Mercurial > hg > octave-lyh
annotate scripts/linear-algebra/condest.m @ 9307:c2923c27c877
Various documentation improvements
| author | Rik <rdrider0-list@yahoo.com> |
|---|---|
| date | Sat, 06 Jun 2009 18:30:52 -0700 |
| parents | 1bf0ce0930be |
| children | 72d6e0de76c7 |
| rev | line source |
|---|---|
| 8920 | 1 ## Copyright (C) 2007, 2008, 2009 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 -*- | |
| 7191 | 20 ## @deftypefn {Function File} {[@var{est}, @var{v}] =} condest (@var{a}, @var{t}) |
| 21 ## @deftypefnx {Function File} {[@var{est}, @var{v}] =} condest (@var{a}, @var{solve}, @var{solve_t}, @var{t}) | |
| 7189 | 22 ## @deftypefnx {Function File} {[@var{est}, @var{v}] =} condest (@var{apply}, @var{apply_t}, @var{solve}, @var{solve_t}, @var{n}, @var{t}) |
| 23 ## | |
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24 ## Estimate the 1-norm condition number of a matrix @var{A} |
| 7189 | 25 ## using @var{t} test vectors using a randomized 1-norm estimator. |
| 26 ## If @var{t} exceeds 5, then only 5 test vectors are used. | |
| 27 ## | |
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Rik <rdrider0-list@yahoo.com>
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28 ## If the matrix is not explicit, e.g., when estimating the condition |
| 7191 | 29 ## number of @var{a} given an LU factorization, @code{condest} uses the |
| 7189 | 30 ## following functions: |
| 31 ## | |
| 32 ## @table @var | |
| 33 ## @item apply | |
| 34 ## @code{A*x} for a matrix @code{x} of size @var{n} by @var{t}. | |
| 35 ## @item apply_t | |
| 36 ## @code{A'*x} for a matrix @code{x} of size @var{n} by @var{t}. | |
| 37 ## @item solve | |
| 38 ## @code{A \ b} for a matrix @code{b} of size @var{n} by @var{t}. | |
| 39 ## @item solve_t | |
| 40 ## @code{A' \ b} for a matrix @code{b} of size @var{n} by @var{t}. | |
| 41 ## @end table | |
| 42 ## | |
| 43 ## The implicit version requires an explicit dimension @var{n}. | |
| 44 ## | |
| 45 ## @code{condest} uses a randomized algorithm to approximate | |
| 46 ## the 1-norms. | |
| 47 ## | |
| 48 ## @code{condest} returns the 1-norm condition estimate @var{est} and | |
| 7191 | 49 ## a vector @var{v} satisfying @code{norm (A*v, 1) == norm (A, 1) * norm |
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Rik <rdrider0-list@yahoo.com>
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changeset
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50 ## (@var{v}, 1) / @var{est}}. When @var{est} is large, @var{v} is an |
| 7191 | 51 ## approximate null vector. |
| 7189 | 52 ## |
| 53 ## References: | |
| 54 ## @itemize | |
| 55 ## @item Nicholas J. Higham and Françoise Tisseur, "A Block Algorithm | |
| 56 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm | |
| 57 ## Pseudospectra." SIMAX vol 21, no 4, pp 1185-1201. | |
| 58 ## @url{http://dx.doi.org/10.1137/S0895479899356080} | |
| 59 ## @item Nicholas J. Higham and Françoise Tisseur, "A Block Algorithm | |
| 60 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm | |
| 61 ## Pseudospectra." @url{http://citeseer.ist.psu.edu/223007.html} | |
| 62 ## @end itemize | |
| 63 ## | |
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9307
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Rik <rdrider0-list@yahoo.com>
parents:
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64 ## @seealso{cond, norm, onenormest} |
| 7189 | 65 ## @end deftypefn |
| 66 | |
| 67 ## Code originally licensed under | |
| 68 ## | |
| 69 ## Copyright (c) 2007, Regents of the University of California | |
| 70 ## All rights reserved. | |
| 7191 | 71 ## |
| 7189 | 72 ## Redistribution and use in source and binary forms, with or without |
| 7191 | 73 ## modification, are permitted provided that the following conditions |
| 74 ## are met: | |
| 7189 | 75 ## |
| 76 ## * Redistributions of source code must retain the above copyright | |
| 77 ## notice, this list of conditions and the following disclaimer. | |
| 7191 | 78 ## |
| 79 ## * Redistributions in binary form must reproduce the above | |
| 80 ## copyright notice, this list of conditions and the following | |
| 81 ## disclaimer in the documentation and/or other materials provided | |
| 82 ## with the distribution. | |
| 83 ## | |
| 84 ## * Neither the name of the University of California, Berkeley nor | |
| 85 ## the names of its contributors may be used to endorse or promote | |
| 86 ## products derived from this software without specific prior | |
| 87 ## written permission. | |
| 7189 | 88 ## |
| 7191 | 89 ## THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' |
| 90 ## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED | |
| 91 ## TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A | |
| 92 ## PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS AND | |
| 93 ## CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, | |
| 94 ## SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT | |
| 95 ## LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF | |
| 96 ## USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND | |
| 97 ## ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, | |
| 98 ## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT | |
| 99 ## OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF | |
| 7189 | 100 ## SUCH DAMAGE. |
| 101 | |
| 102 ## Author: Jason Riedy <ejr@cs.berkeley.edu> | |
| 103 ## Keywords: linear-algebra norm estimation | |
| 104 ## Version: 0.2 | |
| 105 | |
| 106 function [est, v] = condest (varargin) | |
| 7191 | 107 |
| 7309 | 108 if (nargin < 1 || nargin > 6) |
| 7191 | 109 print_usage (); |
| 7189 | 110 endif |
| 111 | |
| 112 default_t = 5; | |
| 113 | |
| 7312 | 114 have_A = false; |
| 115 have_t = false; | |
| 116 have_solve = false; | |
| 117 | |
| 7189 | 118 if (ismatrix (varargin{1})) |
| 7310 | 119 A = varargin{1}; |
| 120 n = issquare (A); | |
| 121 if (! n) | |
| 8664 | 122 error ("condest: matrix must be square"); |
| 7189 | 123 endif |
| 7312 | 124 have_A = true; |
| 7189 | 125 |
| 7312 | 126 if (nargin > 1) |
| 127 if (isscalar (varargin{2})) | |
| 128 t = varargin{2}; | |
| 129 have_t = true; | |
| 130 elseif (nargin > 2) | |
| 131 solve = varargin{2}; | |
| 132 solve_t = varargin{3}; | |
| 133 have_solve = true; | |
| 134 if (nargin > 3) | |
| 135 t = varargin{4}; | |
| 136 have_t = true; | |
| 137 endif | |
| 138 else | |
| 8664 | 139 error ("condest: must supply both solve and solve_t"); |
| 7189 | 140 endif |
| 141 endif | |
| 7311 | 142 elseif (nargin > 4) |
| 7189 | 143 apply = varargin{1}; |
| 144 apply_t = varargin{2}; | |
| 145 solve = varargin{3}; | |
| 146 solve_t = varargin{4}; | |
| 7312 | 147 have_solve = true; |
| 7189 | 148 n = varargin{5}; |
| 149 if (! isscalar (n)) | |
| 8664 | 150 error ("condest: dimension argument of implicit form must be scalar"); |
| 7189 | 151 endif |
| 7309 | 152 if (nargin > 5) |
| 7189 | 153 t = varargin{6}; |
| 7312 | 154 have_t = true; |
| 7189 | 155 endif |
| 7311 | 156 else |
| 8664 | 157 error ("condest: implicit form of condest requires at least 5 arguments"); |
| 7189 | 158 endif |
| 159 | |
| 7312 | 160 if (! have_t) |
| 7189 | 161 t = min (n, default_t); |
| 162 endif | |
| 163 | |
| 7312 | 164 if (! have_solve) |
| 7189 | 165 if (issparse (A)) |
| 8912 | 166 [L, U, P, Pc] = lu (A); |
| 7312 | 167 solve = @(x) Pc' * (U \ (L \ (P * x))); |
| 168 solve_t = @(x) P' * (L' \ (U' \ (Pc * x))); | |
| 7189 | 169 else |
| 170 [L, U, P] = lu (A); | |
| 7312 | 171 solve = @(x) U \ (L \ (P*x)); |
| 172 solve_t = @(x) P' * (L' \ (U' \ x)); | |
| 7189 | 173 endif |
| 174 endif | |
| 175 | |
| 7312 | 176 if (have_A) |
| 7189 | 177 Anorm = norm (A, 1); |
| 178 else | |
| 179 Anorm = onenormest (apply, apply_t, n, t); | |
| 180 endif | |
| 181 | |
| 182 [Ainv_norm, v, w] = onenormest (solve, solve_t, n, t); | |
| 183 | |
| 184 est = Anorm * Ainv_norm; | |
| 185 v = w / norm (w, 1); | |
| 186 | |
| 187 endfunction | |
| 188 | |
| 189 %!demo | |
| 190 %! N = 100; | |
| 191 %! A = randn (N) + eye (N); | |
| 192 %! condest (A) | |
| 193 %! [L,U,P] = lu (A); | |
| 194 %! condest (A, @(x) U\ (L\ (P*x)), @(x) P'*(L'\ (U'\x))) | |
| 195 %! condest (@(x) A*x, @(x) A'*x, @(x) U\ (L\ (P*x)), @(x) P'*(L'\ (U'\x)), N) | |
| 196 %! norm (inv (A), 1) * norm (A, 1) | |
| 197 | |
| 198 ## Yes, these test bounds are really loose. There's | |
| 199 ## enough randomization to trigger odd cases with hilb(). | |
| 200 | |
| 201 %!test | |
| 202 %! N = 6; | |
| 203 %! A = hilb (N); | |
| 204 %! cA = condest (A); | |
| 205 %! cA_test = norm (inv (A), 1) * norm (A, 1); | |
| 8912 | 206 %! assert (cA, cA_test, -2^-8); |
| 7189 | 207 |
| 208 %!test | |
| 209 %! N = 6; | |
| 210 %! A = hilb (N); | |
| 211 %! solve = @(x) A\x; solve_t = @(x) A'\x; | |
| 212 %! cA = condest (A, solve, solve_t); | |
| 213 %! cA_test = norm (inv (A), 1) * norm (A, 1); | |
| 8912 | 214 %! assert (cA, cA_test, -2^-8); |
| 7189 | 215 |
| 216 %!test | |
| 217 %! N = 6; | |
| 218 %! A = hilb (N); | |
| 219 %! apply = @(x) A*x; apply_t = @(x) A'*x; | |
| 220 %! solve = @(x) A\x; solve_t = @(x) A'\x; | |
| 221 %! cA = condest (apply, apply_t, solve, solve_t, N); | |
| 222 %! cA_test = norm (inv (A), 1) * norm (A, 1); | |
| 8912 | 223 %! assert (cA, cA_test, -2^-6); |
| 7189 | 224 |
| 225 %!test | |
| 226 %! N = 12; | |
| 227 %! A = hilb (N); | |
| 228 %! [rcondA, v] = condest (A); | |
| 229 %! x = A*v; | |
| 230 %! assert (norm(x, inf), 0, eps); |