Mercurial > octave
annotate scripts/linear-algebra/condest.m @ 33625:d213a148b3f1 default tip @
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| author | Torsten Lilge <ttl-octave@mailbox.org> |
|---|---|
| date | Sun, 26 May 2024 02:29:44 +0200 |
| parents | c6ef7981b6f1 |
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| rev | line source |
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1 ######################################################################## |
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2 ## |
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3 ## Copyright (C) 2007-2024 The Octave Project Developers |
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4 ## |
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5 ## See the file COPYRIGHT.md in the top-level directory of this |
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6 ## distribution or <https://octave.org/copyright/>. |
| 7189 | 7 ## |
| 8 ## This file is part of Octave. | |
| 9 ## | |
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10 ## Octave is free software: you can redistribute it and/or modify it |
| 7189 | 11 ## under the terms of the GNU General Public License as published by |
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12 ## the Free Software Foundation, either version 3 of the License, or |
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13 ## (at your option) any later version. |
| 7189 | 14 ## |
| 15 ## Octave is distributed in the hope that it will be useful, but | |
| 16 ## WITHOUT ANY WARRANTY; without even the implied warranty of | |
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17 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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18 ## GNU General Public License for more details. |
| 7189 | 19 ## |
| 20 ## You should have received a copy of the GNU General Public License | |
| 21 ## along with Octave; see the file COPYING. If not, see | |
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22 ## <https://www.gnu.org/licenses/>. |
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23 ## |
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24 ######################################################################## |
| 7189 | 25 |
| 26 ## -*- texinfo -*- | |
| 22824 | 27 ## @deftypefn {} {@var{cest} =} condest (@var{A}) |
| 28 ## @deftypefnx {} {@var{cest} =} condest (@var{A}, @var{t}) | |
| 28158 | 29 ## @deftypefnx {} {@var{cest} =} condest (@var{A}, @var{Ainvfcn}) |
| 30 ## @deftypefnx {} {@var{cest} =} condest (@var{A}, @var{Ainvfcn}, @var{t}) | |
| 31 ## @deftypefnx {} {@var{cest} =} condest (@var{A}, @var{Ainvfcn}, @var{t}, @var{p1}, @var{p2}, @dots{}) | |
| 32 ## @deftypefnx {} {@var{cest} =} condest (@var{Afcn}, @var{Ainvfcn}) | |
| 33 ## @deftypefnx {} {@var{cest} =} condest (@var{Afcn}, @var{Ainvfcn}, @var{t}) | |
| 34 ## @deftypefnx {} {@var{cest} =} condest (@var{Afcn}, @var{Ainvfcn}, @var{t}, @var{p1}, @var{p2}, @dots{}) | |
| 22824 | 35 ## @deftypefnx {} {[@var{cest}, @var{v}] =} condest (@dots{}) |
| 7189 | 36 ## |
| 22824 | 37 ## Estimate the 1-norm condition number of a square matrix @var{A} using |
| 38 ## @var{t} test vectors and a randomized 1-norm estimator. | |
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39 ## |
| 22824 | 40 ## The optional input @var{t} specifies the number of test vectors (default 5). |
| 7189 | 41 ## |
| 28158 | 42 ## The input may be a matrix @var{A} (the algorithm is particularly |
| 43 ## appropriate for large, sparse matrices). Alternatively, the behavior of | |
| 44 ## the matrix can be defined implicitly by functions. When using an implicit | |
| 45 ## definition, @code{condest} requires the following functions: | |
| 7189 | 46 ## |
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47 ## @itemize @minus |
| 28158 | 48 ## @item @code{@var{Afcn} (@var{flag}, @var{x})} which must return |
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49 ## |
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50 ## @itemize @bullet |
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51 ## @item |
| 28158 | 52 ## the dimension @var{n} of @var{A}, if @var{flag} is @qcode{"dim"} |
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53 ## |
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54 ## @item |
| 28158 | 55 ## true if @var{A} is a real operator, if @var{flag} is @qcode{"real"} |
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56 ## |
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57 ## @item |
| 28158 | 58 ## the result @code{@var{A} * @var{x}}, if @var{flag} is "notransp" |
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59 ## |
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60 ## @item |
| 28158 | 61 ## the result @code{@var{A}' * @var{x}}, if @var{flag} is "transp" |
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62 ## @end itemize |
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63 ## |
| 28158 | 64 ## @item @code{@var{Ainvfcn} (@var{flag}, @var{x})} which must return |
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65 ## |
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66 ## @itemize @bullet |
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67 ## @item |
| 28158 | 68 ## the dimension @var{n} of @code{inv (@var{A})}, if @var{flag} is |
| 69 ## @qcode{"dim"} | |
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70 ## |
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71 ## @item |
| 28158 | 72 ## true if @code{inv (@var{A})} is a real operator, if @var{flag} is |
| 73 ## @qcode{"real"} | |
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74 ## |
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75 ## @item |
| 28158 | 76 ## the result @code{inv (@var{A}) * @var{x}}, if @var{flag} is "notransp" |
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77 ## |
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78 ## @item |
| 28158 | 79 ## the result @code{inv (@var{A})' * @var{x}}, if @var{flag} is "transp" |
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80 ## @end itemize |
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81 ## @end itemize |
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82 ## |
| 28158 | 83 ## Any parameters @var{p1}, @var{p2}, @dots{} are additional arguments of |
| 22824 | 84 ## @code{@var{Afcn} (@var{flag}, @var{x}, @var{p1}, @var{p2}, @dots{})} |
| 28158 | 85 ## and @code{@var{Ainvfcn} (@var{flag}, @var{x}, @var{p1}, @var{p2}, @dots{})}. |
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86 ## |
| 22824 | 87 ## The principal output is the 1-norm condition number estimate @var{cest}. |
| 7189 | 88 ## |
| 28158 | 89 ## The optional second output @var{v} is an approximate null vector; it |
| 90 ## satisfies the equation @code{norm (@var{A}*@var{v}, 1) == | |
| 91 ## norm (@var{A}, 1) * norm (@var{v}, 1) / @var{cest}}. | |
| 22824 | 92 ## |
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93 ## Algorithm Note: @code{condest} uses a randomized algorithm to approximate |
| 22824 | 94 ## the 1-norms. Therefore, if consistent results are required, the |
| 95 ## @qcode{"state"} of the random generator should be fixed before invoking | |
| 96 ## @code{condest}. | |
| 7189 | 97 ## |
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98 ## References: |
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99 ## |
| 7189 | 100 ## @itemize |
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101 ## @item |
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102 ## @nospell{N.J. Higham and F. Tisseur}, @cite{A Block Algorithm |
| 7189 | 103 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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104 ## Pseudospectra}. SIMAX vol 21, no 4, pp 1185--1201. |
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105 ## @url{https://dx.doi.org/10.1137/S0895479899356080} |
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106 ## |
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107 ## @item |
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108 ## @nospell{N.J. Higham and F. Tisseur}, @cite{A Block Algorithm |
| 7189 | 109 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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110 ## Pseudospectra}. @url{https://citeseer.ist.psu.edu/223007.html} |
| 7189 | 111 ## @end itemize |
| 112 ## | |
| 28158 | 113 ## @seealso{cond, rcond, norm, normest1, normest} |
| 7189 | 114 ## @end deftypefn |
| 115 | |
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116 ## Code originally licensed under: |
| 7189 | 117 ## |
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118 ## Copyright (c) 2007, Regents of the University of California |
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119 ## All rights reserved. |
| 7191 | 120 ## |
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121 ## Redistribution and use in source and binary forms, with or without |
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122 ## modification, are permitted provided that the following conditions |
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123 ## are met: |
| 7189 | 124 ## |
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125 ## * Redistributions of source code must retain the above copyright |
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126 ## notice, this list of conditions and the following disclaimer. |
| 7191 | 127 ## |
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128 ## * Redistributions in binary form must reproduce the above |
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129 ## copyright notice, this list of conditions and the following |
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130 ## disclaimer in the documentation and/or other materials provided |
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131 ## with the distribution. |
| 7191 | 132 ## |
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133 ## * Neither the name of the University of California, Berkeley nor |
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134 ## the names of its contributors may be used to endorse or promote |
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135 ## products derived from this software without specific prior |
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136 ## written permission. |
| 7189 | 137 ## |
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138 ## THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' |
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139 ## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED |
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140 ## TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A |
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141 ## PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS AND |
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142 ## CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, |
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143 ## SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT |
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144 ## LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF |
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145 ## USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND |
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146 ## ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, |
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147 ## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT |
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148 ## OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF |
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149 ## SUCH DAMAGE. |
| 7189 | 150 |
| 22826 | 151 function [cest, v] = condest (varargin) |
| 7191 | 152 |
| 7309 | 153 if (nargin < 1 || nargin > 6) |
| 7191 | 154 print_usage (); |
| 7189 | 155 endif |
| 156 | |
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157 have_A = false; |
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158 have_t = false; |
| 28158 | 159 have_Afcn = false; |
| 160 have_Ainvfcn = false; | |
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161 |
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162 if (isnumeric (varargin{1})) |
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163 A = varargin{1}; |
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164 if (! issquare (A)) |
| 22826 | 165 error ("condest: A must be square"); |
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166 endif |
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167 have_A = true; |
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168 n = rows (A); |
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169 if (nargin > 1) |
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170 if (is_function_handle (varargin{2})) |
| 28158 | 171 Ainvfcn = varargin{2}; |
| 172 have_Ainvfcn = true; | |
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173 if (nargin > 2) |
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174 t = varargin{3}; |
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175 have_t = true; |
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176 endif |
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177 else |
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178 t = varargin{2}; |
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179 have_t = true; |
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180 endif |
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181 endif |
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182 elseif (is_function_handle (varargin{1})) |
| 22826 | 183 if (nargin == 1) |
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184 error ("condest: must provide AINVFCN when using AFCN"); |
| 22826 | 185 endif |
| 28158 | 186 Afcn = varargin{1}; |
| 187 have_Afcn = true; | |
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188 if (! is_function_handle (varargin{2})) |
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189 error ("condest: AINVFCN must be a function handle"); |
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190 endif |
| 28158 | 191 Ainvfcn = varargin{2}; |
| 192 have_Ainvfcn = true; | |
| 193 n = Afcn ("dim", [], varargin{4:end}); | |
| 22826 | 194 if (nargin > 2) |
| 195 t = varargin{3}; | |
| 196 have_t = true; | |
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197 endif |
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198 else |
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199 error ("condest: first argument must be a square matrix or function handle"); |
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200 endif |
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201 |
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202 if (! have_t) |
| 22826 | 203 t = min (n, 5); |
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204 endif |
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205 |
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206 ## Disable warnings which may be emitted during calculation process. |
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207 warning ("off", "Octave:nearly-singular-matrix", "local"); |
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208 |
| 28158 | 209 if (! have_Ainvfcn) |
| 210 ## Prepare Ainvfcn in normest1 form | |
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211 if (issparse (A)) |
| 28158 | 212 [L, U, P, Q] = lu (A); |
| 213 Ainvfcn = @inv_sparse_fcn; | |
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214 else |
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215 [L, U, P] = lu (A); |
| 28158 | 216 Q = []; |
| 217 Ainvfcn = @inv_full_fcn; | |
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218 endif |
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219 |
| 28158 | 220 ## Check for singular matrices before continuing (bug #46737) |
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221 if (any (diag (U) == 0)) |
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222 cest = Inf; |
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223 v = []; |
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224 return; |
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225 endif |
| 28158 | 226 |
| 227 ## Initialize solver | |
| 228 Ainvfcn ("init", A, L, U, P, Q); | |
| 229 clear L U P Q; | |
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230 endif |
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231 |
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232 if (have_A) |
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233 Anorm = norm (A, 1); |
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234 else |
| 28158 | 235 Anorm = normest1 (Afcn, t, [], varargin{4:end}); |
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236 endif |
| 28158 | 237 [Ainv_norm, v, w] = normest1 (Ainvfcn, t, [], varargin{4:end}); |
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238 |
| 22826 | 239 cest = Anorm * Ainv_norm; |
|
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240 if (nargout > 1) |
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241 v = w / norm (w, 1); |
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242 endif |
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243 |
| 28158 | 244 if (! have_Ainvfcn) |
| 245 Ainvfcn ("clear"); # clear persistent memory in subfunction | |
| 246 endif | |
| 247 | |
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248 endfunction |
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249 |
| 28158 | 250 function retval = inv_sparse_fcn (flag, x, varargin) |
|
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251 |
| 28158 | 252 ## FIXME: Sparse algorithm is less accurate than full matrix version. |
| 253 ## See BIST test for asymmetric matrix where relative tolerance | |
|
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254 ## of 1e-12 is used for sparse, but 4e-16 for full matrix. |
| 28158 | 255 ## BUT, does it really matter for an "estimate"? |
| 256 persistent Ainv Ainvt n isreal_op; | |
| 257 | |
|
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258 switch (flag) |
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259 case "dim" |
| 28158 | 260 retval = n; |
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261 case "real" |
| 28158 | 262 retval = isreal_op; |
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263 case "notransp" |
| 28158 | 264 retval = Ainv * x; |
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265 case "transp" |
| 28158 | 266 retval = Ainvt * x; |
| 267 case "init" | |
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268 n = rows (x); |
| 28158 | 269 isreal_op = isreal (x); |
| 270 [L, U, P, Q] = deal (varargin{1:4}); | |
| 271 Ainv = Q * (U \ (L \ P)); | |
| 272 Ainvt = P' * (L' \ (U' \ Q')); | |
| 273 case "clear" # called to free memory at end of condest function | |
| 274 clear Ainv Ainvt n isreal_op; | |
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275 endswitch |
| 28158 | 276 |
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277 endfunction |
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278 |
| 28158 | 279 function retval = inv_full_fcn (flag, x, varargin) |
| 280 persistent Ainv Ainvt n isreal_op; | |
| 281 | |
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282 switch (flag) |
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283 case "dim" |
| 28158 | 284 retval = n; |
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285 case "real" |
| 28158 | 286 retval = isreal_op; |
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287 case "notransp" |
| 28158 | 288 retval = Ainv * x; |
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289 case "transp" |
| 28158 | 290 retval = Ainvt \ x; |
| 291 case "init" | |
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292 n = rows (x); |
| 28158 | 293 isreal_op = isreal (x); |
| 294 [L, U, P] = deal (varargin{1:3}); | |
| 295 Ainv = U \ (L \ P); | |
| 296 Ainvt = P' * (L' \ U'); | |
| 297 case "clear" # called to free memory at end of condest function | |
| 298 clear Ainv Ainvt n isreal_op; | |
|
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299 endswitch |
| 28158 | 300 |
|
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301 endfunction |
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302 |
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303 |
| 22826 | 304 ## Note: These test bounds are very loose. There is enough randomization to |
| 305 ## trigger odd cases with hilb(). | |
| 7189 | 306 |
| 28158 | 307 %!function retval = __Afcn__ (flag, x, A, m) |
|
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308 %! if (nargin == 3) |
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309 %! m = 1; |
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310 %! endif |
| 22826 | 311 %! switch (flag) |
|
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312 %! case "dim" |
| 28158 | 313 %! retval = length (A); |
|
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314 %! case "real" |
| 28158 | 315 %! retval = isreal (A); |
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316 %! case "notransp" |
| 28158 | 317 %! retval = x; for i = 1:m, retval = A * retval;, endfor |
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318 %! case "transp" |
| 28158 | 319 %! retval = x; for i = 1:m, retval = A' * retval;, endfor |
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320 %! endswitch |
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321 %!endfunction |
| 28158 | 322 %!function retval = __Ainvfcn__ (flag, x, A, m) |
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323 %! if (nargin == 3) |
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324 %! m = 1; |
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325 %! endif |
| 22826 | 326 %! switch (flag) |
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327 %! case "dim" |
| 28158 | 328 %! retval = length (A); |
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329 %! case "real" |
| 28158 | 330 %! retval = isreal (A); |
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331 %! case "notransp" |
| 28158 | 332 %! retval = x; for i = 1:m, retval = A \ retval;, endfor |
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333 %! case "transp" |
| 28158 | 334 %! retval = x; for i = 1:m, retval = A' \ retval;, endfor |
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335 %! endswitch |
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336 %!endfunction |
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337 |
| 7189 | 338 %!test |
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339 %! N = 6; |
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340 %! A = hilb (N); |
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341 %! cA = condest (A); |
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342 %! cA_test = norm (inv (A), 1) * norm (A, 1); |
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343 %! assert (cA, cA_test, -2^-8); |
| 7189 | 344 |
| 345 %!test | |
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346 %! N = 12; |
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347 %! A = hilb (N); |
| 22826 | 348 %! [~, v] = condest (A); |
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349 %! x = A*v; |
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350 %! assert (norm (x, inf), 0, eps); |
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351 |
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352 %!test |
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353 %! N = 6; |
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354 %! A = hilb (N); |
| 28158 | 355 %! Ainvfcn = @(flag, x) __Ainvfcn__ (flag, x, A); |
| 356 %! cA = condest (A, Ainvfcn); | |
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357 %! cA_test = norm (inv (A), 1) * norm (A, 1); |
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358 %! assert (cA, cA_test, -2^-6); |
| 22826 | 359 |
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360 %!test |
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361 %! N = 6; |
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362 %! A = hilb (N); |
| 28158 | 363 %! Afcn = @(flag, x) __Afcn__ (flag, x, A); |
| 364 %! Ainvfcn = @(flag, x) __Ainvfcn__ (flag, x, A); | |
| 365 %! cA = condest (Afcn, Ainvfcn); | |
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366 %! cA_test = norm (inv (A), 1) * norm (A, 1); |
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367 %! assert (cA, cA_test, -2^-6); |
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368 |
| 28158 | 369 %!test # parameters for apply and Ainvfcn functions |
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370 %! N = 6; |
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371 %! A = hilb (N); |
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372 %! m = 2; |
| 28158 | 373 %! cA = condest (@__Afcn__, @__Ainvfcn__, [], A, m); |
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374 %! cA_test = norm (inv (A^2), 1) * norm (A^2, 1); |
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375 %! assert (cA, cA_test, -2^-6); |
| 22826 | 376 |
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377 ## Test singular matrices |
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378 %!test <*46737> |
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379 %! A = [ 0 0 0 |
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380 %! 0 3.33333 0.0833333 |
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381 %! 0 0.0833333 1.66667]; |
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382 %! [cest, v] = condest (A); |
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383 %! assert (cest, Inf); |
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384 %! assert (v, []); |
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385 |
| 28158 | 386 ## Test asymmetric matrices |
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387 %!test <*57968> |
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388 %! A = reshape (sqrt (0:15), 4, 4); |
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389 %! cexp = norm (A, 1) * norm (inv (A), 1); |
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390 %! cest = condest (A); |
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391 %! assert (cest, cexp, -2*eps); |
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392 |
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393 %!testif HAVE_UMFPACK <*57968> |
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394 %! As = sparse (reshape (sqrt (0:15), 4, 4)); |
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395 %! cexp = norm (As, 1) * norm (inv (As), 1); |
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396 %! cest = condest (As); |
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397 %! assert (cest, cexp, -1e-12); |
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398 |
| 22826 | 399 ## Test input validation |
| 28158 | 400 %!error <Invalid call> condest () |
| 401 %!error <Invalid call> condest (1,2,3,4,5,6,7) | |
| 402 %!error <A must be square> condest ([1, 2]) | |
| 403 %!error <must provide AINVFCN when using AFCN> condest (@sin) | |
| 404 %!error <AINVFCN must be a function handle> condest (@sin, 1) | |
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405 %!error <argument must be a square matrix or function handle> condest ({1}) |