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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}, @var{w}, @var{iter}] =} onenormest (@var{a}, @var{t}) |
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21 ## @deftypefnx {Function File} {[@var{est}, @var{v}, @var{w}, @var{iter}] =} onenormest (@var{apply}, @var{apply_t}, @var{n}, @var{t}) |
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22 ## |
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23 ## Apply Higham and Tisseur's randomized block 1-norm estimator to |
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24 ## matrix @var{a} using @var{t} test vectors. If @var{t} exceeds 5, then |
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25 ## only 5 test vectors are used. |
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26 ## |
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27 ## If the matrix is not explicit, e.g. when estimating the norm of |
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28 ## @code{inv (@var{A})} given an LU factorization, @code{onenormest} applies |
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29 ## @var{A} and its conjugate transpose through a pair of functions |
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30 ## @var{apply} and @var{apply_t}, respectively, to a dense matrix of size |
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31 ## @var{n} by @var{t}. The implicit version requires an explicit dimension |
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32 ## @var{n}. |
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33 ## |
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34 ## Returns the norm estimate @var{est}, two vectors @var{v} and |
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35 ## @var{w} related by norm |
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36 ## @code{(@var{w}, 1) = @var{est} * norm (@var{v}, 1)}, |
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37 ## and the number of iterations @var{iter}. The number of |
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38 ## iterations is limited to 10 and is at least 2. |
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39 ## |
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40 ## References: |
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41 ## @itemize |
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42 ## @item Nicholas J. Higham and Françoise Tisseur, "A Block Algorithm |
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43 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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44 ## Pseudospectra." SIMAX vol 21, no 4, pp 1185-1201. |
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45 ## @url{http://dx.doi.org/10.1137/S0895479899356080} |
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46 ## @item Nicholas J. Higham and Françoise Tisseur, "A Block Algorithm |
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47 ## for Matrix 1-Norm Estimation, with an Application to 1-Norm |
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48 ## Pseudospectra." @url{http://citeseer.ist.psu.edu/223007.html} |
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49 ## @end itemize |
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50 ## |
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51 ## @seealso{condest, norm, cond} |
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52 ## @end deftypefn |
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53 |
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54 ## Code originally licensed under |
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55 ## |
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56 ## Copyright (c) 2007, Regents of the University of California |
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57 ## All rights reserved. |
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58 ## |
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59 ## Redistribution and use in source and binary forms, with or without |
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60 ## modification, are permitted provided that the following conditions |
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61 ## are met: |
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62 ## |
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63 ## * Redistributions of source code must retain the above copyright |
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64 ## notice, this list of conditions and the following disclaimer. |
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65 ## |
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66 ## * Redistributions in binary form must reproduce the above |
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67 ## copyright notice, this list of conditions and the following |
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68 ## disclaimer in the documentation and/or other materials provided |
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69 ## with the distribution. |
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70 ## |
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71 ## * Neither the name of the University of California, Berkeley nor |
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72 ## the names of its contributors may be used to endorse or promote |
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73 ## products derived from this software without specific prior |
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74 ## written permission. |
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75 ## |
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76 ## THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' |
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77 ## AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED |
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78 ## TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A |
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79 ## PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS AND |
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80 ## CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, |
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81 ## SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT |
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82 ## LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF |
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83 ## USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND |
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84 ## ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, |
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85 ## OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT |
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86 ## OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF |
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87 ## SUCH DAMAGE. |
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88 |
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89 ## Author: Jason Riedy <ejr@cs.berkeley.edu> |
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90 ## Keywords: linear-algebra norm estimation |
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91 ## Version: 0.2 |
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92 |
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93 function [est, v, w, iter] = onenormest (varargin) |
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94 |
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95 if (size (varargin, 2) < 1 || size (varargin, 2) > 4) |
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96 print_usage (); |
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97 endif |
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98 |
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99 default_t = 5; |
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100 itmax = 10; |
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101 |
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102 if (ismatrix (varargin{1})) |
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103 n = size (varargin{1}, 1); |
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104 if n != size (varargin{1}, 2), |
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105 error ("onenormest: matrix must be square."); |
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106 endif |
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107 apply = @(x) varargin{1} * x; |
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108 apply_t = @(x) varargin{1}' * x; |
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109 if (size (varargin) > 1) |
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110 t = varargin{2}; |
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111 else |
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112 t = min (n, default_t); |
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113 endif |
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114 else |
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115 if (size (varargin, 2) < 3) |
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116 print_usage(); |
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117 endif |
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118 n = varargin{3}; |
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119 apply = varargin{1}; |
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120 apply_t = varargin{2}; |
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121 if (size (varargin) > 3) |
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122 t = varargin{4}; |
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123 else |
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124 t = default_t; |
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125 endif |
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126 endif |
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127 |
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128 ## Initial test vectors X. |
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129 X = rand (n, t); |
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130 X = X ./ (ones (n,1) * sum (abs (X), 1)); |
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131 |
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132 been_there = zeros (n, 1); # Track if a vertex has been visited. |
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133 est_old = 0; # To check if the estimate has increased. |
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134 S = zeros (n, t); # Normalized vector of signs. The normalization is |
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135 |
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136 for iter = 1 : itmax + 1 |
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137 Y = feval (apply, X); |
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138 |
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139 ## Find the initial estimate as the largest A*x. |
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140 [est, ind_best] = max (sum (abs (Y), 1)); |
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141 if (est > est_old || iter == 2) |
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142 w = Y(:,ind_best); |
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143 endif |
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144 if (iter >= 2 && est < est_old) |
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145 ## No improvement, so stop. |
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146 est = est_old; |
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147 break; |
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148 endif |
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149 |
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150 est_old = est; |
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151 S_old = S; |
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152 if (iter > itmax), |
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153 ## Gone too far. Stop. |
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154 break; |
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155 endif |
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156 |
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157 S = sign (Y); |
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158 |
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159 ## Test if any of S are approximately parallel to previous S |
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160 ## vectors or current S vectors. If everything is parallel, |
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161 ## stop. Otherwise, replace any parallel vectors with |
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162 ## rand{-1,+1}. |
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163 partest = any (abs (S_old' * S - n) < 4*eps*n); |
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164 if (all (partest)) |
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165 ## All the current vectors are parallel to old vectors. |
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166 ## We've hit a cycle, so stop. |
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167 break; |
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168 endif |
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169 if (any (partest)) |
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170 ## Some vectors are parallel to old ones and are cycling, |
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171 ## but not all of them. Replace the parallel vectors with |
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172 ## rand{-1,+1}. |
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173 numpar = sum (partest); |
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174 replacements = 2*(rand (n,numpar) < 0.5) - 1; |
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175 S(:,partest) = replacements; |
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176 endif |
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177 ## Now test for parallel vectors within S. |
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178 partest = any ( (S' * S - eye (t)) == n ); |
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179 if (any (partest)) |
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180 numpar = sum (partest); |
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181 replacements = 2*(rand (n,numpar) < 0.5) - 1; |
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182 S(:,partest) = replacements; |
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183 endif |
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184 |
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185 Z = feval (apply_t, S); |
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186 |
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187 ## Now find the largest non-previously-visted index per |
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188 ## vector. |
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189 h = max (abs (Z),2); |
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190 [mh, mhi] = max (h); |
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191 if (iter >= 2 && mhi == ind_best) |
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192 ## Hit a cycle, stop. |
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193 break; |
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194 endif |
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195 [h, ind] = sort (h, 'descend'); |
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196 if (t > 1) |
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197 firstind = ind(1:t); |
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198 if (all (been_there(firstind))) |
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199 ## Visited all these before, so stop. |
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200 break; |
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201 endif |
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202 ind = ind (!been_there (ind)); |
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203 if (length (ind) < t) |
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204 ## There aren't enough new vectors, so we're practically |
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205 ## in a cycle. Stop. |
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206 break; |
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207 endif |
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208 endif |
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209 |
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210 ## Visit the new indices. |
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211 X = zeros (n, t); |
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212 for zz = 1 : t |
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213 X(ind(zz),zz) = 1; |
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214 endfor |
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215 been_there (ind (1 : t)) = 1; |
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216 endfor |
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217 |
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218 ## The estimate est and vector w are set in the loop above. The |
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219 ## vector v selects the ind_best column of A. |
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220 v = zeros (n, 1); |
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221 v(ind_best) = 1; |
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222 endfunction |
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223 |
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224 %!demo |
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225 %! N = 100; |
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226 %! A = randn(N) + eye(N); |
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227 %! [L,U,P] = lu(A); |
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228 %! nm1inv = onenormest(@(x) U\(L\(P*x)), @(x) P'*(L'\(U'\x)), N, 30) |
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229 %! norm(inv(A), 1) |
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230 |
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231 %!test |
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232 %! N = 10; |
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233 %! A = ones (N); |
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234 %! [nm1, v1, w1] = onenormest (A); |
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235 %! [nminf, vinf, winf] = onenormest (A', 6); |
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236 %! assert (nm1, N, -2*eps); |
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237 %! assert (nminf, N, -2*eps); |
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238 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps) |
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239 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps) |
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240 |
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241 %!test |
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242 %! N = 10; |
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243 %! A = ones (N); |
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244 %! [nm1, v1, w1] = onenormest (@(x) A*x, @(x) A'*x, N, 3); |
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245 %! [nminf, vinf, winf] = onenormest (@(x) A'*x, @(x) A*x, N, 3); |
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246 %! assert (nm1, N, -2*eps); |
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247 %! assert (nminf, N, -2*eps); |
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248 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps) |
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249 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps) |
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250 |
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251 %!test |
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252 %! N = 5; |
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253 %! A = hilb (N); |
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254 %! [nm1, v1, w1] = onenormest (A); |
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255 %! [nminf, vinf, winf] = onenormest (A', 6); |
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256 %! assert (nm1, norm (A, 1), -2*eps); |
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257 %! assert (nminf, norm (A, inf), -2*eps); |
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258 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps) |
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259 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps) |
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260 |
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261 ## Only likely to be within a factor of 10. |
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262 %!test |
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263 %! N = 100; |
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264 %! A = rand (N); |
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265 %! [nm1, v1, w1] = onenormest (A); |
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266 %! [nminf, vinf, winf] = onenormest (A', 6); |
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267 %! assert (nm1, norm (A, 1), -.1); |
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268 %! assert (nminf, norm (A, inf), -.1); |
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269 %! assert (norm (w1, 1), nm1 * norm (v1, 1), -2*eps) |
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270 %! assert (norm (winf, 1), nminf * norm (vinf, 1), -2*eps) |
