annotate scripts/sparse/svds.m @ 14363:f3d52523cde1

Use Octave coding conventions in all m-file %!test blocks * wavread.m, acosd.m, acot.m, acotd.m, acoth.m, acsc.m, acscd.m, acsch.m, asec.m, asecd.m, asech.m, asind.m, atand.m, cosd.m, cot.m, cotd.m, coth.m, csc.m, cscd.m, csch.m, sec.m, secd.m, sech.m, sind.m, tand.m, accumarray.m, accumdim.m, bitcmp.m, bitget.m, bitset.m, blkdiag.m, cart2pol.m, cart2sph.m, celldisp.m, chop.m, circshift.m, colon.m, common_size.m, cplxpair.m, cumtrapz.m, curl.m, dblquad.m, deal.m, divergence.m, flipdim.m, fliplr.m, flipud.m, genvarname.m, gradient.m, idivide.m, int2str.m, interp1.m, interp1q.m, interp2.m, interp3.m, interpft.m, interpn.m, isa.m, isdir.m, isequal.m, isequalwithequalnans.m, issquare.m, logspace.m, nargchk.m, narginchk.m, nargoutchk.m, nextpow2.m, nthargout.m, num2str.m, pol2cart.m, polyarea.m, postpad.m, prepad.m, profile.m, profshow.m, quadgk.m, quadv.m, randi.m, rat.m, repmat.m, rot90.m, rotdim.m, shift.m, shiftdim.m, sph2cart.m, structfun.m, trapz.m, triplequad.m, convhull.m, dsearch.m, dsearchn.m, griddata3.m, griddatan.m, rectint.m, tsearchn.m, __makeinfo__.m, doc.m, get_first_help_sentence.m, help.m, type.m, unimplemented.m, which.m, imread.m, imwrite.m, dlmwrite.m, fileread.m, is_valid_file_id.m, strread.m, textread.m, textscan.m, commutation_matrix.m, cond.m, condest.m, cross.m, duplication_matrix.m, expm.m, housh.m, isdefinite.m, ishermitian.m, issymmetric.m, logm.m, normest.m, null.m, onenormest.m, orth.m, planerot.m, qzhess.m, rank.m, rref.m, trace.m, vech.m, ans.m, bincoeff.m, bug_report.m, bzip2.m, comma.m, compare_versions.m, computer.m, edit.m, fileparts.m, fullfile.m, getfield.m, gzip.m, info.m, inputname.m, isappdata.m, isdeployed.m, ismac.m, ispc.m, isunix.m, list_primes.m, ls.m, mexext.m, namelengthmax.m, news.m, orderfields.m, paren.m, recycle.m, rmappdata.m, semicolon.m, setappdata.m, setfield.m, substruct.m, symvar.m, ver.m, version.m, warning_ids.m, xor.m, fminbnd.m, fsolve.m, fzero.m, lsqnonneg.m, optimset.m, pqpnonneg.m, sqp.m, matlabroot.m, __gnuplot_drawnow__.m, __plt_get_axis_arg__.m, ancestor.m, cla.m, clf.m, close.m, colorbar.m, colstyle.m, comet3.m, contourc.m, figure.m, gca.m, gcbf.m, gcbo.m, gcf.m, ginput.m, graphics_toolkit.m, gtext.m, hggroup.m, hist.m, hold.m, isfigure.m, ishghandle.m, ishold.m, isocolors.m, isonormals.m, isosurface.m, isprop.m, legend.m, line.m, loglog.m, loglogerr.m, meshgrid.m, ndgrid.m, newplot.m, orient.m, patch.m, plot3.m, plotyy.m, __print_parse_opts__.m, quiver3.m, refreshdata.m, ribbon.m, semilogx.m, semilogxerr.m, semilogy.m, stem.m, stem3.m, subplot.m, title.m, uigetfile.m, view.m, whitebg.m, compan.m, conv.m, deconv.m, mkpp.m, mpoles.m, pchip.m, poly.m, polyaffine.m, polyder.m, polyfit.m, polygcd.m, polyint.m, polyout.m, polyval.m, polyvalm.m, ppder.m, ppint.m, ppjumps.m, ppval.m, residue.m, roots.m, spline.m, intersect.m, ismember.m, powerset.m, setdiff.m, setxor.m, union.m, unique.m, autoreg_matrix.m, bartlett.m, blackman.m, detrend.m, fftconv.m, fftfilt.m, fftshift.m, freqz.m, hamming.m, hanning.m, ifftshift.m, sinc.m, sinetone.m, sinewave.m, unwrap.m, bicg.m, bicgstab.m, gmres.m, gplot.m, nonzeros.m, pcg.m, pcr.m, spaugment.m, spconvert.m, spdiags.m, speye.m, spfun.m, spones.m, sprand.m, sprandsym.m, spstats.m, spy.m, svds.m, treelayout.m, bessel.m, beta.m, betaln.m, factor.m, factorial.m, isprime.m, lcm.m, legendre.m, nchoosek.m, nthroot.m, perms.m, pow2.m, primes.m, reallog.m, realpow.m, realsqrt.m, hadamard.m, hankel.m, hilb.m, invhilb.m, magic.m, rosser.m, vander.m, __finish__.m, center.m, cloglog.m, corr.m, cov.m, gls.m, histc.m, iqr.m, kendall.m, kurtosis.m, logit.m, mahalanobis.m, mean.m, meansq.m, median.m, mode.m, moment.m, ols.m, ppplot.m, prctile.m, probit.m, quantile.m, range.m, ranks.m, run_count.m, runlength.m, skewness.m, spearman.m, statistics.m, std.m, table.m, var.m, zscore.m, betacdf.m, betainv.m, betapdf.m, betarnd.m, binocdf.m, binoinv.m, binopdf.m, binornd.m, cauchy_cdf.m, cauchy_inv.m, cauchy_pdf.m, cauchy_rnd.m, chi2cdf.m, chi2inv.m, chi2pdf.m, chi2rnd.m, discrete_cdf.m, discrete_inv.m, discrete_pdf.m, discrete_rnd.m, empirical_cdf.m, empirical_inv.m, empirical_pdf.m, empirical_rnd.m, expcdf.m, expinv.m, exppdf.m, exprnd.m, fcdf.m, finv.m, fpdf.m, frnd.m, gamcdf.m, gaminv.m, gampdf.m, gamrnd.m, geocdf.m, geoinv.m, geopdf.m, geornd.m, hygecdf.m, hygeinv.m, hygepdf.m, hygernd.m, kolmogorov_smirnov_cdf.m, laplace_cdf.m, laplace_inv.m, laplace_pdf.m, laplace_rnd.m, logistic_cdf.m, logistic_inv.m, logistic_pdf.m, logistic_rnd.m, logncdf.m, logninv.m, lognpdf.m, lognrnd.m, nbincdf.m, nbininv.m, nbinpdf.m, nbinrnd.m, normcdf.m, norminv.m, normpdf.m, normrnd.m, poisscdf.m, poissinv.m, poisspdf.m, poissrnd.m, stdnormal_cdf.m, stdnormal_inv.m, stdnormal_pdf.m, stdnormal_rnd.m, tcdf.m, tinv.m, tpdf.m, trnd.m, unidcdf.m, unidinv.m, unidpdf.m, unidrnd.m, unifcdf.m, unifinv.m, unifpdf.m, unifrnd.m, wblcdf.m, wblinv.m, wblpdf.m, wblrnd.m, kolmogorov_smirnov_test.m, kruskal_wallis_test.m, base2dec.m, bin2dec.m, blanks.m, cstrcat.m, deblank.m, dec2base.m, dec2bin.m, dec2hex.m, findstr.m, hex2dec.m, index.m, isletter.m, mat2str.m, rindex.m, str2num.m, strcat.m, strjust.m, strmatch.m, strsplit.m, strtok.m, strtrim.m, strtrunc.m, substr.m, validatestring.m, demo.m, example.m, fail.m, speed.m, addtodate.m, asctime.m, clock.m, ctime.m, date.m, datenum.m, datetick.m, datevec.m, eomday.m, etime.m, is_leap_year.m, now.m: Use Octave coding conventions in all m-file %!test blocks
author Rik <octave@nomad.inbox5.com>
date Mon, 13 Feb 2012 07:29:44 -0800
parents 4d917a6a858b
children 5d3a684236b0
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1 ## Copyright (C) 2006-2012 David Bateman
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2 ##
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3 ## This file is part of Octave.
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4 ##
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5 ## Octave is free software; you can redistribute it and/or modify it
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6 ## under the terms of the GNU General Public License as published by
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7 ## the Free Software Foundation; either version 3 of the License, or (at
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8 ## your option) any later version.
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9 ##
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10 ## Octave is distributed in the hope that it will be useful, but
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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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
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16 ## along with Octave; see the file COPYING. If not, see
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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}. The singular values
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28 ## are calculated using
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29 ##
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30 ## @example
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31 ## @group
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32 ## [@var{m}, @var{n}] = size (@var{A});
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33 ## @var{s} = eigs ([sparse(@var{m}, @var{m}), @var{A};
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34 ## @var{A}', sparse(@var{n}, @var{n})])
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35 ## @end group
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36 ## @end example
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37 ##
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38 ## The eigenvalues returned by @code{eigs} correspond to the singular values
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39 ## of @var{A}. The number of singular values to calculate is given by @var{k}
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40 ## and defaults to 6.
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41 ##
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42 ## The argument @var{sigma} specifies which singular values to find. When
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43 ## @var{sigma} is the string 'L', the default, the largest singular values of
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44 ## @var{A} are found. Otherwise, @var{sigma} must be a real scalar and the
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45 ## singular values closest to @var{sigma} are found. As a corollary,
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46 ## @code{@var{sigma} = 0} finds the smallest singular values. Note that for
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47 ## relatively small values of @var{sigma}, there is a chance that the requested
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48 ## number of singular values will not be found. In that case @var{sigma}
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49 ## should be increased.
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50 ##
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51 ## @var{opts} is a structure defining options that @code{svds} will pass
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52 ## to @code{eigs}. The possible fields of this structure are documented in
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53 ## @code{eigs}. By default, @code{svds} sets the following three fields:
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54 ##
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55 ## @table @code
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56 ## @item tol
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57 ## The required convergence tolerance for the singular values. The default
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58 ## value is 1e-10. @code{eigs} is passed @code{@var{tol} / sqrt(2)}.
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59 ##
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60 ## @item maxit
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61 ## The maximum number of iterations. The default is 300.
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62 ##
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63 ## @item disp
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64 ## The level of diagnostic printout (0|1|2). If @code{disp} is 0 then
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65 ## diagnostics are disabled. The default value is 0.
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66 ## @end table
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67 ##
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68 ## If more than one output is requested then @code{svds} will return an
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69 ## approximation of the singular value decomposition of @var{A}
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70 ##
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71 ## @example
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72 ## @var{A}_approx = @var{u}*@var{s}*@var{v}'
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73 ## @end example
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74 ##
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75 ## @noindent
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76 ## where @var{A}_approx is a matrix of size @var{A} but only rank @var{k}.
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77 ##
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78 ## @var{flag} returns 0 if the algorithm has succesfully converged, and 1
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79 ## otherwise. The test for convergence is
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80 ##
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81 ## @example
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82 ## @group
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83 ## norm (@var{A}*@var{v} - @var{u}*@var{s}, 1) <= @var{tol} * norm (@var{A}, 1)
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84 ## @end group
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85 ## @end example
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86 ##
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87 ## @code{svds} is best for finding only a few singular values from a large
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88 ## sparse matrix. Otherwise, @code{svd (full(@var{A}))} will likely be more
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diff changeset
89 ## efficient.
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parents:
diff changeset
90 ## @end deftypefn
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diff changeset
91 ## @seealso{svd, eigs}
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diff changeset
92
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diff changeset
93 function [u, s, v, flag] = svds (A, k, sigma, opts)
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parents:
diff changeset
94
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diff changeset
95 persistent root2 = sqrt (2);
cadc73247d65 style fixes
John W. Eaton <jwe@octave.org>
parents: 8417
diff changeset
96
8417
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diff changeset
97 if (nargin < 1 || nargin > 4)
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parents: 8417
diff changeset
98 print_usage ();
8417
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parents:
diff changeset
99 endif
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parents:
diff changeset
100
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diff changeset
101 if (ndims(A) > 2)
11589
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diff changeset
102 error ("svds: A must be a 2D matrix");
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103 endif
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diff changeset
104
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105 if (nargin < 4)
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106 opts.tol = 1e-10 / root2;
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parents:
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107 opts.disp = 0;
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parents:
diff changeset
108 opts.maxit = 300;
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parents:
diff changeset
109 else
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110 if (!isstruct (opts))
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111 error ("svds: OPTS must be a structure");
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parents:
diff changeset
112 endif
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113 if (!isfield (opts, "tol"))
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114 opts.tol = 1e-10 / root2;
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115 else
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116 opts.tol = opts.tol / root2;
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117 endif
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118 if (isfield (opts, "v0"))
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119 if (!isvector (opts.v0) || (length (opts.v0) != sum (size (A))))
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120 error ("svds: OPTS.v0 must be a vector with rows(A)+columns(A) entries");
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121 endif
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122 endif
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123 endif
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diff changeset
124
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125 if (nargin < 3 || strcmp (sigma, "L"))
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diff changeset
126 if (isreal (A))
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127 sigma = "LA";
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128 else
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129 sigma = "LR";
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130 endif
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131 elseif (isscalar (sigma) && isnumeric (sigma) && isreal (sigma))
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132 if (sigma < 0)
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133 error ("svds: SIGMA must be a positive real value");
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134 endif
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diff changeset
135 else
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diff changeset
136 error ("svds: SIGMA must be a positive real value or the string 'L'");
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137 endif
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diff changeset
138
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diff changeset
139 [m, n] = size (A);
994e2a93a8e2 Use uppercase 'A' to refer to matrix inputs in m-files.
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diff changeset
140 max_a = max (abs (A(:)));
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141 if (max_a == 0)
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diff changeset
142 s = zeros (k, 1); # special case of zero matrix
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143 else
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diff changeset
144 if (nargin < 2)
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diff changeset
145 k = min ([6, m, n]);
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146 else
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diff changeset
147 k = min ([k, m, n]);
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148 endif
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diff changeset
149
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diff changeset
150 ## Scale everything by the 1-norm to make things more stable.
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diff changeset
151 b = A / max_a;
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diff changeset
152 b_opts = opts;
12649
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153 ## Call to eigs is always a symmetric matrix by construction
8579e7a6e6e0 Improve performance (2X) of svds by avoiding issymmetric calculation (bug #33073)
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diff changeset
154 b_opts.issym = true;
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155 b_opts.tol = opts.tol / max_a;
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diff changeset
156 b_sigma = sigma;
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diff changeset
157 if (!ischar (b_sigma))
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diff changeset
158 b_sigma = b_sigma / max_a;
8417
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diff changeset
159 endif
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diff changeset
160
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diff changeset
161 if (b_sigma == 0)
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162 ## Find the smallest eigenvalues
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163 ## The eigenvalues returns by eigs for sigma=0 are symmetric about 0.
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164 ## As we are only interested in the positive eigenvalues, we have to
11587
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165 ## double k and then throw out the k negative eigenvalues.
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diff changeset
166 ## Separately, if sigma is non-zero, but smaller than the smallest
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diff changeset
167 ## singular value, ARPACK may not return k eigenvalues. However, as
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168 ## computation scales with k we'd like to avoid doubling k for all
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169 ## scalar values of sigma.
11587
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diff changeset
170 b_k = 2 * k;
8417
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diff changeset
171 else
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diff changeset
172 b_k = k; # Normal case, find just the k largest eigenvalues
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diff changeset
173 endif
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diff changeset
174
f5f9bc8e83fc svds.m: Overhaul code while fixing bug #29721.
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diff changeset
175 if (nargout > 1)
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diff changeset
176 [V, s, flag] = eigs ([sparse(m,m), b; b', sparse(n,n)],
10671
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diff changeset
177 b_k, b_sigma, b_opts);
12517
c2e8c9d9e284 Fix bug #32818, nonconformant arguments in svds.
Rik <octave@nomad.inbox5.com>
parents: 12329
diff changeset
178 s = diag (s);
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diff changeset
179 else
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diff changeset
180 s = eigs ([sparse(m,m), b; b', sparse(n,n)], b_k, b_sigma, b_opts);
8417
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181 endif
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parents:
diff changeset
182
8507
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diff changeset
183 if (ischar (sigma))
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diff changeset
184 norma = max (s);
8417
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185 else
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diff changeset
186 norma = normest (A);
8417
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187 endif
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diff changeset
188 ## We wish to exclude all eigenvalues that are less than zero as these
11587
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189 ## are artifacts of the way the matrix passed to eigs is formed. There
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diff changeset
190 ## is also the possibility that the value of sigma chosen is exactly
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diff changeset
191 ## a singular value, and in that case we're dead!! So have to rely on
8417
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diff changeset
192 ## the warning from eigs. We exclude the singular values which are
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parents:
diff changeset
193 ## less than or equal to zero to within some tolerance scaled by the
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194 ## norm since if we don't we might end up with too many singular
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diff changeset
195 ## values.
8417
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diff changeset
196 tol = norma * opts.tol;
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diff changeset
197 ind = find(s > tol);
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198 if (length (ind) < k)
11587
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199 ## Too few eigenvalues returned. Add in any zero eigenvalues of B,
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diff changeset
200 ## including the nominally negative ones.
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diff changeset
201 zind = find (abs (s) <= tol);
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John W. Eaton <jwe@octave.org>
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diff changeset
202 p = min (length (zind), k - length (ind));
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diff changeset
203 ind = [ind; zind(1:p)];
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diff changeset
204 elseif (length (ind) > k)
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205 ## Too many eigenvalues returned. Select according to criterium.
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diff changeset
206 if (b_sigma == 0)
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diff changeset
207 ind = ind(end+1-k:end); # smallest eigenvalues
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diff changeset
208 else
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diff changeset
209 ind = ind(1:k); # largest eigenvalues
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diff changeset
210 endif
8417
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diff changeset
211 endif
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parents:
diff changeset
212 s = s(ind);
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parents:
diff changeset
213
8507
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214 if (length (s) < k)
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diff changeset
215 warning ("returning fewer singular values than requested");
cadc73247d65 style fixes
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diff changeset
216 if (!ischar (sigma))
10549
95c3e38098bf Untabify .m scripts
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217 warning ("try increasing the value of sigma");
8417
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218 endif
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219 endif
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diff changeset
220
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221 s = s * max_a;
8417
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parents:
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222 endif
654bcfb937bf Add the eigs and svds functions
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parents:
diff changeset
223
654bcfb937bf Add the eigs and svds functions
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diff changeset
224 if (nargout < 2)
654bcfb937bf Add the eigs and svds functions
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diff changeset
225 u = s;
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226 else
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diff changeset
227 if (max_a == 0)
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diff changeset
228 u = eye (m, k);
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229 s = diag (s);
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230 v = eye (n, k);
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231 else
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232 u = root2 * V(1:m,ind);
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233 s = diag (s);
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234 v = root2 * V(m+1:end,ind);
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235 endif
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236
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237 if (nargout > 3)
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238 flag = norm (A*v - u*s, 1) > root2 * opts.tol * norm (A, 1);
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239 endif
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240 endif
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241
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242 endfunction
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243
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244
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245 %!shared n, k, A, u, s, v, opts, rand_state, randn_state
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246 %! n = 100;
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247 %! k = 7;
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248 %! 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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249 %! [u,s,v] = svd (full (A));
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250 %! s = diag (s);
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251 %! [~, idx] = sort (abs(s));
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252 %! s = s(idx);
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253 %! u = u(:, idx);
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254 %! v = v(:, idx);
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255 %! randn_state = randn ("state");
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256 %! rand_state = rand ("state");
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257 %! randn ("state", 42); % Initialize to make normest function reproducible
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258 %! rand ("state", 42);
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259 %! opts.v0 = rand (2*n,1); % Initialize eigs ARPACK starting vector
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260 %! % to guarantee reproducible results
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261 %!
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262 %!testif HAVE_ARPACK
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263 %! [u2,s2,v2,flag] = svds (A,k);
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264 %! s2 = diag (s2);
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265 %! assert (flag, !1);
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266 %! assert (s2, s(end:-1:end-k+1), 1e-10);
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267 %!
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268 %!testif HAVE_ARPACK, HAVE_UMFPACK
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269 %! [u2,s2,v2,flag] = svds (A,k,0,opts);
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270 %! s2 = diag (s2);
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271 %! assert (flag, !1);
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272 %! assert (s2, s(k:-1:1), 1e-10);
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273 %!
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274 %!testif HAVE_ARPACK, HAVE_UMFPACK
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275 %! idx = floor(n/2);
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276 %! % Don't put sigma right on a singular value or there are convergence issues
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277 %! sigma = 0.99*s(idx) + 0.01*s(idx+1);
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278 %! [u2,s2,v2,flag] = svds (A,k,sigma,opts);
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279 %! s2 = diag (s2);
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280 %! assert (flag, !1);
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281 %! assert (s2, s((idx+floor(k/2)):-1:(idx-floor(k/2))), 1e-10);
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282 %!
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283 %!testif HAVE_ARPACK
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284 %! [u2,s2,v2,flag] = svds (zeros (10), k);
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285 %! assert (u2, eye (10, k));
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286 %! assert (s2, zeros (k));
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287 %! assert (v2, eye (10, 7));
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288 %!
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289 %!testif HAVE_ARPACK
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290 %! s = svds (speye (10));
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291 %! assert (s, ones (6, 1), 2*eps);
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292
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293 %!test
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294 %! ## Restore random number generator seeds at end of tests
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295 %! rand ("state", rand_state);
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296 %! randn ("state", randn_state);
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297