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1 ## Copyright (C) 2007, Regents of the University of California |
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2 ## |
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3 ## This file is part of Octave. |
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4 ## |
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5 ## Octave is free software; you can redistribute it and/or modify it |
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6 ## under the terms of the GNU General Public License as published by |
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7 ## the Free Software Foundation; either version 3 of the License, or (at |
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8 ## your option) any later version. |
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9 ## |
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10 ## Octave is distributed in the hope that it will be useful, but |
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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13 ## General Public License for more details. |
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14 ## |
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15 ## You should have received a copy of the GNU General Public License |
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16 ## along with Octave; see the file COPYING. If not, see |
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17 ## <http://www.gnu.org/licenses/>. |
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18 |
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19 ## -*- texinfo -*- |
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20 ## @deftypefn {Function File} {[@var{est}, @var{v}] =} condest (@var{a}, @var{t}) |
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21 ## @deftypefnx {Function File} {[@var{est}, @var{v}] =} condest (@var{a}, @var{solve}, @var{solve_t}, @var{t}) |
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22 ## @deftypefnx {Function File} {[@var{est}, @var{v}] =} condest (@var{apply}, @var{apply_t}, @var{solve}, @var{solve_t}, @var{n}, @var{t}) |
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23 ## |
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24 ## Estimate the 1-norm condition number of a matrix matrix @var{A} |
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25 ## using @var{t} test vectors using a randomized 1-norm estimator. |
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26 ## If @var{t} exceeds 5, then only 5 test vectors are used. |
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27 ## |
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28 ## If the matrix is not explicit, e.g. when estimating the condition |
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29 ## number of @var{a} given an LU factorization, @code{condest} uses the |
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30 ## following functions: |
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31 ## |
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32 ## @table @var |
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33 ## @item apply |
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34 ## @code{A*x} for a matrix @code{x} of size @var{n} by @var{t}. |
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35 ## @item apply_t |
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36 ## @code{A'*x} for a matrix @code{x} of size @var{n} by @var{t}. |
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37 ## @item solve |
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38 ## @code{A \ b} for a matrix @code{b} of size @var{n} by @var{t}. |
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39 ## @item solve_t |
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40 ## @code{A' \ b} for a matrix @code{b} of size @var{n} by @var{t}. |
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41 ## @end table |
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42 ## |
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43 ## The implicit version requires an explicit dimension @var{n}. |
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44 ## |
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45 ## @code{condest} uses a randomized algorithm to approximate |
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46 ## the 1-norms. |
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47 ## |
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48 ## @code{condest} returns the 1-norm condition estimate @var{est} and |
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49 ## a vector @var{v} satisfying @code{norm (A*v, 1) == norm (A, 1) * norm |
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50 ## (@var{v}, 1) / @var{est}}. When @var{est} is large, @var{v} is an |
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51 ## approximate null vector. |
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52 ## |
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53 ## References: |
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54 ## @itemize |
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55 ## @item Nicholas J. Higham and Françoise Tisseur, "A Block Algorithm |
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56 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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57 ## Pseudospectra." SIMAX vol 21, no 4, pp 1185-1201. |
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58 ## @url{http://dx.doi.org/10.1137/S0895479899356080} |
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59 ## @item Nicholas J. Higham and Françoise Tisseur, "A Block Algorithm |
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60 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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61 ## Pseudospectra." @url{http://citeseer.ist.psu.edu/223007.html} |
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62 ## @end itemize |
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63 ## |
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64 ## @seealso{norm, cond, onenormest} |
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65 ## @end deftypefn |
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66 |
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67 ## Code originally licensed under |
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68 ## |
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69 ## Copyright (c) 2007, Regents of the University of California |
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70 ## All rights reserved. |
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71 ## |
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72 ## Redistribution and use in source and binary forms, with or without |
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73 ## modification, are permitted provided that the following conditions |
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74 ## are met: |
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75 ## |
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76 ## * Redistributions of source code must retain the above copyright |
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77 ## notice, this list of conditions and the following disclaimer. |
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78 ## |
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79 ## * Redistributions in binary form must reproduce the above |
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80 ## copyright notice, this list of conditions and the following |
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81 ## disclaimer in the documentation and/or other materials provided |
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82 ## with the distribution. |
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83 ## |
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84 ## * Neither the name of the University of California, Berkeley nor |
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85 ## the names of its contributors may be used to endorse or promote |
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86 ## products derived from this software without specific prior |
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87 ## written permission. |
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88 ## |
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89 ## THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' |
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90 ## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED |
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91 ## TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A |
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92 ## PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS AND |
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93 ## CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, |
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94 ## SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT |
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95 ## LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF |
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96 ## USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND |
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97 ## ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, |
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98 ## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT |
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99 ## OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF |
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100 ## SUCH DAMAGE. |
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101 |
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102 ## Author: Jason Riedy <ejr@cs.berkeley.edu> |
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103 ## Keywords: linear-algebra norm estimation |
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104 ## Version: 0.2 |
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105 |
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106 function [est, v] = condest (varargin) |
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107 |
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108 if (nargin < 1 || nargin > 6) |
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109 print_usage (); |
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110 endif |
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111 |
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112 default_t = 5; |
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113 |
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114 if (ismatrix (varargin{1})) |
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115 n = size (varargin{1}, 1); |
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116 if (n != size (varargin{1}, 2)) |
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117 error ("condest: matrix must be square."); |
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118 endif |
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119 A = varargin{1}; |
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120 |
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121 if (nargin > 1) |
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122 if (isscalar (varargin{2})) |
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123 t = varargin{2}; |
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124 else |
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125 if (nargin < 3) |
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126 error ("condest: must supply both solve and solve_t."); |
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127 else |
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128 solve = varargin{2}; |
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129 solve_t = varargin{3}; |
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130 if (nargin > 3) |
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131 t = varargin{4}; |
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132 endif |
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133 endif |
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134 endif |
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135 endif |
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136 else |
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137 if (nargin < 5) |
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138 error ("condest: implicit form of condest requires at least 5 arguments."); |
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139 endif |
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140 apply = varargin{1}; |
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141 apply_t = varargin{2}; |
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142 solve = varargin{3}; |
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143 solve_t = varargin{4}; |
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144 n = varargin{5}; |
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145 if (! isscalar (n)) |
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146 error ("condest: dimension argument of implicit form must be scalar."); |
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147 endif |
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148 if (nargin > 5) |
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149 t = varargin{6}; |
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150 endif |
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151 endif |
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152 |
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153 if (! exist ("t", "var")) |
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154 t = min (n, default_t); |
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155 endif |
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156 |
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157 if (! exist ("solve", "var")) |
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158 if (issparse (A)) |
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159 [L, U, P, Pc] = splu (A); |
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160 solve = @(x) Pc' * (U\ (L\ (P*x))); |
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161 solve_t = @(x) P'*(L'\ (U'\ (Pc*x))); |
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162 else |
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163 [L, U, P] = lu (A); |
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164 solve = @(x) U\ (L\ (P*x)); |
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165 solve_t = @(x) P' * (L'\ (U'\x)); |
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166 endif |
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167 endif |
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168 |
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169 if (exist ("A", "var")) |
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170 Anorm = norm (A, 1); |
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171 else |
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172 Anorm = onenormest (apply, apply_t, n, t); |
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173 endif |
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174 |
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175 [Ainv_norm, v, w] = onenormest (solve, solve_t, n, t); |
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176 |
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177 est = Anorm * Ainv_norm; |
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178 v = w / norm (w, 1); |
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179 |
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180 endfunction |
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181 |
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182 %!demo |
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183 %! N = 100; |
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184 %! A = randn (N) + eye (N); |
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185 %! condest (A) |
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186 %! [L,U,P] = lu (A); |
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187 %! condest (A, @(x) U\ (L\ (P*x)), @(x) P'*(L'\ (U'\x))) |
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188 %! condest (@(x) A*x, @(x) A'*x, @(x) U\ (L\ (P*x)), @(x) P'*(L'\ (U'\x)), N) |
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189 %! norm (inv (A), 1) * norm (A, 1) |
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190 |
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191 ## Yes, these test bounds are really loose. There's |
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192 ## enough randomization to trigger odd cases with hilb(). |
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193 |
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194 %!test |
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195 %! N = 6; |
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196 %! A = hilb (N); |
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197 %! cA = condest (A); |
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198 %! cA_test = norm (inv (A), 1) * norm (A, 1); |
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199 %! assert (cA, cA_test, 2^-8); |
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200 |
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201 %!test |
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202 %! N = 6; |
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203 %! A = hilb (N); |
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204 %! solve = @(x) A\x; solve_t = @(x) A'\x; |
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205 %! cA = condest (A, solve, solve_t); |
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206 %! cA_test = norm (inv (A), 1) * norm (A, 1); |
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207 %! assert (cA, cA_test, 2^-8); |
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208 |
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209 %!test |
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210 %! N = 6; |
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211 %! A = hilb (N); |
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212 %! apply = @(x) A*x; apply_t = @(x) A'*x; |
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213 %! solve = @(x) A\x; solve_t = @(x) A'\x; |
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214 %! cA = condest (apply, apply_t, solve, solve_t, N); |
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215 %! cA_test = norm (inv (A), 1) * norm (A, 1); |
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216 %! assert (cA, cA_test, 2^-6); |
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217 |
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218 %!test |
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219 %! N = 12; |
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220 %! A = hilb (N); |
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221 %! [rcondA, v] = condest (A); |
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222 %! x = A*v; |
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223 %! assert (norm(x, inf), 0, eps); |
