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Add the eigs and svds functions
author David Bateman <dbateman@free.fr>
date Tue, 23 Dec 2008 08:28:23 +0100
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8416:0a56e5c21c29 8417:654bcfb937bf
1 ## Copyright (C) 2006 David Bateman
2 ##
3 ## This program is free software; you can redistribute it and/or modify
4 ## it under the terms of the GNU General Public License as published by
5 ## the Free Software Foundation; either version 2 of the License, or
6 ## (at your option) any later version.
7 ##
8 ## This program is distributed in the hope that it will be useful,
9 ## but WITHOUT ANY WARRANTY; without even the implied warranty of
10 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
11 ## GNU General Public License for more details.
12 ##
13 ## You should have received a copy of the GNU General Public License
14 ## along with this program; If not, see <http://www.gnu.org/licenses/>.
15
16 ## -*- texinfo -*-
17 ## @deftypefn {Function File} {@var{s} =} svds (@var{a})
18 ## @deftypefnx {Function File} {@var{s} =} svds (@var{a}, @var{k})
19 ## @deftypefnx {Function File} {@var{s} =} svds (@var{a}, @var{k}, @var{sigma})
20 ## @deftypefnx {Function File} {@var{s} =} svds (@var{a}, @var{k}, @var{sigma}, @var{opts})
21 ## @deftypefnx {Function File} {[@var{u}, @var{s}, @var{v}, @var{flag}] =} svds (@dots{})
22 ##
23 ## Find a few singular values of the matrix @var{a}. The singular values
24 ## are calculated using
25 ##
26 ## @example
27 ## @group
28 ## [@var{m}, @var{n}] = size(@var{a})
29 ## @var{s} = eigs([sparse(@var{m}, @var{m}), @var{a}; @dots{}
30 ## @var{a}', sparse(@var{n}, @var{n})])
31 ## @end group
32 ## @end example
33 ##
34 ## The eigenvalues returned by @code{eigs} correspond to the singular
35 ## values of @var{a}. The number of singular values to calculate is given
36 ## by @var{k}, whose default value is 6.
37 ##
38 ## The argument @var{sigma} can be used to specify which singular values
39 ## to find. @var{sigma} can be either the string 'L', the default, in
40 ## which case the largest singular values of @var{a} are found. Otherwise
41 ## @var{sigma} should be a real scalar, in which case the singular values
42 ## closest to @var{sigma} are found. Note that for relatively small values
43 ## of @var{sigma}, there is the chance that the requested number of singular
44 ## values are not returned. In that case @var{sigma} should be increased.
45 ##
46 ## If @var{opts} is given, then it is a structure that defines options
47 ## that @code{svds} will pass to @var{eigs}. The possible fields of this
48 ## structure are therefore determined by @code{eigs}. By default three
49 ## fields of this structure are set by @code{svds}.
50 ##
51 ## @table @code
52 ## @item tol
53 ## The required convergence tolerance for the singular values. @code{eigs}
54 ## is passed @var{tol} divided by @code{sqrt(2)}. The default value is
55 ## 1e-10.
56 ##
57 ## @item maxit
58 ## The maximum number of iterations. The defaut is 300.
59 ##
60 ## @item disp
61 ## The level of diagnostic printout. If @code{disp} is 0 then there is no
62 ## printout. The default value is 0.
63 ## @end table
64 ##
65 ## If more than one output argument is given, then @code{svds} also
66 ## calculates the left and right singular vectors of @var{a}. @var{flag}
67 ## is used to signal the convergence of @code{svds}. If @code{svds}
68 ## converges to the desired tolerance, then @var{flag} given by
69 ##
70 ## @example
71 ## @group
72 ## norm (@var{a} * @var{v} - @var{u} * @var{s}, 1) <= @dots{}
73 ## @var{tol} * norm (@var{a}, 1)
74 ## @end group
75 ## @end example
76 ##
77 ## will be zero.
78 ## @end deftypefn
79 ## @seealso{eigs}
80
81 function [u, s, v, flag] = svds (a, k, sigma, opts)
82
83 if (nargin < 1 || nargin > 4)
84 error ("Incorrect number of arguments");
85 endif
86
87 if (nargin < 4)
88 opts.tol = 1e-10 / sqrt(2);
89 opts.disp = 0;
90 opts.maxit = 300;
91 else
92 if (!isstruct(opts))
93 error("opts must be a structure");
94 endif
95 if (!isfield(opts,"tol"))
96 opts.tol = 1e-10 / sqrt(2);
97 endif
98 endif
99
100 if (nargin < 3 || strcmp(sigma,"L"))
101 if (isreal(a))
102 sigma = "LA";
103 else
104 sigma = "LR";
105 endif
106 elseif (isscalar(sigma) && isreal(sigma))
107 if ((sigma < 0))
108 error ("sigma must be a positive real value");
109 endif
110 else
111 error ("sigma must be a positive real value or the string 'L'");
112 endif
113
114 maxA = max(max(abs(a)));
115 if (maxA == 0)
116 u = eye(m, k);
117 s = zeros(k, k);
118 v = eye(n, k);
119 else
120 [m, n] = size(a);
121 if (nargin < 2)
122 k = min([6, m, n]);
123 else
124 k = min([k, m, n]);
125 endif
126
127 ## Scale everything by the 1-norm to make things more stable.
128 B = a / maxA;
129 Bopts = opts;
130 Bopts.tol = opts.tol / maxA;
131 Bsigma = sigma;
132 if (!ischar(Bsigma))
133 Bsigma = Bsigma / maxA;
134 endif
135
136 if (!ischar(Bsigma) && Bsigma == 0)
137 ## The eigenvalues returns by eigs are symmetric about 0. As we
138 ## are only interested in the positive eigenvalues, we have to
139 ## double k. If sigma is smaller than the smallest singular value
140 ## this can also be an issue. However, we'd like to avoid double
141 ## k for all scalar value of sigma...
142 [V, s, flag] = eigs ([sparse(m,m), B; B', sparse(n,n)],
143 2 * k, Bsigma, Bopts);
144 else
145 [V, s, flag] = eigs ([sparse(m,m), B; B', sparse(n,n)],
146 k, Bsigma, Bopts);
147 endif
148 s = diag(s);
149
150 if (ischar(sigma))
151 norma = max(s);
152 else
153 norma = normest(a);
154 endif
155 V = sqrt(2) * V;
156 u = V(1:m,:);
157 v = V(m+1:end,:);
158
159 ## We wish to exclude all eigenvalues that are less than zero as these
160 ## are artifacts of the way the matrix passed to eigs is formed. There
161 ## is also the possibility that the value of sigma chosen is exactly
162 ## a singular value, and in that case we're dead!! So have to rely on
163 ## the warning from eigs. We exclude the singular values which are
164 ## less than or equal to zero to within some tolerance scaled by the
165 ## norm since if we don't we might end up with too many singular
166 ## values. What is appropriate for the tolerance?
167 tol = norma * opts.tol;
168 ind = find(s > tol);
169 if (length(ind) < k)
170 ## Find the zero eigenvalues of B, Ignore the eigenvalues that are
171 ## nominally negative.
172 zind = find(abs(s) <= tol);
173 p = min(length(zind), k-length(ind));
174 ind = [ind;zind(1:p)];
175 elseif (length(ind) > k)
176 ind = ind(1:k);
177 endif
178 u = u(:,ind);
179 s = s(ind);
180 v = v(:,ind);
181
182 if (length(s) < k)
183 warning("returning fewer singular values than requested.");
184 if (!ischar(sigma))
185 warning("try increasing the value of sigma");
186 endif
187 endif
188
189 s = s * maxA;
190 endif
191
192 if (nargout < 2)
193 u = s;
194 else
195 s = diag(s);
196 if (nargout > 3)
197 flag = norm(a*v - u*s, 1) > sqrt(2) * opts.tol * norm(a, 1);
198 endif
199 endif
200 endfunction
201
202 %!shared n, k, a, u, s, v, opts
203 %! n = 100;
204 %! k = 7;
205 %! 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)]);
206 %! %%a = sparse([3:n,1:n,1:(n-2)],[1:(n-2),1:n,3:n],[ones(1,n-2),1:n,-ones(1,n-2)]);
207 %! [u,s,v] = svd(full(a));
208 %! s = diag(s);
209 %! [dum, idx] = sort(abs(s));
210 %! s = s(idx);
211 %! u = u(:,idx);
212 %! v = v(:,idx);
213 %! randn('state',42)
214 %!test
215 %! [u2,s2,v2,flag] = svds(a,k);
216 %! s2 = diag(s2);
217 %! assert(flag,!1);
218 %! assert(s(end:-1:end-k+1), s2, 1e-10);
219 %!test
220 %! [u2,s2,v2,flag] = svds(a,k,0);
221 %! s2 = diag(s2);
222 %! assert(flag,!1);
223 %! assert(s(k:-1:1), s2, 1e-10);
224 %!test
225 %! idx = floor(n/2);
226 %! % Don't put sigma right on a singular value or there are convergence
227 %! sigma = 0.99*s(idx) + 0.01*s(idx+1);
228 %! [u2,s2,v2,flag] = svds(a,k,sigma);
229 %! s2 = diag(s2);
230 %! assert(flag,!1);
231 %! assert(s((idx+floor(k/2)):-1:(idx-floor(k/2))), s2, 1e-10);