |
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130 sigma = "LR"; |
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131 endif |
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132 elseif (isscalar (sigma) && isnumeric (sigma) && isreal (sigma)) |
8507 | 133 if (sigma < 0) |
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134 error ("svds: SIGMA must be a positive real value"); |
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135 endif |
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136 else |
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137 error ("svds: SIGMA must be a positive real value or the string 'L'"); |
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138 endif |
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139 |
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140 [m, n] = size (A); |
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141 max_a = max (abs (nonzeros (A))); |
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142 if (isempty (max_a)) |
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143 max_a = 0; |
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144 endif |
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145 ## Must initialize variable value, otherwise it may appear to interpreter |
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146 ## that code is trying to call flag() colormap function. |
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147 flag = 0; |
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148 |
8507 | 149 if (max_a == 0) |
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150 s = zeros (k, 1); # special case of zero matrix |
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151 else |
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152 if (nargin < 2) |
8507 | 153 k = min ([6, m, n]); |
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154 else |
8507 | 155 k = min ([k, m, n]); |
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156 endif |
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157 |
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158 ## Scale everything by the 1-norm to make things more stable. |
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159 b = A / max_a; |
8507 | 160 b_opts = opts; |
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161 ## Call to eigs is always a symmetric matrix by construction |
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162 b_opts.issym = true; |
8507 | 163 b_sigma = sigma; |
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164 if (! ischar (b_sigma)) |
8507 | 165 b_sigma = b_sigma / max_a; |
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166 endif |
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167 |
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168 if (b_sigma == 0) |
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169 ## Find the smallest eigenvalues |
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170 ## The eigenvalues returns by eigs for sigma=0 are symmetric about 0. |
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171 ## As we are only interested in the positive eigenvalues, we have to |
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172 ## double k and then throw out the k negative eigenvalues. |
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173 ## Separately, if sigma is nonzero, but smaller than the smallest |
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174 ## singular value, ARPACK may not return k eigenvalues. However, as |
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175 ## computation scales with k we'd like to avoid doubling k for all |
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176 ## scalar values of sigma. |
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177 b_k = 2 * k; |
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178 else |
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179 b_k = k; # Normal case, find just the k largest eigenvalues |
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180 endif |
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181 |
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182 if (nargout > 1) |
8507 | 183 [V, s, flag] = eigs ([sparse(m,m), b; b', sparse(n,n)], |
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184 b_k, b_sigma, b_opts); |
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185 s = diag (s); |
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186 else |
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187 s = eigs ([sparse(m,m), b; b', sparse(n,n)], b_k, b_sigma, b_opts); |
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188 endif |
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189 |
8507 | 190 if (ischar (sigma)) |
191 norma = max (s); | |
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192 else |
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193 norma = normest (A); |
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194 endif |
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195 ## We wish to exclude all eigenvalues that are less than zero as these |
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196 ## are artifacts of the way the matrix passed to eigs is formed. There |
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197 ## is also the possibility that the value of sigma chosen is exactly |
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198 ## a singular value, and in that case we're dead!! So have to rely on |
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199 ## the warning from eigs. We exclude the singular values which are |
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200 ## less than or equal to zero to within some tolerance scaled by the |
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201 ## norm since if we don't we might end up with too many singular |
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202 ## values. |
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203 tol = norma * opts.tol; |
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204 ind = find (s > tol); |
8507 | 205 if (length (ind) < k) |
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206 ## Too few eigenvalues returned. Add in any zero eigenvalues of B, |
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207 ## including the nominally negative ones. |
8507 | 208 zind = find (abs (s) <= tol); |
209 p = min (length (zind), k - length (ind)); | |
210 ind = [ind; zind(1:p)]; | |
211 elseif (length (ind) > k) | |
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212 ## Too many eigenvalues returned. Select according to criterium. |
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213 if (b_sigma == 0) |
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214 ind = ind(end+1-k:end); # smallest eigenvalues |
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215 else |
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216 ind = ind(1:k); # largest eigenvalues |
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217 endif |
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218 endif |
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219 s = s(ind); |
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220 |
8507 | 221 if (length (s) < k) |
222 warning ("returning fewer singular values than requested"); | |
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223 if (! ischar (sigma)) |
10549 | 224 warning ("try increasing the value of sigma"); |
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225 endif |
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226 endif |
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227 |
8507 | 228 s = s * max_a; |
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229 endif |
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230 |
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231 if (nargout < 2) |
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232 u = s; |
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233 else |
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234 if (max_a == 0) |
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235 u = eye (m, k); |
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236 s = diag (s); |
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237 v = eye (n, k); |
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238 else |
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239 u = root2 * V(1:m,ind); |
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240 s = diag (s); |
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241 v = root2 * V(m+1:end,ind); |
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242 endif |
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243 |
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244 if (nargout > 3) |
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245 flag = (flag != 0); |
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246 endif |
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247 endif |
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248 |
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249 endfunction |
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250 |
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251 |
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252 %!shared n, k, A, u, s, v, opts, rand_state, randn_state, tol |
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253 %! n = 100; |
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254 %! k = 7; |
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255 %! A = sparse ([3:n,1:n,1:(n-2)],[1:(n-2),1:n,3:n],[ones(1,n-2),0.4*n*ones(1,n),ones(1,n-2)]); |
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256 %! [u,s,v] = svd (full (A)); |
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257 %! s = diag (s); |
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258 %! [~, idx] = sort (abs (s)); |
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259 %! s = s(idx); |
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260 %! u = u(:, idx); |
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261 %! v = v(:, idx); |
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262 %! randn_state = randn ("state"); |
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263 %! rand_state = rand ("state"); |
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264 %! randn ("state", 42); % Initialize to make normest function reproducible |
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265 %! rand ("state", 42); |
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266 %! opts.v0 = rand (2*n,1); % Initialize eigs ARPACK starting vector |
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267 %! % to guarantee reproducible results |
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268 %! |
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269 %!testif HAVE_ARPACK |
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270 %! [u2,s2,v2,flag] = svds (A,k); |
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271 %! s2 = diag (s2); |
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272 %! assert (flag, !1); |
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273 %! tol = 10 * eps() * norm(s2, 1); |
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274 %! assert (s2, s(end:-1:end-k+1), tol); |
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275 %! |
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276 %!testif HAVE_ARPACK, HAVE_UMFPACK |
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277 %! [u2,s2,v2,flag] = svds (A,k,0,opts); |
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278 %! s2 = diag (s2); |
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279 %! assert (flag, !1); |
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280 %! tol = 100 * eps() * norm(s2, 1); |
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281 %! assert (s2, s(length(s2):-1:1), tol); |
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282 %! |
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283 %!testif HAVE_ARPACK, HAVE_UMFPACK |
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284 %! idx = floor (n/2); |
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285 %! % Don't put sigma right on a singular value or there are convergence issues |
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286 %! sigma = 0.99*s(idx) + 0.01*s(idx+1); |
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287 %! [u2,s2,v2,flag] = svds (A,k,sigma,opts); |
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288 %! s2 = diag (s2); |
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289 %! assert (flag, !1); |
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290 %! tol = 10 * eps() * norm(s2, 1); |
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291 %! assert (s2, s((idx+floor(k/2)):-1:(idx-floor(k/2))), tol); |
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292 %! |
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293 %!testif HAVE_ARPACK |
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294 %! [u2,s2,v2,flag] = svds (zeros (10), k); |
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295 %! assert (u2, eye (10, k)); |
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296 %! assert (s2, zeros (k)); |
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297 %! assert (v2, eye (10, 7)); |
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298 %! |
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299 %!testif HAVE_ARPACK |
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300 %! s = svds (speye (10)); |
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301 %! assert (s, ones (6, 1), 8*eps); |
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302 |
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303 %!test |
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304 %! ## Restore random number generator seeds at end of tests |
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305 %! rand ("state", rand_state); |
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306 %! randn ("state", randn_state); |
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307 |