Mercurial > hg > octave-lyh
annotate scripts/linear-algebra/condest.m @ 16466:ac332eb727dd
Simplify calculation of colormaps
* scripts/image/copper.m: Replace slower linspace with range operator.
* scripts/image/hot.m: Add test for n==2 to allow elimination of subsequent
if conditional tests. Replace mod calculation with simpler expression.
* scripts/image/rainbow.m: Replace slower linspace with range operator.
Align columns of calculation for better code readability.
| author | Rik <rik@octave.org> |
|---|---|
| date | Sun, 07 Apr 2013 22:24:34 -0700 |
| parents | f739e30494c8 |
| children | 12005245b645 |
| rev | line source |
|---|---|
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1 ## Copyright (C) 2007-2012 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: |
| 7189 | 59 ## @itemize |
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60 ## @item |
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61 ## N.J. Higham and F. Tisseur, @cite{A Block Algorithm |
| 7189 | 62 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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63 ## Pseudospectra}. SIMAX vol 21, no 4, pp 1185-1201. |
| 7189 | 64 ## @url{http://dx.doi.org/10.1137/S0895479899356080} |
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65 ## |
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66 ## @item |
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67 ## N.J. Higham and F. Tisseur, @cite{A Block Algorithm |
| 7189 | 68 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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69 ## Pseudospectra}. @url{http://citeseer.ist.psu.edu/223007.html} |
| 7189 | 70 ## @end itemize |
| 71 ## | |
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72 ## @seealso{cond, norm, onenormest} |
| 7189 | 73 ## @end deftypefn |
| 74 | |
| 75 ## Code originally licensed under | |
| 76 ## | |
| 77 ## Copyright (c) 2007, Regents of the University of California | |
| 78 ## All rights reserved. | |
| 7191 | 79 ## |
| 7189 | 80 ## Redistribution and use in source and binary forms, with or without |
| 7191 | 81 ## modification, are permitted provided that the following conditions |
| 82 ## are met: | |
| 7189 | 83 ## |
| 84 ## * Redistributions of source code must retain the above copyright | |
| 85 ## notice, this list of conditions and the following disclaimer. | |
| 7191 | 86 ## |
| 87 ## * Redistributions in binary form must reproduce the above | |
| 88 ## copyright notice, this list of conditions and the following | |
| 89 ## disclaimer in the documentation and/or other materials provided | |
| 90 ## with the distribution. | |
| 91 ## | |
| 92 ## * Neither the name of the University of California, Berkeley nor | |
| 93 ## the names of its contributors may be used to endorse or promote | |
| 94 ## products derived from this software without specific prior | |
| 95 ## written permission. | |
| 7189 | 96 ## |
| 7191 | 97 ## THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' |
| 98 ## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED | |
| 99 ## TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A | |
| 100 ## PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS AND | |
| 101 ## CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, | |
| 102 ## SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT | |
| 103 ## LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF | |
| 104 ## USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND | |
| 105 ## ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, | |
| 106 ## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT | |
| 107 ## OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF | |
| 7189 | 108 ## SUCH DAMAGE. |
| 109 | |
| 110 ## Author: Jason Riedy <ejr@cs.berkeley.edu> | |
| 111 ## Keywords: linear-algebra norm estimation | |
| 112 ## Version: 0.2 | |
| 113 | |
| 114 function [est, v] = condest (varargin) | |
| 7191 | 115 |
| 7309 | 116 if (nargin < 1 || nargin > 6) |
| 7191 | 117 print_usage (); |
| 7189 | 118 endif |
| 119 | |
| 120 default_t = 5; | |
| 121 | |
| 7312 | 122 have_A = false; |
| 123 have_t = false; | |
| 124 have_solve = false; | |
| 125 | |
| 7189 | 126 if (ismatrix (varargin{1})) |
| 7310 | 127 A = varargin{1}; |
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128 if (! issquare (A)) |
| 8664 | 129 error ("condest: matrix must be square"); |
| 7189 | 130 endif |
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131 n = rows (A); |
| 7312 | 132 have_A = true; |
| 7189 | 133 |
| 7312 | 134 if (nargin > 1) |
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135 if (! is_function_handle (varargin{2})) |
| 10549 | 136 t = varargin{2}; |
| 137 have_t = true; | |
| 7312 | 138 elseif (nargin > 2) |
| 10549 | 139 solve = varargin{2}; |
| 140 solve_t = varargin{3}; | |
| 141 have_solve = true; | |
| 142 if (nargin > 3) | |
| 143 t = varargin{4}; | |
| 144 have_t = true; | |
| 145 endif | |
| 7312 | 146 else |
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147 error ("condest: must supply both SOLVE and SOLVE_T"); |
| 7189 | 148 endif |
| 149 endif | |
| 7311 | 150 elseif (nargin > 4) |
| 7189 | 151 apply = varargin{1}; |
| 152 apply_t = varargin{2}; | |
| 153 solve = varargin{3}; | |
| 154 solve_t = varargin{4}; | |
| 7312 | 155 have_solve = true; |
| 7189 | 156 n = varargin{5}; |
| 157 if (! isscalar (n)) | |
| 8664 | 158 error ("condest: dimension argument of implicit form must be scalar"); |
| 7189 | 159 endif |
| 7309 | 160 if (nargin > 5) |
| 7189 | 161 t = varargin{6}; |
| 7312 | 162 have_t = true; |
| 7189 | 163 endif |
| 7311 | 164 else |
| 8664 | 165 error ("condest: implicit form of condest requires at least 5 arguments"); |
| 7189 | 166 endif |
| 167 | |
| 7312 | 168 if (! have_t) |
| 7189 | 169 t = min (n, default_t); |
| 170 endif | |
| 171 | |
| 7312 | 172 if (! have_solve) |
| 7189 | 173 if (issparse (A)) |
| 8912 | 174 [L, U, P, Pc] = lu (A); |
| 7312 | 175 solve = @(x) Pc' * (U \ (L \ (P * x))); |
| 176 solve_t = @(x) P' * (L' \ (U' \ (Pc * x))); | |
| 7189 | 177 else |
| 178 [L, U, P] = lu (A); | |
| 7312 | 179 solve = @(x) U \ (L \ (P*x)); |
| 180 solve_t = @(x) P' * (L' \ (U' \ x)); | |
| 7189 | 181 endif |
| 182 endif | |
| 183 | |
| 7312 | 184 if (have_A) |
| 7189 | 185 Anorm = norm (A, 1); |
| 186 else | |
| 187 Anorm = onenormest (apply, apply_t, n, t); | |
| 188 endif | |
| 189 | |
| 190 [Ainv_norm, v, w] = onenormest (solve, solve_t, n, t); | |
| 191 | |
| 192 est = Anorm * Ainv_norm; | |
| 193 v = w / norm (w, 1); | |
| 194 | |
| 195 endfunction | |
| 196 | |
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197 |
| 7189 | 198 %!demo |
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199 %! N = 100; |
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200 %! A = randn (N) + eye (N); |
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201 %! condest (A) |
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202 %! [L,U,P] = lu (A); |
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203 %! condest (A, @(x) U \ (L \ (P*x)), @(x) P'*(L' \ (U'\x))) |
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204 %! condest (@(x) A*x, @(x) A'*x, @(x) U \ (L \ (P*x)), @(x) P'*(L' \ (U'\x)), N) |
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205 %! norm (inv (A), 1) * norm (A, 1) |
| 7189 | 206 |
| 207 ## Yes, these test bounds are really loose. There's | |
| 208 ## enough randomization to trigger odd cases with hilb(). | |
| 209 | |
| 210 %!test | |
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211 %! N = 6; |
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212 %! A = hilb (N); |
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213 %! cA = condest (A); |
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214 %! cA_test = norm (inv (A), 1) * norm (A, 1); |
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215 %! assert (cA, cA_test, -2^-8); |
| 7189 | 216 |
| 217 %!test | |
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218 %! N = 6; |
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219 %! A = hilb (N); |
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220 %! solve = @(x) A\x; solve_t = @(x) A'\x; |
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221 %! cA = condest (A, solve, solve_t); |
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222 %! cA_test = norm (inv (A), 1) * norm (A, 1); |
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223 %! assert (cA, cA_test, -2^-8); |
| 7189 | 224 |
| 225 %!test | |
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226 %! N = 6; |
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227 %! A = hilb (N); |
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228 %! apply = @(x) A*x; apply_t = @(x) A'*x; |
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229 %! solve = @(x) A\x; solve_t = @(x) A'\x; |
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230 %! cA = condest (apply, apply_t, solve, solve_t, N); |
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231 %! cA_test = norm (inv (A), 1) * norm (A, 1); |
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232 %! assert (cA, cA_test, -2^-6); |
| 7189 | 233 |
| 234 %!test | |
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235 %! N = 12; |
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236 %! A = hilb (N); |
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237 %! [rcondA, v] = condest (A); |
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238 %! x = A*v; |
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239 %! assert (norm (x, inf), 0, eps); |
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240 |