Mercurial > hg > octave-lyh
annotate scripts/linear-algebra/condest.m @ 10687:a8ce6bdecce5
Improve documentation strings.
| author | Rik <octave@nomad.inbox5.com> |
|---|---|
| date | Tue, 08 Jun 2010 20:22:38 -0700 |
| parents | 95c3e38098bf |
| children | 3140cb7a05a1 |
| rev | line source |
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| 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 -*- | |
|
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20 ## @deftypefn {Function File} {} condest (@var{a}) |
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21 ## @deftypefnx {Function File} {} condest (@var{a}, @var{t}) |
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22 ## @deftypefnx {Function File} {[@var{est}, @var{v}] =} condest (@dots{}) |
| 7191 | 23 ## @deftypefnx {Function File} {[@var{est}, @var{v}] =} condest (@var{a}, @var{solve}, @var{solve_t}, @var{t}) |
| 7189 | 24 ## @deftypefnx {Function File} {[@var{est}, @var{v}] =} condest (@var{apply}, @var{apply_t}, @var{solve}, @var{solve_t}, @var{n}, @var{t}) |
| 25 ## | |
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26 ## Estimate the 1-norm condition number of a matrix @var{A} |
| 7189 | 27 ## using @var{t} test vectors using a randomized 1-norm estimator. |
| 28 ## If @var{t} exceeds 5, then only 5 test vectors are used. | |
| 29 ## | |
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30 ## If the matrix is not explicit, e.g., when estimating the condition |
| 7191 | 31 ## number of @var{a} given an LU factorization, @code{condest} uses the |
| 7189 | 32 ## following functions: |
| 33 ## | |
| 34 ## @table @var | |
| 35 ## @item apply | |
| 36 ## @code{A*x} for a matrix @code{x} of size @var{n} by @var{t}. | |
| 37 ## @item apply_t | |
| 38 ## @code{A'*x} for a matrix @code{x} of size @var{n} by @var{t}. | |
| 39 ## @item solve | |
| 40 ## @code{A \ b} for a matrix @code{b} of size @var{n} by @var{t}. | |
| 41 ## @item solve_t | |
| 42 ## @code{A' \ b} for a matrix @code{b} of size @var{n} by @var{t}. | |
| 43 ## @end table | |
| 44 ## | |
| 45 ## The implicit version requires an explicit dimension @var{n}. | |
| 46 ## | |
| 47 ## @code{condest} uses a randomized algorithm to approximate | |
| 48 ## the 1-norms. | |
| 49 ## | |
| 50 ## @code{condest} returns the 1-norm condition estimate @var{est} and | |
| 7191 | 51 ## a vector @var{v} satisfying @code{norm (A*v, 1) == norm (A, 1) * norm |
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52 ## (@var{v}, 1) / @var{est}}. When @var{est} is large, @var{v} is an |
| 7191 | 53 ## approximate null vector. |
| 7189 | 54 ## |
| 55 ## References: | |
| 56 ## @itemize | |
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57 ## @item Nicholas J. Higham and Francoise Tisseur, "A Block Algorithm |
| 7189 | 58 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
| 59 ## Pseudospectra." SIMAX vol 21, no 4, pp 1185-1201. | |
| 60 ## @url{http://dx.doi.org/10.1137/S0895479899356080} | |
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61 ## @item Nicholas J. Higham and Francoise Tisseur, "A Block Algorithm |
| 7189 | 62 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
| 63 ## Pseudospectra." @url{http://citeseer.ist.psu.edu/223007.html} | |
| 64 ## @end itemize | |
| 65 ## | |
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66 ## @seealso{cond, norm, onenormest} |
| 7189 | 67 ## @end deftypefn |
| 68 | |
| 69 ## Code originally licensed under | |
| 70 ## | |
| 71 ## Copyright (c) 2007, Regents of the University of California | |
| 72 ## All rights reserved. | |
| 7191 | 73 ## |
| 7189 | 74 ## Redistribution and use in source and binary forms, with or without |
| 7191 | 75 ## modification, are permitted provided that the following conditions |
| 76 ## are met: | |
| 7189 | 77 ## |
| 78 ## * Redistributions of source code must retain the above copyright | |
| 79 ## notice, this list of conditions and the following disclaimer. | |
| 7191 | 80 ## |
| 81 ## * Redistributions in binary form must reproduce the above | |
| 82 ## copyright notice, this list of conditions and the following | |
| 83 ## disclaimer in the documentation and/or other materials provided | |
| 84 ## with the distribution. | |
| 85 ## | |
| 86 ## * Neither the name of the University of California, Berkeley nor | |
| 87 ## the names of its contributors may be used to endorse or promote | |
| 88 ## products derived from this software without specific prior | |
| 89 ## written permission. | |
| 7189 | 90 ## |
| 7191 | 91 ## THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' |
| 92 ## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED | |
| 93 ## TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A | |
| 94 ## PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS AND | |
| 95 ## CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, | |
| 96 ## SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT | |
| 97 ## LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF | |
| 98 ## USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND | |
| 99 ## ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, | |
| 100 ## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT | |
| 101 ## OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF | |
| 7189 | 102 ## SUCH DAMAGE. |
| 103 | |
| 104 ## Author: Jason Riedy <ejr@cs.berkeley.edu> | |
| 105 ## Keywords: linear-algebra norm estimation | |
| 106 ## Version: 0.2 | |
| 107 | |
| 108 function [est, v] = condest (varargin) | |
| 7191 | 109 |
| 7309 | 110 if (nargin < 1 || nargin > 6) |
| 7191 | 111 print_usage (); |
| 7189 | 112 endif |
| 113 | |
| 114 default_t = 5; | |
| 115 | |
| 7312 | 116 have_A = false; |
| 117 have_t = false; | |
| 118 have_solve = false; | |
| 119 | |
| 7189 | 120 if (ismatrix (varargin{1})) |
| 7310 | 121 A = varargin{1}; |
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122 if (! issquare (A)) |
| 8664 | 123 error ("condest: matrix must be square"); |
| 7189 | 124 endif |
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125 n = rows (A); |
| 7312 | 126 have_A = true; |
| 7189 | 127 |
| 7312 | 128 if (nargin > 1) |
| 129 if (isscalar (varargin{2})) | |
| 10549 | 130 t = varargin{2}; |
| 131 have_t = true; | |
| 7312 | 132 elseif (nargin > 2) |
| 10549 | 133 solve = varargin{2}; |
| 134 solve_t = varargin{3}; | |
| 135 have_solve = true; | |
| 136 if (nargin > 3) | |
| 137 t = varargin{4}; | |
| 138 have_t = true; | |
| 139 endif | |
| 7312 | 140 else |
| 10549 | 141 error ("condest: must supply both solve and solve_t"); |
| 7189 | 142 endif |
| 143 endif | |
| 7311 | 144 elseif (nargin > 4) |
| 7189 | 145 apply = varargin{1}; |
| 146 apply_t = varargin{2}; | |
| 147 solve = varargin{3}; | |
| 148 solve_t = varargin{4}; | |
| 7312 | 149 have_solve = true; |
| 7189 | 150 n = varargin{5}; |
| 151 if (! isscalar (n)) | |
| 8664 | 152 error ("condest: dimension argument of implicit form must be scalar"); |
| 7189 | 153 endif |
| 7309 | 154 if (nargin > 5) |
| 7189 | 155 t = varargin{6}; |
| 7312 | 156 have_t = true; |
| 7189 | 157 endif |
| 7311 | 158 else |
| 8664 | 159 error ("condest: implicit form of condest requires at least 5 arguments"); |
| 7189 | 160 endif |
| 161 | |
| 7312 | 162 if (! have_t) |
| 7189 | 163 t = min (n, default_t); |
| 164 endif | |
| 165 | |
| 7312 | 166 if (! have_solve) |
| 7189 | 167 if (issparse (A)) |
| 8912 | 168 [L, U, P, Pc] = lu (A); |
| 7312 | 169 solve = @(x) Pc' * (U \ (L \ (P * x))); |
| 170 solve_t = @(x) P' * (L' \ (U' \ (Pc * x))); | |
| 7189 | 171 else |
| 172 [L, U, P] = lu (A); | |
| 7312 | 173 solve = @(x) U \ (L \ (P*x)); |
| 174 solve_t = @(x) P' * (L' \ (U' \ x)); | |
| 7189 | 175 endif |
| 176 endif | |
| 177 | |
| 7312 | 178 if (have_A) |
| 7189 | 179 Anorm = norm (A, 1); |
| 180 else | |
| 181 Anorm = onenormest (apply, apply_t, n, t); | |
| 182 endif | |
| 183 | |
| 184 [Ainv_norm, v, w] = onenormest (solve, solve_t, n, t); | |
| 185 | |
| 186 est = Anorm * Ainv_norm; | |
| 187 v = w / norm (w, 1); | |
| 188 | |
| 189 endfunction | |
| 190 | |
| 191 %!demo | |
| 192 %! N = 100; | |
| 193 %! A = randn (N) + eye (N); | |
| 194 %! condest (A) | |
| 195 %! [L,U,P] = lu (A); | |
| 196 %! condest (A, @(x) U\ (L\ (P*x)), @(x) P'*(L'\ (U'\x))) | |
| 197 %! condest (@(x) A*x, @(x) A'*x, @(x) U\ (L\ (P*x)), @(x) P'*(L'\ (U'\x)), N) | |
| 198 %! norm (inv (A), 1) * norm (A, 1) | |
| 199 | |
| 200 ## Yes, these test bounds are really loose. There's | |
| 201 ## enough randomization to trigger odd cases with hilb(). | |
| 202 | |
| 203 %!test | |
| 204 %! N = 6; | |
| 205 %! A = hilb (N); | |
| 206 %! cA = condest (A); | |
| 207 %! cA_test = norm (inv (A), 1) * norm (A, 1); | |
| 8912 | 208 %! assert (cA, cA_test, -2^-8); |
| 7189 | 209 |
| 210 %!test | |
| 211 %! N = 6; | |
| 212 %! A = hilb (N); | |
| 213 %! solve = @(x) A\x; solve_t = @(x) A'\x; | |
| 214 %! cA = condest (A, solve, solve_t); | |
| 215 %! cA_test = norm (inv (A), 1) * norm (A, 1); | |
| 8912 | 216 %! assert (cA, cA_test, -2^-8); |
| 7189 | 217 |
| 218 %!test | |
| 219 %! N = 6; | |
| 220 %! A = hilb (N); | |
| 221 %! apply = @(x) A*x; apply_t = @(x) A'*x; | |
| 222 %! solve = @(x) A\x; solve_t = @(x) A'\x; | |
| 223 %! cA = condest (apply, apply_t, solve, solve_t, N); | |
| 224 %! cA_test = norm (inv (A), 1) * norm (A, 1); | |
| 8912 | 225 %! assert (cA, cA_test, -2^-6); |
| 7189 | 226 |
| 227 %!test | |
| 228 %! N = 12; | |
| 229 %! A = hilb (N); | |
| 230 %! [rcondA, v] = condest (A); | |
| 231 %! x = A*v; | |
| 232 %! assert (norm(x, inf), 0, eps); |