Mercurial > hg > octave-nkf
annotate scripts/linear-algebra/condest.m @ 10793:be55736a0783
Grammarcheck the documentation from m-files.
| author | Rik <octave@nomad.inbox5.com> |
|---|---|
| date | Sun, 18 Jul 2010 20:35:16 -0700 |
| parents | 3140cb7a05a1 |
| children | 693e22af08ae |
| 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 |
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58 ## N.J. Higham and F. Tisseur, @cite{A Block Algorithm |
| 7189 | 59 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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60 ## Pseudospectra}. SIMAX vol 21, no 4, pp 1185-1201. |
| 7189 | 61 ## @url{http://dx.doi.org/10.1137/S0895479899356080} |
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62 ## @item |
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63 ## N.J. Higham and F. Tisseur, @cite{A Block Algorithm |
| 7189 | 64 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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65 ## Pseudospectra}. @url{http://citeseer.ist.psu.edu/223007.html} |
| 7189 | 66 ## @end itemize |
| 67 ## | |
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68 ## @seealso{cond, norm, onenormest} |
| 7189 | 69 ## @end deftypefn |
| 70 | |
| 71 ## Code originally licensed under | |
| 72 ## | |
| 73 ## Copyright (c) 2007, Regents of the University of California | |
| 74 ## All rights reserved. | |
| 7191 | 75 ## |
| 7189 | 76 ## Redistribution and use in source and binary forms, with or without |
| 7191 | 77 ## modification, are permitted provided that the following conditions |
| 78 ## are met: | |
| 7189 | 79 ## |
| 80 ## * Redistributions of source code must retain the above copyright | |
| 81 ## notice, this list of conditions and the following disclaimer. | |
| 7191 | 82 ## |
| 83 ## * Redistributions in binary form must reproduce the above | |
| 84 ## copyright notice, this list of conditions and the following | |
| 85 ## disclaimer in the documentation and/or other materials provided | |
| 86 ## with the distribution. | |
| 87 ## | |
| 88 ## * Neither the name of the University of California, Berkeley nor | |
| 89 ## the names of its contributors may be used to endorse or promote | |
| 90 ## products derived from this software without specific prior | |
| 91 ## written permission. | |
| 7189 | 92 ## |
| 7191 | 93 ## THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' |
| 94 ## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED | |
| 95 ## TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A | |
| 96 ## PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS AND | |
| 97 ## CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, | |
| 98 ## SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT | |
| 99 ## LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF | |
| 100 ## USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND | |
| 101 ## ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, | |
| 102 ## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT | |
| 103 ## OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF | |
| 7189 | 104 ## SUCH DAMAGE. |
| 105 | |
| 106 ## Author: Jason Riedy <ejr@cs.berkeley.edu> | |
| 107 ## Keywords: linear-algebra norm estimation | |
| 108 ## Version: 0.2 | |
| 109 | |
| 110 function [est, v] = condest (varargin) | |
| 7191 | 111 |
| 7309 | 112 if (nargin < 1 || nargin > 6) |
| 7191 | 113 print_usage (); |
| 7189 | 114 endif |
| 115 | |
| 116 default_t = 5; | |
| 117 | |
| 7312 | 118 have_A = false; |
| 119 have_t = false; | |
| 120 have_solve = false; | |
| 121 | |
| 7189 | 122 if (ismatrix (varargin{1})) |
| 7310 | 123 A = varargin{1}; |
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124 if (! issquare (A)) |
| 8664 | 125 error ("condest: matrix must be square"); |
| 7189 | 126 endif |
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127 n = rows (A); |
| 7312 | 128 have_A = true; |
| 7189 | 129 |
| 7312 | 130 if (nargin > 1) |
| 131 if (isscalar (varargin{2})) | |
| 10549 | 132 t = varargin{2}; |
| 133 have_t = true; | |
| 7312 | 134 elseif (nargin > 2) |
| 10549 | 135 solve = varargin{2}; |
| 136 solve_t = varargin{3}; | |
| 137 have_solve = true; | |
| 138 if (nargin > 3) | |
| 139 t = varargin{4}; | |
| 140 have_t = true; | |
| 141 endif | |
| 7312 | 142 else |
| 10549 | 143 error ("condest: must supply both solve and solve_t"); |
| 7189 | 144 endif |
| 145 endif | |
| 7311 | 146 elseif (nargin > 4) |
| 7189 | 147 apply = varargin{1}; |
| 148 apply_t = varargin{2}; | |
| 149 solve = varargin{3}; | |
| 150 solve_t = varargin{4}; | |
| 7312 | 151 have_solve = true; |
| 7189 | 152 n = varargin{5}; |
| 153 if (! isscalar (n)) | |
| 8664 | 154 error ("condest: dimension argument of implicit form must be scalar"); |
| 7189 | 155 endif |
| 7309 | 156 if (nargin > 5) |
| 7189 | 157 t = varargin{6}; |
| 7312 | 158 have_t = true; |
| 7189 | 159 endif |
| 7311 | 160 else |
| 8664 | 161 error ("condest: implicit form of condest requires at least 5 arguments"); |
| 7189 | 162 endif |
| 163 | |
| 7312 | 164 if (! have_t) |
| 7189 | 165 t = min (n, default_t); |
| 166 endif | |
| 167 | |
| 7312 | 168 if (! have_solve) |
| 7189 | 169 if (issparse (A)) |
| 8912 | 170 [L, U, P, Pc] = lu (A); |
| 7312 | 171 solve = @(x) Pc' * (U \ (L \ (P * x))); |
| 172 solve_t = @(x) P' * (L' \ (U' \ (Pc * x))); | |
| 7189 | 173 else |
| 174 [L, U, P] = lu (A); | |
| 7312 | 175 solve = @(x) U \ (L \ (P*x)); |
| 176 solve_t = @(x) P' * (L' \ (U' \ x)); | |
| 7189 | 177 endif |
| 178 endif | |
| 179 | |
| 7312 | 180 if (have_A) |
| 7189 | 181 Anorm = norm (A, 1); |
| 182 else | |
| 183 Anorm = onenormest (apply, apply_t, n, t); | |
| 184 endif | |
| 185 | |
| 186 [Ainv_norm, v, w] = onenormest (solve, solve_t, n, t); | |
| 187 | |
| 188 est = Anorm * Ainv_norm; | |
| 189 v = w / norm (w, 1); | |
| 190 | |
| 191 endfunction | |
| 192 | |
| 193 %!demo | |
| 194 %! N = 100; | |
| 195 %! A = randn (N) + eye (N); | |
| 196 %! condest (A) | |
| 197 %! [L,U,P] = lu (A); | |
| 198 %! condest (A, @(x) U\ (L\ (P*x)), @(x) P'*(L'\ (U'\x))) | |
| 199 %! condest (@(x) A*x, @(x) A'*x, @(x) U\ (L\ (P*x)), @(x) P'*(L'\ (U'\x)), N) | |
| 200 %! norm (inv (A), 1) * norm (A, 1) | |
| 201 | |
| 202 ## Yes, these test bounds are really loose. There's | |
| 203 ## enough randomization to trigger odd cases with hilb(). | |
| 204 | |
| 205 %!test | |
| 206 %! N = 6; | |
| 207 %! A = hilb (N); | |
| 208 %! cA = condest (A); | |
| 209 %! cA_test = norm (inv (A), 1) * norm (A, 1); | |
| 8912 | 210 %! assert (cA, cA_test, -2^-8); |
| 7189 | 211 |
| 212 %!test | |
| 213 %! N = 6; | |
| 214 %! A = hilb (N); | |
| 215 %! solve = @(x) A\x; solve_t = @(x) A'\x; | |
| 216 %! cA = condest (A, solve, solve_t); | |
| 217 %! cA_test = norm (inv (A), 1) * norm (A, 1); | |
| 8912 | 218 %! assert (cA, cA_test, -2^-8); |
| 7189 | 219 |
| 220 %!test | |
| 221 %! N = 6; | |
| 222 %! A = hilb (N); | |
| 223 %! apply = @(x) A*x; apply_t = @(x) A'*x; | |
| 224 %! solve = @(x) A\x; solve_t = @(x) A'\x; | |
| 225 %! cA = condest (apply, apply_t, solve, solve_t, N); | |
| 226 %! cA_test = norm (inv (A), 1) * norm (A, 1); | |
| 8912 | 227 %! assert (cA, cA_test, -2^-6); |
| 7189 | 228 |
| 229 %!test | |
| 230 %! N = 12; | |
| 231 %! A = hilb (N); | |
| 232 %! [rcondA, v] = condest (A); | |
| 233 %! x = A*v; | |
| 234 %! assert (norm(x, inf), 0, eps); |