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1 ## Copyright (C) 2004, 2006, 2007 Piotr Krzyzanowski |
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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{x} =} pcr (@var{a}, @var{b}, @var{tol}, @var{maxit}, @var{m}, @var{x0}, @dots{}) |
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21 ## @deftypefnx {Function File} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}] =} pcr (@dots{}) |
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22 ## |
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23 ## Solves the linear system of equations @code{@var{a} * @var{x} = |
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24 ## @var{b}} by means of the Preconditioned Conjugate Residuals iterative |
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25 ## method. The input arguments are |
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26 ## |
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27 ## @itemize |
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28 ## @item |
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29 ## @var{a} can be either a square (preferably sparse) matrix or a |
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30 ## function handle, inline function or string containing the name |
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31 ## of a function which computes @code{@var{a} * @var{x}}. In principle |
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32 ## @var{a} should be symmetric and non-singular; if @code{pcr} |
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33 ## finds @var{a} to be numerically singular, you will get a warning |
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34 ## message and the @var{flag} output parameter will be set. |
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35 ## |
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36 ## @item |
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37 ## @var{b} is the right hand side vector. |
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38 ## |
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39 ## @item |
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40 ## @var{tol} is the required relative tolerance for the residual error, |
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41 ## @code{@var{b} - @var{a} * @var{x}}. The iteration stops if @code{norm |
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42 ## (@var{b} - @var{a} * @var{x}) <= @var{tol} * norm (@var{b} - @var{a} * |
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43 ## @var{x0})}. If @var{tol} is empty or is omitted, the function sets |
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44 ## @code{@var{tol} = 1e-6} by default. |
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45 ## |
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46 ## @item |
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47 ## @var{maxit} is the maximum allowable number of iterations; if |
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48 ## @code{[]} is supplied for @code{maxit}, or @code{pcr} has less |
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49 ## arguments, a default value equal to 20 is used. |
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50 ## |
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51 ## @item |
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52 ## @var{m} is the (left) preconditioning matrix, so that the iteration is |
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53 ## (theoretically) equivalent to solving by @code{pcr} @code{@var{P} * |
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54 ## @var{x} = @var{m} \ @var{b}}, with @code{@var{P} = @var{m} \ @var{a}}. |
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55 ## Note that a proper choice of the preconditioner may dramatically |
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56 ## improve the overall performance of the method. Instead of matrix |
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57 ## @var{m}, the user may pass a function which returns the results of |
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58 ## applying the inverse of @var{m} to a vector (usually this is the |
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59 ## preferred way of using the preconditioner). If @code{[]} is supplied |
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60 ## for @var{m}, or @var{m} is omitted, no preconditioning is applied. |
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61 ## |
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62 ## @item |
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63 ## @var{x0} is the initial guess. If @var{x0} is empty or omitted, the |
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64 ## function sets @var{x0} to a zero vector by default. |
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65 ## @end itemize |
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66 ## |
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67 ## The arguments which follow @var{x0} are treated as parameters, and |
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68 ## passed in a proper way to any of the functions (@var{a} or @var{m}) |
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69 ## which are passed to @code{pcr}. See the examples below for further |
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70 ## details. The output arguments are |
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71 ## |
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72 ## @itemize |
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73 ## @item |
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74 ## @var{x} is the computed approximation to the solution of |
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75 ## @code{@var{a} * @var{x} = @var{b}}. |
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76 ## |
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77 ## @item |
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78 ## @var{flag} reports on the convergence. @code{@var{flag} = 0} means |
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79 ## the solution converged and the tolerance criterion given by @var{tol} |
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80 ## is satisfied. @code{@var{flag} = 1} means that the @var{maxit} limit |
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81 ## for the iteration count was reached. @code{@var{flag} = 3} reports t |
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82 ## @code{pcr} breakdown, see [1] for details. |
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83 ## |
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84 ## @item |
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85 ## @var{relres} is the ratio of the final residual to its initial value, |
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86 ## measured in the Euclidean norm. |
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87 ## |
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88 ## @item |
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89 ## @var{iter} is the actual number of iterations performed. |
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90 ## |
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91 ## @item |
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92 ## @var{resvec} describes the convergence history of the method, |
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93 ## so that @code{@var{resvec} (i)} contains the Euclidean norms of the |
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94 ## residual after the (@var{i}-1)-th iteration, @code{@var{i} = |
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95 ## 1,2, @dots{}, @var{iter}+1}. |
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96 ## @end itemize |
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97 ## |
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98 ## Let us consider a trivial problem with a diagonal matrix (we exploit the |
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99 ## sparsity of A) |
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100 ## |
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101 ## @example |
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102 ## @group |
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103 ## N = 10; |
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104 ## A = diag([1:N]); A = sparse(A); |
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105 ## b = rand(N,1); |
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106 ## @end group |
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107 ## @end example |
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108 ## |
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109 ## @sc{Example 1:} Simplest use of @code{pcr} |
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110 ## |
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111 ## @example |
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112 ## x = pcr(A, b) |
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113 ## @end example |
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114 ## |
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115 ## @sc{Example 2:} @code{pcr} with a function which computes |
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116 ## @code{@var{a} * @var{x}}. |
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117 ## |
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118 ## @example |
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119 ## @group |
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120 ## function y = applyA(x) |
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121 ## y = [1:10]'.*x; |
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122 ## endfunction |
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123 ## |
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124 ## x = pcr('applyA',b) |
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125 ## @end group |
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126 ## @end example |
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127 ## |
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128 ## @sc{Example 3:} Preconditioned iteration, with full diagnostics. The |
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129 ## preconditioner (quite strange, because even the original matrix |
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130 ## @var{a} is trivial) is defined as a function |
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131 ## |
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132 ## @example |
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133 ## @group |
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134 ## function y = applyM(x) |
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135 ## K = floor(length(x)-2); |
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136 ## y = x; |
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137 ## y(1:K) = x(1:K)./[1:K]'; |
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138 ## endfunction |
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139 ## |
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140 ## [x, flag, relres, iter, resvec] = ... |
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141 ## pcr(A, b, [], [], 'applyM') |
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142 ## semilogy([1:iter+1], resvec); |
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143 ## @end group |
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144 ## @end example |
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145 ## |
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146 ## @sc{Example 4:} Finally, a preconditioner which depends on a |
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147 ## parameter @var{k}. |
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148 ## |
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149 ## @example |
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150 ## @group |
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151 ## function y = applyM(x, varargin) |
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152 ## K = varargin@{1@}; |
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153 ## y = x; y(1:K) = x(1:K)./[1:K]'; |
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154 ## endfunction |
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155 ## |
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156 ## [x, flag, relres, iter, resvec] = ... |
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157 ## pcr(A, b, [], [], 'applyM', [], 3) |
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158 ## @end group |
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159 ## @end example |
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160 ## |
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161 ## @sc{References} |
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162 ## |
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163 ## [1] W. Hackbusch, "Iterative Solution of Large Sparse Systems of |
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164 ## Equations", section 9.5.4; Springer, 1994 |
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165 ## |
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166 ## @seealso{sparse, pcg} |
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167 ## @end deftypefn |
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168 |
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169 ## Author: Piotr Krzyzanowski <piotr.krzyzanowski@mimuw.edu.pl> |
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170 |
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171 function [x, flag, relres, iter, resvec] = pcr (A, b, tol, maxit, M, x0, varargin) |
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172 |
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173 breakdown = false; |
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174 |
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175 if (nargin < 6 || isempty (x0)) |
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176 x = zeros (size (b)); |
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177 else |
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178 x = x0; |
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179 endif |
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180 |
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181 if (nargin < 5) |
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182 M = []; |
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183 endif |
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184 |
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185 if (nargin < 4 || isempty (maxit)) |
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186 maxit = 20; |
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187 endif |
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188 |
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189 maxit += 2; |
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190 |
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191 if (nargin < 3 || isempty (tol)) |
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192 tol = 1e-6; |
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193 endif |
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194 |
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195 if (nargin < 2) |
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196 print_usage (); |
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197 endif |
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198 |
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199 ## init |
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200 if (isnumeric (A)) # is A a matrix? |
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201 r = b - A*x; |
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202 else # then A should be a function! |
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203 r = b - feval (A, x, varargin{:}); |
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204 endif |
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205 |
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206 if (isnumeric (M)) # is M a matrix? |
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207 if (isempty (M)) # if M is empty, use no precond |
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208 p = r; |
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209 else # otherwise, apply the precond |
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210 p = M \ r; |
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211 endif |
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212 else # then M should be a function! |
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213 p = feval (M, r, varargin{:}); |
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214 endif |
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215 |
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216 iter = 2; |
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217 |
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218 b_bot_old = 1; |
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219 q_old = p_old = s_old = zeros (size (x)); |
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220 |
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221 if (isnumeric (A)) # is A a matrix? |
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222 q = A * p; |
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223 else # then A should be a function! |
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224 q = feval (A, p, varargin{:}); |
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225 endif |
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226 |
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227 resvec(1) = abs (norm (r)); |
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228 |
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229 ## iteration |
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230 while (resvec(iter-1) > tol*resvec(1) && iter < maxit) |
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231 |
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232 if (isnumeric (M)) # is M a matrix? |
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233 if (isempty (M)) # if M is empty, use no precond |
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234 s = q; |
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235 else # otherwise, apply the precond |
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236 s = M \ q; |
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237 endif |
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238 else # then M should be a function! |
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239 s = feval (M, q, varargin{:}); |
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240 endif |
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241 b_top = r' * s; |
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242 b_bot = q' * s; |
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243 |
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244 if (b_bot == 0.0) |
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245 breakdown = true; |
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246 break; |
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247 endif |
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248 lambda = b_top / b_bot; |
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249 |
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250 x += lambda*p; |
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251 r -= lambda*q; |
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252 |
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253 if (isnumeric(A)) # is A a matrix? |
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254 t = A*s; |
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255 else # then A should be a function! |
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256 t = feval (A, s, varargin{:}); |
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257 endif |
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258 |
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259 alpha0 = (t'*s) / b_bot; |
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260 alpha1 = (t'*s_old) / b_bot_old; |
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261 |
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262 p_temp = p; |
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263 q_temp = q; |
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264 |
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265 p = s - alpha0*p - alpha1*p_old; |
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266 q = t - alpha0*q - alpha1*q_old; |
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267 |
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268 s_old = s; |
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269 p_old = p_temp; |
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270 q_old = q_temp; |
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271 b_bot_old = b_bot; |
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272 |
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273 resvec(iter) = abs (norm (r)); |
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274 iter++; |
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275 endwhile |
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276 |
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277 flag = 0; |
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278 relres = resvec(iter-1) ./ resvec(1); |
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279 iter -= 2; |
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280 if (iter >= maxit-2) |
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281 flag = 1; |
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282 if (nargout < 2) |
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283 warning ("pcr: maximum number of iterations (%d) reached\n", iter); |
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284 warning ("the initial residual norm was reduced %g times.\n", 1.0/relres); |
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285 endif |
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286 elseif (nargout < 2 && ! breakdown) |
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287 fprintf (stderr, "pcr: converged in %d iterations. \n", iter); |
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288 fprintf (stderr, "the initial residual norm was reduced %g times.\n", |
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289 1.0/relres); |
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290 endif |
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291 |
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292 if (breakdown) |
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293 flag = 3; |
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294 if (nargout < 2) |
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295 warning ("pcr: breakdown occurred:\n"); |
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296 warning ("system matrix singular or preconditioner indefinite?\n"); |
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297 endif |
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298 endif |
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299 |
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300 endfunction |
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301 |
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302 %!demo |
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303 %! |
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304 %! # Simplest usage of PCR (see also 'help pcr') |
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305 %! |
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306 %! N = 20; |
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307 %! A = diag(linspace(-3.1,3,N)); b = rand(N,1); y = A\b; #y is the true solution |
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308 %! x = pcr(A,b); |
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309 %! printf('The solution relative error is %g\n', norm(x-y)/norm(y)); |
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310 %! |
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311 %! # You shouldn't be afraid if PCR issues some warning messages in this |
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312 %! # example: watch out in the second example, why it takes N iterations |
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313 %! # of PCR to converge to (a very accurate, by the way) solution |
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314 %!demo |
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315 %! |
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316 %! # Full output from PCR |
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317 %! # We use this output to plot the convergence history |
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318 %! |
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319 %! N = 20; |
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320 %! A = diag(linspace(-3.1,30,N)); b = rand(N,1); X = A\b; #X is the true solution |
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321 %! [x, flag, relres, iter, resvec] = pcr(A,b); |
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322 %! printf('The solution relative error is %g\n', norm(x-X)/norm(X)); |
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323 %! title('Convergence history'); xlabel('Iteration'); ylabel('log(||b-Ax||/||b||)'); |
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324 %! semilogy([0:iter],resvec/resvec(1),'o-g;relative residual;'); |
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325 %!demo |
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326 %! |
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327 %! # Full output from PCR |
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328 %! # We use indefinite matrix based on the Hilbert matrix, with one |
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329 %! # strongly negative eigenvalue |
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330 %! # Hilbert matrix is extremely ill conditioned, so is ours, |
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331 %! # and that's why PCR WILL have problems |
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332 %! |
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333 %! N = 10; |
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334 %! A = hilb(N); A(1,1)=-A(1,1); b = rand(N,1); X = A\b; #X is the true solution |
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335 %! printf('Condition number of A is %g\n', cond(A)); |
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336 %! [x, flag, relres, iter, resvec] = pcr(A,b,[],200); |
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337 %! if (flag == 3) |
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338 %! printf('PCR breakdown. System matrix is [close to] singular\n'); |
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339 %! end |
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340 %! title('Convergence history'); xlabel('Iteration'); ylabel('log(||b-Ax||)'); |
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341 %! semilogy([0:iter],resvec,'o-g;absolute residual;'); |
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342 %!demo |
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343 %! |
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344 %! # Full output from PCR |
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345 %! # We use an indefinite matrix based on the 1-D Laplacian matrix for A, |
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346 %! # and here we have cond(A) = O(N^2) |
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347 %! # That's the reason we need some preconditioner; here we take |
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348 %! # a very simple and not powerful Jacobi preconditioner, |
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349 %! # which is the diagonal of A |
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350 %! |
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351 %! # Note that we use here indefinite preconditioners! |
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352 %! |
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353 %! N = 100; |
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354 %! A = zeros(N,N); |
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355 %! for i=1:N-1 # form 1-D Laplacian matrix |
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356 %! A(i:i+1,i:i+1) = [2 -1; -1 2]; |
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357 %! endfor |
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358 %! A = [A, zeros(size(A)); zeros(size(A)), -A]; |
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359 %! b = rand(2*N,1); X = A\b; #X is the true solution |
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360 %! maxit = 80; |
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361 %! printf('System condition number is %g\n',cond(A)); |
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362 %! # No preconditioner: the convergence is very slow! |
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363 %! |
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364 %! [x, flag, relres, iter, resvec] = pcr(A,b,[],maxit); |
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365 %! title('Convergence history'); xlabel('Iteration'); ylabel('log(||b-Ax||)'); |
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366 %! semilogy([0:iter],resvec,'o-g;NO preconditioning: absolute residual;'); |
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367 %! |
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368 %! pause(1); |
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369 %! # Test Jacobi preconditioner: it will not help much!!! |
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370 %! |
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371 %! M = diag(diag(A)); # Jacobi preconditioner |
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372 %! [x, flag, relres, iter, resvec] = pcr(A,b,[],maxit,M); |
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373 %! hold on; |
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374 %! semilogy([0:iter],resvec,'o-r;JACOBI preconditioner: absolute residual;'); |
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375 %! |
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376 %! pause(1); |
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377 %! # Test nonoverlapping block Jacobi preconditioner: this one should give |
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378 %! # some convergence speedup! |
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379 %! |
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380 %! M = zeros(N,N);k=4; |
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381 %! for i=1:k:N # get k x k diagonal blocks of A |
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382 %! M(i:i+k-1,i:i+k-1) = A(i:i+k-1,i:i+k-1); |
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383 %! endfor |
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384 %! M = [M, zeros(size(M)); zeros(size(M)), -M]; |
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385 %! [x, flag, relres, iter, resvec] = pcr(A,b,[],maxit,M); |
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386 %! semilogy([0:iter],resvec,'o-b;BLOCK JACOBI preconditioner: absolute residual;'); |
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387 %! hold off; |
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388 %!test |
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389 %! |
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390 %! #solve small indefinite diagonal system |
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391 %! |
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392 %! N = 10; |
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393 %! A = diag(linspace(-10.1,10,N)); b = ones(N,1); X = A\b; #X is the true solution |
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394 %! [x, flag] = pcr(A,b,[],N+1); |
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395 %! assert(norm(x-X)/norm(X)<1e-10); |
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396 %! assert(flag,0); |
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397 %! |
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398 %!test |
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399 %! |
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400 %! #solve tridiagonal system, do not converge in default 20 iterations |
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401 %! #should perform max allowable default number of iterations |
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402 %! |
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403 %! N = 100; |
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404 %! A = zeros(N,N); |
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405 %! for i=1:N-1 # form 1-D Laplacian matrix |
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406 %! A(i:i+1,i:i+1) = [2 -1; -1 2]; |
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407 %! endfor |
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408 %! b = ones(N,1); X = A\b; #X is the true solution |
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409 %! [x, flag, relres, iter, resvec] = pcr(A,b,1e-12); |
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410 %! assert(flag,1); |
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411 %! assert(relres>0.6); |
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412 %! assert(iter,20); |
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413 %! |
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414 %!test |
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415 %! |
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416 %! #solve tridiagonal system with 'prefect' preconditioner |
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417 %! #converges in one iteration |
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418 %! |
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419 %! N = 100; |
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420 %! A = zeros(N,N); |
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421 %! for i=1:N-1 # form 1-D Laplacian matrix |
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422 %! A(i:i+1,i:i+1) = [2 -1; -1 2]; |
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423 %! endfor |
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424 %! b = ones(N,1); X = A\b; #X is the true solution |
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425 %! [x, flag, relres, iter] = pcr(A,b,[],[],A,b); |
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426 %! assert(norm(x-X)/norm(X)<1e-6); |
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427 %! assert(relres<1e-6); |
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428 %! assert(flag,0); |
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429 %! assert(iter,1); #should converge in one iteration |
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430 %! |
