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annotate scripts/linear-algebra/condest.m @ 19840:c5270263d466 gui-release
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| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Fri, 30 Jan 2015 17:41:50 -0500 |
| parents | d63878346099 |
| children | 0850b5212619 |
| rev | line source |
|---|---|
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1 ## Copyright (C) 2007-2013 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{}) |
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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 |
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31 ## number of @var{A} given an LU@tie{}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}. | |
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37 ## |
| 7189 | 38 ## @item apply_t |
| 39 ## @code{A'*x} for a matrix @code{x} of size @var{n} by @var{t}. | |
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40 ## |
| 7189 | 41 ## @item solve |
| 42 ## @code{A \ b} for a matrix @code{b} of size @var{n} by @var{t}. | |
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43 ## |
| 7189 | 44 ## @item solve_t |
| 45 ## @code{A' \ b} for a matrix @code{b} of size @var{n} by @var{t}. | |
| 46 ## @end table | |
| 47 ## | |
| 48 ## The implicit version requires an explicit dimension @var{n}. | |
| 49 ## | |
| 50 ## @code{condest} uses a randomized algorithm to approximate | |
| 51 ## the 1-norms. | |
| 52 ## | |
| 53 ## @code{condest} returns the 1-norm condition estimate @var{est} and | |
| 7191 | 54 ## a vector @var{v} satisfying @code{norm (A*v, 1) == norm (A, 1) * norm |
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55 ## (@var{v}, 1) / @var{est}}. When @var{est} is large, @var{v} is an |
| 7191 | 56 ## approximate null vector. |
| 7189 | 57 ## |
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58 ## References: |
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59 ## |
| 7189 | 60 ## @itemize |
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61 ## @item |
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62 ## N.J. Higham and F. Tisseur, @cite{A Block Algorithm |
| 7189 | 63 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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64 ## Pseudospectra}. SIMAX vol 21, no 4, pp 1185-1201. |
| 7189 | 65 ## @url{http://dx.doi.org/10.1137/S0895479899356080} |
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66 ## |
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67 ## @item |
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68 ## N.J. Higham and F. Tisseur, @cite{A Block Algorithm |
| 7189 | 69 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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70 ## Pseudospectra}. @url{http://citeseer.ist.psu.edu/223007.html} |
| 7189 | 71 ## @end itemize |
| 72 ## | |
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73 ## @seealso{cond, norm, onenormest} |
| 7189 | 74 ## @end deftypefn |
| 75 | |
| 76 ## Code originally licensed under | |
| 77 ## | |
| 78 ## Copyright (c) 2007, Regents of the University of California | |
| 79 ## All rights reserved. | |
| 7191 | 80 ## |
| 7189 | 81 ## Redistribution and use in source and binary forms, with or without |
| 7191 | 82 ## modification, are permitted provided that the following conditions |
| 83 ## are met: | |
| 7189 | 84 ## |
| 85 ## * Redistributions of source code must retain the above copyright | |
| 86 ## notice, this list of conditions and the following disclaimer. | |
| 7191 | 87 ## |
| 88 ## * Redistributions in binary form must reproduce the above | |
| 89 ## copyright notice, this list of conditions and the following | |
| 90 ## disclaimer in the documentation and/or other materials provided | |
| 91 ## with the distribution. | |
| 92 ## | |
| 93 ## * Neither the name of the University of California, Berkeley nor | |
| 94 ## the names of its contributors may be used to endorse or promote | |
| 95 ## products derived from this software without specific prior | |
| 96 ## written permission. | |
| 7189 | 97 ## |
| 7191 | 98 ## THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' |
| 99 ## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED | |
| 100 ## TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A | |
| 101 ## PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS AND | |
| 102 ## CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, | |
| 103 ## SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT | |
| 104 ## LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF | |
| 105 ## USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND | |
| 106 ## ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, | |
| 107 ## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT | |
| 108 ## OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF | |
| 7189 | 109 ## SUCH DAMAGE. |
| 110 | |
| 111 ## Author: Jason Riedy <ejr@cs.berkeley.edu> | |
| 112 ## Keywords: linear-algebra norm estimation | |
| 113 ## Version: 0.2 | |
| 114 | |
| 115 function [est, v] = condest (varargin) | |
| 7191 | 116 |
| 7309 | 117 if (nargin < 1 || nargin > 6) |
| 7191 | 118 print_usage (); |
| 7189 | 119 endif |
| 120 | |
| 121 default_t = 5; | |
| 122 | |
| 7312 | 123 have_A = false; |
| 124 have_t = false; | |
| 125 have_solve = false; | |
| 126 | |
| 7189 | 127 if (ismatrix (varargin{1})) |
| 7310 | 128 A = varargin{1}; |
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129 if (! issquare (A)) |
| 8664 | 130 error ("condest: matrix must be square"); |
| 7189 | 131 endif |
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132 n = rows (A); |
| 7312 | 133 have_A = true; |
| 7189 | 134 |
| 7312 | 135 if (nargin > 1) |
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136 if (! is_function_handle (varargin{2})) |
| 10549 | 137 t = varargin{2}; |
| 138 have_t = true; | |
| 7312 | 139 elseif (nargin > 2) |
| 10549 | 140 solve = varargin{2}; |
| 141 solve_t = varargin{3}; | |
| 142 have_solve = true; | |
| 143 if (nargin > 3) | |
| 144 t = varargin{4}; | |
| 145 have_t = true; | |
| 146 endif | |
| 7312 | 147 else |
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148 error ("condest: must supply both SOLVE and SOLVE_T"); |
| 7189 | 149 endif |
| 150 endif | |
| 7311 | 151 elseif (nargin > 4) |
| 7189 | 152 apply = varargin{1}; |
| 153 apply_t = varargin{2}; | |
| 154 solve = varargin{3}; | |
| 155 solve_t = varargin{4}; | |
| 7312 | 156 have_solve = true; |
| 7189 | 157 n = varargin{5}; |
| 158 if (! isscalar (n)) | |
| 8664 | 159 error ("condest: dimension argument of implicit form must be scalar"); |
| 7189 | 160 endif |
| 7309 | 161 if (nargin > 5) |
| 7189 | 162 t = varargin{6}; |
| 7312 | 163 have_t = true; |
| 7189 | 164 endif |
| 7311 | 165 else |
| 8664 | 166 error ("condest: implicit form of condest requires at least 5 arguments"); |
| 7189 | 167 endif |
| 168 | |
| 7312 | 169 if (! have_t) |
| 7189 | 170 t = min (n, default_t); |
| 171 endif | |
| 172 | |
| 7312 | 173 if (! have_solve) |
| 7189 | 174 if (issparse (A)) |
| 8912 | 175 [L, U, P, Pc] = lu (A); |
| 7312 | 176 solve = @(x) Pc' * (U \ (L \ (P * x))); |
| 177 solve_t = @(x) P' * (L' \ (U' \ (Pc * x))); | |
| 7189 | 178 else |
| 179 [L, U, P] = lu (A); | |
| 7312 | 180 solve = @(x) U \ (L \ (P*x)); |
| 181 solve_t = @(x) P' * (L' \ (U' \ x)); | |
| 7189 | 182 endif |
| 183 endif | |
| 184 | |
| 7312 | 185 if (have_A) |
| 7189 | 186 Anorm = norm (A, 1); |
| 187 else | |
| 188 Anorm = onenormest (apply, apply_t, n, t); | |
| 189 endif | |
| 190 | |
| 191 [Ainv_norm, v, w] = onenormest (solve, solve_t, n, t); | |
| 192 | |
| 193 est = Anorm * Ainv_norm; | |
| 194 v = w / norm (w, 1); | |
| 195 | |
| 196 endfunction | |
| 197 | |
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198 |
| 7189 | 199 %!demo |
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200 %! N = 100; |
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201 %! A = randn (N) + eye (N); |
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202 %! condest (A) |
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203 %! [L,U,P] = lu (A); |
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204 %! condest (A, @(x) U \ (L \ (P*x)), @(x) P'*(L' \ (U'\x))) |
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205 %! condest (@(x) A*x, @(x) A'*x, @(x) U \ (L \ (P*x)), @(x) P'*(L' \ (U'\x)), N) |
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206 %! norm (inv (A), 1) * norm (A, 1) |
| 7189 | 207 |
| 208 ## Yes, these test bounds are really loose. There's | |
| 209 ## enough randomization to trigger odd cases with hilb(). | |
| 210 | |
| 211 %!test | |
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212 %! N = 6; |
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213 %! A = hilb (N); |
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214 %! cA = condest (A); |
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215 %! cA_test = norm (inv (A), 1) * norm (A, 1); |
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216 %! assert (cA, cA_test, -2^-8); |
| 7189 | 217 |
| 218 %!test | |
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219 %! N = 6; |
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220 %! A = hilb (N); |
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221 %! solve = @(x) A\x; solve_t = @(x) A'\x; |
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222 %! cA = condest (A, solve, solve_t); |
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223 %! cA_test = norm (inv (A), 1) * norm (A, 1); |
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224 %! assert (cA, cA_test, -2^-8); |
| 7189 | 225 |
| 226 %!test | |
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227 %! N = 6; |
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228 %! A = hilb (N); |
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229 %! apply = @(x) A*x; apply_t = @(x) A'*x; |
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230 %! solve = @(x) A\x; solve_t = @(x) A'\x; |
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231 %! cA = condest (apply, apply_t, solve, solve_t, N); |
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232 %! cA_test = norm (inv (A), 1) * norm (A, 1); |
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233 %! assert (cA, cA_test, -2^-6); |
| 7189 | 234 |
| 235 %!test | |
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236 %! N = 12; |
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237 %! A = hilb (N); |
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238 %! [rcondA, v] = condest (A); |
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239 %! x = A*v; |
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240 %! assert (norm (x, inf), 0, eps); |
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241 |