Mercurial > hg > octave-lyh
annotate scripts/linear-algebra/onenormest.m @ 14237:11949c9795a0
Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Add clf() to all demos using plot features to get reproducibility.
Use 64 as input to all colormaps (jet (64)) to get reproducibility.
* bicubic.m, cell2mat.m, celldisp.m, cplxpair.m, interp1.m, interp2.m,
interpft.m, interpn.m, profile.m, profshow.m, convhull.m, delaunay.m,
griddata.m, inpolygon.m, voronoi.m, autumn.m, bone.m, contrast.m, cool.m,
copper.m, flag.m, gmap40.m, gray.m, hot.m, hsv.m, image.m, imshow.m, jet.m,
ocean.m, pink.m, prism.m, rainbow.m, spring.m, summer.m, white.m, winter.m,
condest.m, onenormest.m, axis.m, clabel.m, colorbar.m, comet.m, comet3.m,
compass.m, contour.m, contour3.m, contourf.m, cylinder.m, daspect.m,
ellipsoid.m, errorbar.m, ezcontour.m, ezcontourf.m, ezmesh.m, ezmeshc.m,
ezplot.m, ezplot3.m, ezpolar.m, ezsurf.m, ezsurfc.m, feather.m, fill.m,
fplot.m, grid.m, hold.m, isosurface.m, legend.m, loglog.m, loglogerr.m,
pareto.m, patch.m, pbaspect.m, pcolor.m, pie.m, pie3.m, plot3.m, plotmatrix.m,
plotyy.m, polar.m, quiver.m, quiver3.m, rectangle.m, refreshdata.m, ribbon.m,
rose.m, scatter.m, scatter3.m, semilogx.m, semilogxerr.m, semilogy.m,
semilogyerr.m, shading.m, slice.m, sombrero.m, stairs.m, stem.m, stem3.m,
subplot.m, surf.m, surfc.m, surfl.m, surfnorm.m, text.m, title.m, trimesh.m,
triplot.m, trisurf.m, uigetdir.m, uigetfile.m, uimenu.m, uiputfile.m,
waitbar.m, xlim.m, ylim.m, zlim.m, mkpp.m, pchip.m, polyaffine.m, spline.m,
bicgstab.m, cgs.m, gplot.m, pcg.m, pcr.m, treeplot.m, strtok.m, demo.m,
example.m, rundemos.m, speed.m, test.m, calendar.m, datestr.m, datetick.m,
weekday.m: Revamp %!demos to use Octave coding conventions on spacing, etc.
| author | Rik <octave@nomad.inbox5.com> |
|---|---|
| date | Fri, 20 Jan 2012 12:59:53 -0800 |
| parents | 72c96de7a403 |
| children | f3d52523cde1 |
| rev | line source |
|---|---|
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1 ## Copyright (C) 2007-2012 Regents of the University of California |
| 7189 | 2 ## |
| 3 ## This file is part of Octave. | |
| 4 ## | |
| 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. | |
| 14 ## | |
| 15 ## You should have received a copy of the GNU General Public License | |
| 16 ## along with Octave; see the file COPYING. If not, see | |
| 17 ## <http://www.gnu.org/licenses/>. | |
| 18 | |
| 19 ## -*- texinfo -*- | |
|
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20 ## @deftypefn {Function File} {[@var{est}, @var{v}, @var{w}, @var{iter}] =} onenormest (@var{A}, @var{t}) |
| 7189 | 21 ## @deftypefnx {Function File} {[@var{est}, @var{v}, @var{w}, @var{iter}] =} onenormest (@var{apply}, @var{apply_t}, @var{n}, @var{t}) |
| 22 ## | |
| 23 ## Apply Higham and Tisseur's randomized block 1-norm estimator to | |
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24 ## matrix @var{A} using @var{t} test vectors. If @var{t} exceeds 5, then |
| 7189 | 25 ## only 5 test vectors are used. |
| 26 ## | |
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27 ## If the matrix is not explicit, e.g., when estimating the norm of |
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28 ## @code{inv (@var{A})} given an LU@tie{}factorization, @code{onenormest} |
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29 ## applies @var{A} and its conjugate transpose through a pair of functions |
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30 ## @var{apply} and @var{apply_t}, respectively, to a dense matrix of size |
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31 ## @var{n} by @var{t}. The implicit version requires an explicit dimension |
| 7189 | 32 ## @var{n}. |
| 33 ## | |
| 34 ## Returns the norm estimate @var{est}, two vectors @var{v} and | |
| 35 ## @var{w} related by norm | |
| 36 ## @code{(@var{w}, 1) = @var{est} * norm (@var{v}, 1)}, | |
| 37 ## and the number of iterations @var{iter}. The number of | |
| 38 ## iterations is limited to 10 and is at least 2. | |
| 39 ## | |
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40 ## References: |
| 7189 | 41 ## @itemize |
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42 ## @item |
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43 ## N.J. Higham and F. Tisseur, @cite{A Block Algorithm |
| 7189 | 44 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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45 ## Pseudospectra}. SIMAX vol 21, no 4, pp 1185-1201. |
| 7189 | 46 ## @url{http://dx.doi.org/10.1137/S0895479899356080} |
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47 ## |
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48 ## @item |
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49 ## N.J. Higham and F. Tisseur, @cite{A Block Algorithm |
| 7189 | 50 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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51 ## Pseudospectra}. @url{http://citeseer.ist.psu.edu/223007.html} |
| 7189 | 52 ## @end itemize |
| 53 ## | |
| 54 ## @seealso{condest, norm, cond} | |
| 55 ## @end deftypefn | |
| 56 | |
| 57 ## Code originally licensed under | |
| 58 ## | |
| 59 ## Copyright (c) 2007, Regents of the University of California | |
| 60 ## All rights reserved. | |
| 7191 | 61 ## |
| 7189 | 62 ## Redistribution and use in source and binary forms, with or without |
| 7191 | 63 ## modification, are permitted provided that the following conditions |
| 64 ## are met: | |
| 7189 | 65 ## |
| 66 ## * Redistributions of source code must retain the above copyright | |
| 67 ## notice, this list of conditions and the following disclaimer. | |
| 7191 | 68 ## |
| 69 ## * Redistributions in binary form must reproduce the above | |
| 70 ## copyright notice, this list of conditions and the following | |
| 71 ## disclaimer in the documentation and/or other materials provided | |
| 72 ## with the distribution. | |
| 73 ## | |
| 74 ## * Neither the name of the University of California, Berkeley nor | |
| 75 ## the names of its contributors may be used to endorse or promote | |
| 76 ## products derived from this software without specific prior | |
| 77 ## written permission. | |
| 7189 | 78 ## |
| 7191 | 79 ## THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' |
| 80 ## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED | |
| 81 ## TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A | |
| 82 ## PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS AND | |
| 83 ## CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, | |
| 84 ## SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT | |
| 85 ## LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF | |
| 86 ## USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND | |
| 87 ## ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, | |
| 88 ## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT | |
| 89 ## OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF | |
| 7189 | 90 ## SUCH DAMAGE. |
| 91 | |
| 92 ## Author: Jason Riedy <ejr@cs.berkeley.edu> | |
| 93 ## Keywords: linear-algebra norm estimation | |
| 94 ## Version: 0.2 | |
| 95 | |
| 96 function [est, v, w, iter] = onenormest (varargin) | |
| 97 | |
| 98 if (size (varargin, 2) < 1 || size (varargin, 2) > 4) | |
| 99 print_usage (); | |
| 100 endif | |
| 101 | |
| 102 default_t = 5; | |
| 103 itmax = 10; | |
| 104 | |
| 105 if (ismatrix (varargin{1})) | |
| 106 n = size (varargin{1}, 1); | |
| 107 if n != size (varargin{1}, 2), | |
| 8664 | 108 error ("onenormest: matrix must be square"); |
| 7189 | 109 endif |
| 110 apply = @(x) varargin{1} * x; | |
| 111 apply_t = @(x) varargin{1}' * x; | |
| 112 if (size (varargin) > 1) | |
| 113 t = varargin{2}; | |
| 114 else | |
| 115 t = min (n, default_t); | |
| 116 endif | |
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117 issing = isa (varargin {1}, "single"); |
| 7189 | 118 else |
| 119 if (size (varargin, 2) < 3) | |
| 120 print_usage(); | |
| 121 endif | |
| 122 n = varargin{3}; | |
| 123 apply = varargin{1}; | |
| 124 apply_t = varargin{2}; | |
| 125 if (size (varargin) > 3) | |
| 126 t = varargin{4}; | |
| 127 else | |
| 128 t = default_t; | |
| 129 endif | |
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130 issing = isa (varargin {3}, "single"); |
| 7189 | 131 endif |
| 132 | |
| 133 ## Initial test vectors X. | |
| 134 X = rand (n, t); | |
| 135 X = X ./ (ones (n,1) * sum (abs (X), 1)); | |
| 136 | |
| 8506 | 137 ## Track if a vertex has been visited. |
| 138 been_there = zeros (n, 1); | |
| 139 | |
| 140 ## To check if the estimate has increased. | |
| 141 est_old = 0; | |
| 142 | |
| 143 ## Normalized vector of signs. | |
| 144 S = zeros (n, t); | |
| 7189 | 145 |
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146 if (issing) |
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147 myeps = eps ("single"); |
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148 X = single (X); |
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149 else |
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150 myeps = eps; |
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151 endif |
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152 |
| 7189 | 153 for iter = 1 : itmax + 1 |
| 154 Y = feval (apply, X); | |
| 155 | |
| 156 ## Find the initial estimate as the largest A*x. | |
| 157 [est, ind_best] = max (sum (abs (Y), 1)); | |
| 158 if (est > est_old || iter == 2) | |
| 159 w = Y(:,ind_best); | |
| 160 endif | |
| 161 if (iter >= 2 && est < est_old) | |
| 162 ## No improvement, so stop. | |
| 163 est = est_old; | |
| 164 break; | |
| 165 endif | |
| 166 | |
| 167 est_old = est; | |
| 168 S_old = S; | |
| 169 if (iter > itmax), | |
| 170 ## Gone too far. Stop. | |
| 171 break; | |
| 172 endif | |
| 173 | |
| 174 S = sign (Y); | |
| 175 | |
| 176 ## Test if any of S are approximately parallel to previous S | |
| 177 ## vectors or current S vectors. If everything is parallel, | |
| 178 ## stop. Otherwise, replace any parallel vectors with | |
| 179 ## rand{-1,+1}. | |
| 180 partest = any (abs (S_old' * S - n) < 4*eps*n); | |
| 181 if (all (partest)) | |
| 182 ## All the current vectors are parallel to old vectors. | |
| 183 ## We've hit a cycle, so stop. | |
| 184 break; | |
| 185 endif | |
| 186 if (any (partest)) | |
| 187 ## Some vectors are parallel to old ones and are cycling, | |
| 188 ## but not all of them. Replace the parallel vectors with | |
| 189 ## rand{-1,+1}. | |
| 190 numpar = sum (partest); | |
| 191 replacements = 2*(rand (n,numpar) < 0.5) - 1; | |
| 192 S(:,partest) = replacements; | |
| 193 endif | |
| 194 ## Now test for parallel vectors within S. | |
| 8507 | 195 partest = any ((S' * S - eye (t)) == n); |
| 7189 | 196 if (any (partest)) |
| 197 numpar = sum (partest); | |
| 198 replacements = 2*(rand (n,numpar) < 0.5) - 1; | |
| 199 S(:,partest) = replacements; | |
| 200 endif | |
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201 |
| 7189 | 202 Z = feval (apply_t, S); |
| 203 | |
| 204 ## Now find the largest non-previously-visted index per | |
| 205 ## vector. | |
| 206 h = max (abs (Z),2); | |
| 207 [mh, mhi] = max (h); | |
| 208 if (iter >= 2 && mhi == ind_best) | |
| 209 ## Hit a cycle, stop. | |
| 210 break; | |
| 211 endif | |
| 212 [h, ind] = sort (h, 'descend'); | |
| 213 if (t > 1) | |
| 214 firstind = ind(1:t); | |
| 215 if (all (been_there(firstind))) | |
| 10549 | 216 ## Visited all these before, so stop. |
| 217 break; | |
| 7189 | 218 endif |
| 219 ind = ind (!been_there (ind)); | |
| 220 if (length (ind) < t) | |
| 10549 | 221 ## There aren't enough new vectors, so we're practically |
| 222 ## in a cycle. Stop. | |
| 223 break; | |
| 7189 | 224 endif |
| 225 endif | |
| 226 | |
| 227 ## Visit the new indices. | |
| 228 X = zeros (n, t); | |
| 229 for zz = 1 : t | |
| 230 X(ind(zz),zz) = 1; | |
| 231 endfor | |
| 232 been_there (ind (1 : t)) = 1; | |
| 233 endfor | |
| 234 | |
| 235 ## The estimate est and vector w are set in the loop above. The | |
| 236 ## vector v selects the ind_best column of A. | |
| 237 v = zeros (n, 1); | |
| 238 v(ind_best) = 1; | |
| 239 endfunction | |
| 240 | |
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241 |
| 7189 | 242 %!demo |
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243 %! N = 100; |
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244 %! A = randn (N) + eye (N); |
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245 %! [L,U,P] = lu (A); |
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246 %! nm1inv = onenormest (@(x) U\(L\(P*x)), @(x) P'*(L'\(U'\x)), N, 30) |
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247 %! norm (inv (A), 1) |
| 7189 | 248 |
| 249 %!test | |
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250 %! N = 10; |
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251 %! A = ones (N); |
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252 %! [nm1, v1, w1] = onenormest (A); |
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253 %! [nminf, vinf, winf] = onenormest (A', 6); |
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254 %! assert (nm1, N, -2*eps); |
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255 %! assert (nminf, N, -2*eps); |
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256 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps) |
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257 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps) |
| 7189 | 258 |
| 259 %!test | |
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260 %! N = 10; |
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261 %! A = ones (N); |
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262 %! [nm1, v1, w1] = onenormest (@(x) A*x, @(x) A'*x, N, 3); |
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263 %! [nminf, vinf, winf] = onenormest (@(x) A'*x, @(x) A*x, N, 3); |
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264 %! assert (nm1, N, -2*eps); |
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265 %! assert (nminf, N, -2*eps); |
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266 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps) |
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267 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps) |
| 7189 | 268 |
| 269 %!test | |
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270 %! N = 5; |
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271 %! A = hilb (N); |
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272 %! [nm1, v1, w1] = onenormest (A); |
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273 %! [nminf, vinf, winf] = onenormest (A', 6); |
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274 %! assert (nm1, norm (A, 1), -2*eps); |
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275 %! assert (nminf, norm (A, inf), -2*eps); |
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276 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps) |
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277 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps) |
| 7189 | 278 |
| 279 ## Only likely to be within a factor of 10. | |
| 280 %!test | |
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281 %! old_state = rand ("state"); |
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282 %! restore_state = onCleanup (@() rand ("state", old_state)); |
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283 %! rand ('state', 42); % Initialize to guarantee reproducible results |
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284 %! N = 100; |
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285 %! A = rand (N); |
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286 %! [nm1, v1, w1] = onenormest (A); |
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287 %! [nminf, vinf, winf] = onenormest (A', 6); |
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288 %! assert (nm1, norm (A, 1), -.1); |
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289 %! assert (nminf, norm (A, inf), -.1); |
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290 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps) |
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291 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps) |