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annotate scripts/linear-algebra/onenormest.m @ 11587:c792872f8942
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| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Thu, 20 Jan 2011 17:35:29 -0500 |
| parents | fd0a3ac60b0e |
| children | 1577c6f80926 |
| rev | line source |
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| 11523 | 1 ## Copyright (C) 2007-2011 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 factorization, @code{onenormest} applies |
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29 ## @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 | |
| 241 %!demo | |
| 242 %! N = 100; | |
| 243 %! A = randn(N) + eye(N); | |
| 244 %! [L,U,P] = lu(A); | |
| 245 %! nm1inv = onenormest(@(x) U\(L\(P*x)), @(x) P'*(L'\(U'\x)), N, 30) | |
| 246 %! norm(inv(A), 1) | |
| 247 | |
| 248 %!test | |
| 249 %! N = 10; | |
| 250 %! A = ones (N); | |
| 251 %! [nm1, v1, w1] = onenormest (A); | |
| 252 %! [nminf, vinf, winf] = onenormest (A', 6); | |
| 253 %! assert (nm1, N, -2*eps); | |
| 254 %! assert (nminf, N, -2*eps); | |
| 255 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps) | |
| 256 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps) | |
| 257 | |
| 258 %!test | |
| 259 %! N = 10; | |
| 260 %! A = ones (N); | |
| 261 %! [nm1, v1, w1] = onenormest (@(x) A*x, @(x) A'*x, N, 3); | |
| 262 %! [nminf, vinf, winf] = onenormest (@(x) A'*x, @(x) A*x, N, 3); | |
| 263 %! assert (nm1, N, -2*eps); | |
| 264 %! assert (nminf, N, -2*eps); | |
| 265 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps) | |
| 266 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps) | |
| 267 | |
| 268 %!test | |
| 269 %! N = 5; | |
| 270 %! A = hilb (N); | |
| 271 %! [nm1, v1, w1] = onenormest (A); | |
| 272 %! [nminf, vinf, winf] = onenormest (A', 6); | |
| 273 %! assert (nm1, norm (A, 1), -2*eps); | |
| 274 %! assert (nminf, norm (A, inf), -2*eps); | |
| 275 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps) | |
| 276 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps) | |
| 277 | |
| 278 ## Only likely to be within a factor of 10. | |
| 279 %!test | |
| 280 %! N = 100; | |
| 281 %! A = rand (N); | |
| 282 %! [nm1, v1, w1] = onenormest (A); | |
| 283 %! [nminf, vinf, winf] = onenormest (A', 6); | |
| 284 %! assert (nm1, norm (A, 1), -.1); | |
| 285 %! assert (nminf, norm (A, inf), -.1); | |
| 286 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps) | |
| 287 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps) |