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svds.m: Overhaul code while fixing bug #29721. Return smallest singular values if sigma == 0 (Bug #29721). Avoid calculating U and V matrices unless requested. Correctly handle zero matrix input Improve documentation string.
author Rik <octave@nomad.inbox5.com>
date Sun, 30 May 2010 13:45:50 -0700
parents b90328525985
children a4f482e66b65
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1 ## Copyright (C) 2006, 2008, 2009 David Bateman
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2 ##
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3 ## This program is free software; you can redistribute it and/or modify
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4 ## it under the terms of the GNU General Public License as published by
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5 ## the Free Software Foundation; either version 2 of the License, or
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6 ## (at your option) any later version.
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7 ##
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8 ## This program is distributed in the hope that it will be useful,
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9 ## but WITHOUT ANY WARRANTY; without even the implied warranty of
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10 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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11 ## GNU General Public License for more details.
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12 ##
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13 ## You should have received a copy of the GNU General Public License
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14 ## along with this program; If not, see <http://www.gnu.org/licenses/>.
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15
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16 ## -*- texinfo -*-
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17 ## @deftypefn {Function File} {@var{s} =} svds (@var{a})
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18 ## @deftypefnx {Function File} {@var{s} =} svds (@var{a}, @var{k})
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19 ## @deftypefnx {Function File} {@var{s} =} svds (@var{a}, @var{k}, @var{sigma})
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20 ## @deftypefnx {Function File} {@var{s} =} svds (@var{a}, @var{k}, @var{sigma}, @var{opts})
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21 ## @deftypefnx {Function File} {[@var{u}, @var{s}, @var{v}] =} svds (@dots{})
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22 ## @deftypefnx {Function File} {[@var{u}, @var{s}, @var{v}, @var{flag}] =} svds (@dots{})
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23 ##
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24 ## Find a few singular values of the matrix @var{a}. The singular values
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25 ## are calculated using
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26 ##
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27 ## @example
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28 ## @group
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29 ## [@var{m}, @var{n}] = size(@var{a})
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30 ## @var{s} = eigs([sparse(@var{m}, @var{m}), @var{a};
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31 ## @var{a}', sparse(@var{n}, @var{n})])
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32 ## @end group
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33 ## @end example
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34 ##
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35 ## The eigenvalues returned by @code{eigs} correspond to the singular values
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36 ## of @var{a}. The number of singular values to calculate is given by @var{k}
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37 ## and defaults to 6.
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38 ##
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39 ## The argument @var{sigma} specifies which singular values to find. When
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40 ## @var{sigma} is the string 'L', the default, the largest singular values of
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41 ## @var{a} are found. Otherwise, @var{sigma} must be a real scalar and the
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42 ## singular values closest to @var{sigma} are found. As a corollary,
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43 ## @code{@var{sigma} = 0} finds the smallest singular values. Note that for
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44 ## relatively small values of @var{sigma}, there is a chance that the requested
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45 ## number of singular values will not be found. In that case @var{sigma}
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46 ## should be increased.
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47 ##
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48 ## @var{opts} is a structure defining options that @code{svds} will pass
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49 ## to @code{eigs}. The possible fields of this structure are documented in
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50 ## @code{eigs}. By default, @code{svds} sets the following three fields:
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51 ##
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52 ## @table @code
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53 ## @item tol
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54 ## The required convergence tolerance for the singular values. The default
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55 ## value is 1e-10. @code{eigs} is passed @code{@var{tol} / sqrt(2)}.
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56 ##
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57 ## @item maxit
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58 ## The maximum number of iterations. The default is 300.
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59 ##
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60 ## @item disp
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61 ## The level of diagnostic printout (0|1|2). If @code{disp} is 0 then
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62 ## diagnostics are disabled. The default value is 0.
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63 ## @end table
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64 ##
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65 ## If more than one output is requested then @code{svds} will return an
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66 ## approximation of the singular value decomposition of @var{a}
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67 ##
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68 ## @example
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69 ## @var{a}_approx = @var{u}*@var{s}*@var{v}'
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70 ## @end example
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71 ##
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72 ## where @var{a}_approx is a matrix of size @var{a} but only rank @var{k}.
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73 ##
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74 ## @var{flag} returns 0 if the algorithm has succesfully converged, and 1
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75 ## otherwise. The test for convergence is
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76 ##
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77 ## @example
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78 ## @group
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79 ## norm (@var{a}*@var{v} - @var{u}*@var{s}, 1) <= @var{tol} * norm (@var{a}, 1)
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80 ## @end group
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81 ## @end example
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82 ##
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83 ## @code{svds} is best for finding only a few singular values from a large sparse
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84 ## matrix. Otherwise, @code{svd (full(@var{a}))} will likely be more efficient.
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85 ## @end deftypefn
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86 ## @seealso{svd, eigs}
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87
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88 function [u, s, v, flag] = svds (a, k, sigma, opts)
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89
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90 persistent root2 = sqrt (2);
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91
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92 if (nargin < 1 || nargin > 4)
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93 print_usage ();
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94 endif
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95
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96 if (ndims(a) > 2)
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97 error ("svds: A must be a 2D matrix")
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98 endif
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99
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100 if (nargin < 4)
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101 opts.tol = 1e-10 / root2;
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102 opts.disp = 0;
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103 opts.maxit = 300;
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104 else
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105 if (!isstruct (opts))
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106 error ("svds: OPTS must be a structure");
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107 endif
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108 if (!isfield (opts, "tol"))
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109 opts.tol = 1e-10 / root2;
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110 else
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111 opts.tol = opts.tol / root2;
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112 endif
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113 if (isfield (opts, "v0"))
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114 if (!isvector (opts.v0) || (length (opts.v0) != sum (size (a))))
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115 error ("svds: OPTS.v0 must be a vector with rows(A)+columns(A) entries");
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116 endif
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117 endif
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118 endif
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119
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120 if (nargin < 3 || strcmp (sigma, "L"))
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121 if (isreal (a))
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122 sigma = "LA";
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123 else
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124 sigma = "LR";
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125 endif
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126 elseif (isscalar (sigma) && isnumeric (sigma) && isreal (sigma))
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127 if (sigma < 0)
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128 error ("svds: SIGMA must be a positive real value");
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129 endif
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130 else
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131 error ("svds: SIGMA must be a positive real value or the string 'L'");
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132 endif
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133
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134 [m, n] = size (a);
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135 max_a = max (abs (a(:)));
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136 if (max_a == 0)
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137 s = zeros (k, 1); # special case of zero matrix
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138 else
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139 if (nargin < 2)
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140 k = min ([6, m, n]);
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141 else
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142 k = min ([k, m, n]);
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143 endif
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144
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145 ## Scale everything by the 1-norm to make things more stable.
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146 b = a / max_a;
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147 b_opts = opts;
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148 b_opts.tol = opts.tol / max_a;
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149 b_sigma = sigma;
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150 if (!ischar (b_sigma))
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151 b_sigma = b_sigma / max_a;
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152 endif
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153
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154 if (b_sigma == 0)
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155 ## Find the smallest eigenvalues
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156 ## The eigenvalues returns by eigs for sigma=0 are symmetric about 0.
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157 ## As we are only interested in the positive eigenvalues, we have to
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158 ## double k and then throw out the k negative eigenvalues.
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159 ## Separately, if sigma is non-zero, but smaller than the smallest
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160 ## singular value, ARPACK may not return k eigenvalues. However, as
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161 ## computation scales with k we'd like to avoid doubling k for all
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162 ## scalar values of sigma.
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163 b_k = 2 * k;
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164 else
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165 b_k = k; # Normal case, find just the k largest eigenvalues
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166 endif
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167
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168 if (nargout > 1)
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169 [V, s, flag] = eigs ([sparse(m,m), b; b', sparse(n,n)],
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170 b_k, b_sigma, b_opts);
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171 else
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172 s = eigs ([sparse(m,m), b; b', sparse(n,n)], b_k, b_sigma, b_opts);
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173 endif
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174 s = diag (s);
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175
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176 if (ischar (sigma))
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177 norma = max (s);
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178 else
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179 norma = normest (a);
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180 endif
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181 ## We wish to exclude all eigenvalues that are less than zero as these
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182 ## are artifacts of the way the matrix passed to eigs is formed. There
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183 ## is also the possibility that the value of sigma chosen is exactly
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184 ## a singular value, and in that case we're dead!! So have to rely on
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185 ## the warning from eigs. We exclude the singular values which are
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186 ## less than or equal to zero to within some tolerance scaled by the
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187 ## norm since if we don't we might end up with too many singular
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188 ## values.
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189 tol = norma * opts.tol;
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190 ind = find(s > tol);
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191 if (length (ind) < k)
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192 ## Too few eigenvalues returned. Add in any zero eigenvalues of B,
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193 ## including the nominally negative ones.
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194 zind = find (abs (s) <= tol);
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195 p = min (length (zind), k - length (ind));
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196 ind = [ind; zind(1:p)];
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197 elseif (length (ind) > k)
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198 ## Too many eigenvalues returned. Select according to criterium.
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199 if (b_sigma == 0)
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200 ind = ind(end+1-k:end); # smallest eigenvalues
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201 else
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202 ind = ind(1:k); # largest eigenvalues
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203 endif
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204 endif
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205 s = s(ind);
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206
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207 if (length (s) < k)
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208 warning ("returning fewer singular values than requested");
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209 if (!ischar (sigma))
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210 warning ("try increasing the value of sigma");
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211 endif
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212 endif
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213
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214 s = s * max_a;
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215 endif
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216
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217 if (nargout < 2)
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218 u = s;
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219 else
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220 if (max_a == 0)
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221 u = eye (m, k);
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222 s = diag(s);
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223 v = eye (n, k);
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224 else
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225 u = root2 * V(1:m,ind);
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226 s = diag(s);
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227 v = root2 * V(m+1:end,ind);
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228 endif
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229
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230 if (nargout > 3)
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231 flag = norm (a*v - u*s, 1) > root2 * opts.tol * norm (a, 1);
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232 endif
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233 endif
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234
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235 endfunction
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236
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237 %!shared n, k, a, u, s, v, opts
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238 %! n = 100;
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239 %! k = 7;
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240 %! 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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241 %! [u,s,v] = svd(full(a));
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242 %! s = diag(s);
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243 %! [~, idx] = sort(abs(s));
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244 %! s = s(idx);
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245 %! u = u(:,idx);
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246 %! v = v(:,idx);
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247 %! randn('state',42); % Initialize to make normest function reproducible
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248 %! rand('state',42)
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249 %! opts.v0 = rand (2*n,1); % Initialize eigs ARPACK starting vector
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250 %! % to guarantee reproducible results
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251 %!testif HAVE_ARPACK
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252 %! [u2,s2,v2,flag] = svds(a,k);
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253 %! s2 = diag(s2);
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254 %! assert(flag,!1);
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255 %! assert(s2, s(end:-1:end-k+1), 1e-10);
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256 %!testif HAVE_ARPACK
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257 %! [u2,s2,v2,flag] = svds(a,k,0,opts);
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258 %! s2 = diag(s2);
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259 %! assert(flag,!1);
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260 %! assert(s2, s(k:-1:1), 1e-10);
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261 %!testif HAVE_ARPACK
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262 %! idx = floor(n/2);
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263 %! % Don't put sigma right on a singular value or there are convergence issues
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264 %! sigma = 0.99*s(idx) + 0.01*s(idx+1);
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265 %! [u2,s2,v2,flag] = svds(a,k,sigma,opts);
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266 %! s2 = diag(s2);
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267 %! assert(flag,!1);
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268 %! assert(s2, s((idx+floor(k/2)):-1:(idx-floor(k/2))), 1e-10);
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269 %!testif HAVE_ARPACK
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270 %! [u2,s2,v2,flag] = svds(zeros (10), k);
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271 %! assert (isequal(u2, eye (10, k)) && isequal (s2, zeros(k)) && isequal (v2, eye(10, 7)))