Mercurial > octave-antonio
annotate scripts/linear-algebra/onenormest.m @ 20160:03b9d17a2d95 stable
doc: Update more docstrings to have one sentence summary as first line.
Reviewed io, java, linear-algebra, prefs, and set script directories.
* scripts/io/beep.m, scripts/io/dlmwrite.m, scripts/io/importdata.m,
scripts/io/strread.m, scripts/io/textread.m, scripts/java/javaArray.m,
scripts/java/java_get.m, scripts/java/java_set.m, scripts/java/javaaddpath.m,
scripts/java/javachk.m, scripts/java/javaclasspath.m, scripts/java/javamem.m,
scripts/java/javarmpath.m, scripts/linear-algebra/bandwidth.m,
scripts/linear-algebra/commutation_matrix.m, scripts/linear-algebra/cond.m,
scripts/linear-algebra/condest.m, scripts/linear-algebra/cross.m,
scripts/linear-algebra/duplication_matrix.m, scripts/linear-algebra/expm.m,
scripts/linear-algebra/housh.m, scripts/linear-algebra/isdefinite.m,
scripts/linear-algebra/ishermitian.m, scripts/linear-algebra/issymmetric.m,
scripts/linear-algebra/istril.m, scripts/linear-algebra/istriu.m,
scripts/linear-algebra/krylov.m, scripts/linear-algebra/logm.m,
scripts/linear-algebra/normest.m, scripts/linear-algebra/null.m,
scripts/linear-algebra/onenormest.m, scripts/linear-algebra/orth.m,
scripts/linear-algebra/qzhess.m, scripts/linear-algebra/rank.m,
scripts/linear-algebra/rref.m, scripts/linear-algebra/vech.m,
scripts/path/matlabroot.m, scripts/prefs/addpref.m, scripts/prefs/getpref.m,
scripts/prefs/ispref.m, scripts/prefs/rmpref.m, scripts/prefs/setpref.m,
scripts/set/powerset.m, scripts/set/setdiff.m:
Update more docstrings to have one sentence summary as first line.
| author | Rik <rik@octave.org> |
|---|---|
| date | Sun, 03 May 2015 15:36:23 -0700 |
| parents | 00e31f316a3a |
| children |
| rev | line source |
|---|---|
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1 ## Copyright (C) 2007-2015 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 ## | |
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23 ## Apply @nospell{Higham and Tisseur's} randomized block 1-norm estimator to |
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24 ## matrix @var{A} using @var{t} test vectors. |
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25 ## |
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26 ## If @var{t} exceeds 5, then only 5 test vectors are used. |
| 7189 | 27 ## |
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28 ## If the matrix is not explicit, e.g., when estimating the norm of |
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29 ## @code{inv (@var{A})} given an LU@tie{}factorization, @code{onenormest} |
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30 ## applies @var{A} and its conjugate transpose through a pair of functions |
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31 ## @var{apply} and @var{apply_t}, respectively, to a dense matrix of size |
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32 ## @var{n} by @var{t}. The implicit version requires an explicit dimension |
| 7189 | 33 ## @var{n}. |
| 34 ## | |
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35 ## Returns the norm estimate @var{est}, two vectors @var{v} and @var{w} related |
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36 ## by norm @code{(@var{w}, 1) = @var{est} * norm (@var{v}, 1)}, and the number |
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37 ## of iterations @var{iter}. The number of iterations is limited to 10 and is |
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38 ## at least 2. |
| 7189 | 39 ## |
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40 ## References: |
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41 ## |
| 7189 | 42 ## @itemize |
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43 ## @item |
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44 ## @nospell{N.J. Higham and F. Tisseur}, @cite{A Block Algorithm |
| 7189 | 45 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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46 ## Pseudospectra}. SIMAX vol 21, no 4, pp 1185-1201. |
| 7189 | 47 ## @url{http://dx.doi.org/10.1137/S0895479899356080} |
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48 ## |
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49 ## @item |
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50 ## @nospell{N.J. Higham and F. Tisseur}, @cite{A Block Algorithm |
| 7189 | 51 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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52 ## Pseudospectra}. @url{http://citeseer.ist.psu.edu/223007.html} |
| 7189 | 53 ## @end itemize |
| 54 ## | |
| 55 ## @seealso{condest, norm, cond} | |
| 56 ## @end deftypefn | |
| 57 | |
| 58 ## Code originally licensed under | |
| 59 ## | |
| 60 ## Copyright (c) 2007, Regents of the University of California | |
| 61 ## All rights reserved. | |
| 7191 | 62 ## |
| 7189 | 63 ## Redistribution and use in source and binary forms, with or without |
| 7191 | 64 ## modification, are permitted provided that the following conditions |
| 65 ## are met: | |
| 7189 | 66 ## |
| 67 ## * Redistributions of source code must retain the above copyright | |
| 68 ## notice, this list of conditions and the following disclaimer. | |
| 7191 | 69 ## |
| 70 ## * Redistributions in binary form must reproduce the above | |
| 71 ## copyright notice, this list of conditions and the following | |
| 72 ## disclaimer in the documentation and/or other materials provided | |
| 73 ## with the distribution. | |
| 74 ## | |
| 75 ## * Neither the name of the University of California, Berkeley nor | |
| 76 ## the names of its contributors may be used to endorse or promote | |
| 77 ## products derived from this software without specific prior | |
| 78 ## written permission. | |
| 7189 | 79 ## |
| 7191 | 80 ## THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' |
| 81 ## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED | |
| 82 ## TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A | |
| 83 ## PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS AND | |
| 84 ## CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, | |
| 85 ## SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT | |
| 86 ## LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF | |
| 87 ## USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND | |
| 88 ## ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, | |
| 89 ## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT | |
| 90 ## OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF | |
| 7189 | 91 ## SUCH DAMAGE. |
| 92 | |
| 93 ## Author: Jason Riedy <ejr@cs.berkeley.edu> | |
| 94 ## Keywords: linear-algebra norm estimation | |
| 95 ## Version: 0.2 | |
| 96 | |
| 97 function [est, v, w, iter] = onenormest (varargin) | |
| 98 | |
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99 if (nargin < 1 || nargin > 4) |
| 7189 | 100 print_usage (); |
| 101 endif | |
| 102 | |
| 103 default_t = 5; | |
| 104 itmax = 10; | |
| 105 | |
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106 if (isnumeric (varargin{1})) |
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107 [n, nc] = size (varargin{1}); |
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108 if (n != nc) |
| 8664 | 109 error ("onenormest: matrix must be square"); |
| 7189 | 110 endif |
| 111 apply = @(x) varargin{1} * x; | |
| 112 apply_t = @(x) varargin{1}' * x; | |
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113 if (nargin > 1) |
| 7189 | 114 t = varargin{2}; |
| 115 else | |
| 116 t = min (n, default_t); | |
| 117 endif | |
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118 issing = isa (varargin{1}, "single"); |
| 7189 | 119 else |
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120 if (nargin < 3) |
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121 print_usage (); |
| 7189 | 122 endif |
| 123 apply = varargin{1}; | |
| 124 apply_t = varargin{2}; | |
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125 n = varargin{3}; |
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126 if (nargin > 3) |
| 7189 | 127 t = varargin{4}; |
| 128 else | |
| 129 t = default_t; | |
| 130 endif | |
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131 issing = isa (n, "single"); |
| 7189 | 132 endif |
| 133 | |
| 134 ## Initial test vectors X. | |
| 135 X = rand (n, t); | |
| 136 X = X ./ (ones (n,1) * sum (abs (X), 1)); | |
| 137 | |
| 8506 | 138 ## Track if a vertex has been visited. |
| 139 been_there = zeros (n, 1); | |
| 140 | |
| 141 ## To check if the estimate has increased. | |
| 142 est_old = 0; | |
| 143 | |
| 144 ## Normalized vector of signs. | |
| 145 S = zeros (n, t); | |
| 7189 | 146 |
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147 if (issing) |
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148 myeps = eps ("single"); |
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149 X = single (X); |
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150 else |
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151 myeps = eps; |
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152 endif |
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153 |
| 7189 | 154 for iter = 1 : itmax + 1 |
| 155 Y = feval (apply, X); | |
| 156 | |
| 157 ## Find the initial estimate as the largest A*x. | |
| 158 [est, ind_best] = max (sum (abs (Y), 1)); | |
| 159 if (est > est_old || iter == 2) | |
| 160 w = Y(:,ind_best); | |
| 161 endif | |
| 162 if (iter >= 2 && est < est_old) | |
| 163 ## No improvement, so stop. | |
| 164 est = est_old; | |
| 165 break; | |
| 166 endif | |
| 167 | |
| 168 est_old = est; | |
| 169 S_old = S; | |
| 170 if (iter > itmax), | |
| 171 ## Gone too far. Stop. | |
| 172 break; | |
| 173 endif | |
| 174 | |
| 175 S = sign (Y); | |
| 176 | |
| 177 ## Test if any of S are approximately parallel to previous S | |
| 178 ## vectors or current S vectors. If everything is parallel, | |
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179 ## stop. Otherwise, replace any parallel vectors with |
| 7189 | 180 ## rand{-1,+1}. |
| 181 partest = any (abs (S_old' * S - n) < 4*eps*n); | |
| 182 if (all (partest)) | |
| 183 ## All the current vectors are parallel to old vectors. | |
| 184 ## We've hit a cycle, so stop. | |
| 185 break; | |
| 186 endif | |
| 187 if (any (partest)) | |
| 188 ## Some vectors are parallel to old ones and are cycling, | |
| 189 ## but not all of them. Replace the parallel vectors with | |
| 190 ## rand{-1,+1}. | |
| 191 numpar = sum (partest); | |
| 192 replacements = 2*(rand (n,numpar) < 0.5) - 1; | |
| 193 S(:,partest) = replacements; | |
| 194 endif | |
| 195 ## Now test for parallel vectors within S. | |
| 8507 | 196 partest = any ((S' * S - eye (t)) == n); |
| 7189 | 197 if (any (partest)) |
| 198 numpar = sum (partest); | |
| 199 replacements = 2*(rand (n,numpar) < 0.5) - 1; | |
| 200 S(:,partest) = replacements; | |
| 201 endif | |
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202 |
| 7189 | 203 Z = feval (apply_t, S); |
| 204 | |
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205 ## Now find the largest non-previously-visted index per vector. |
| 7189 | 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 |
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219 ind = ind(! been_there(ind)); |
| 7189 | 220 if (length (ind) < t) |
| 10549 | 221 ## There aren't enough new vectors, so we're practically |
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222 ## in a cycle. Stop. |
| 10549 | 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 | |
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232 been_there(ind(1 : t)) = 1; |
| 7189 | 233 endfor |
| 234 | |
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235 ## The estimate est and vector w are set in the loop above. |
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236 ## The vector v selects the ind_best column of A. |
| 7189 | 237 v = zeros (n, 1); |
| 238 v(ind_best) = 1; | |
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239 |
| 7189 | 240 endfunction |
| 241 | |
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242 |
| 7189 | 243 %!demo |
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244 %! N = 100; |
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245 %! A = randn (N) + eye (N); |
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246 %! [L,U,P] = lu (A); |
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247 %! nm1inv = onenormest (@(x) U\(L\(P*x)), @(x) P'*(L'\(U'\x)), N, 30) |
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248 %! norm (inv (A), 1) |
| 7189 | 249 |
| 250 %!test | |
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251 %! N = 10; |
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252 %! A = ones (N); |
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253 %! [nm1, v1, w1] = onenormest (A); |
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254 %! [nminf, vinf, winf] = onenormest (A', 6); |
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255 %! assert (nm1, N, -2*eps); |
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256 %! assert (nminf, N, -2*eps); |
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257 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps); |
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258 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps); |
| 7189 | 259 |
| 260 %!test | |
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261 %! N = 10; |
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262 %! A = ones (N); |
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263 %! [nm1, v1, w1] = onenormest (@(x) A*x, @(x) A'*x, N, 3); |
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264 %! [nminf, vinf, winf] = onenormest (@(x) A'*x, @(x) A*x, N, 3); |
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265 %! assert (nm1, N, -2*eps); |
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266 %! assert (nminf, N, -2*eps); |
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267 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps); |
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268 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps); |
| 7189 | 269 |
| 270 %!test | |
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271 %! N = 5; |
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272 %! A = hilb (N); |
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273 %! [nm1, v1, w1] = onenormest (A); |
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274 %! [nminf, vinf, winf] = onenormest (A', 6); |
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275 %! assert (nm1, norm (A, 1), -2*eps); |
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276 %! assert (nminf, norm (A, inf), -2*eps); |
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277 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps); |
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278 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps); |
| 7189 | 279 |
| 280 ## Only likely to be within a factor of 10. | |
| 281 %!test | |
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282 %! old_state = rand ("state"); |
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283 %! restore_state = onCleanup (@() rand ("state", old_state)); |
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284 %! rand ('state', 42); % Initialize to guarantee reproducible results |
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285 %! N = 100; |
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286 %! A = rand (N); |
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287 %! [nm1, v1, w1] = onenormest (A); |
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288 %! [nminf, vinf, winf] = onenormest (A', 6); |
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289 %! assert (nm1, norm (A, 1), -.1); |
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290 %! assert (nminf, norm (A, inf), -.1); |
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291 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps); |
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292 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps); |
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293 |