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ensure exp. terminal widget has focus at startup * main-window.cc (main_window): call focus_command_window only if event loop is idle by using a single shot timer * main-window.h: make focus_command_window a public slot
author Torsten Lilge <ttl-octave@mailbox.org>
date Sun, 26 May 2024 02:29:44 +0200
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1 ########################################################################
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2 ##
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3 ## Copyright (C) 2004-2024 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/>.
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7 ##
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8 ## This file is part of Octave.
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9 ##
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10 ## Octave is free software: you can redistribute it and/or modify it
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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.
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14 ##
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15 ## Octave is distributed in the hope that it will be useful, but
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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.
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19 ##
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20 ## You should have received a copy of the GNU General Public License
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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 ########################################################################
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25
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26 ## -*- texinfo -*-
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27 ## @deftypefn {} {@var{x} =} pcr (@var{A}, @var{b}, @var{tol}, @var{maxit}, @var{m}, @var{x0}, @dots{})
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28 ## @deftypefnx {} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}] =} pcr (@dots{})
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29 ##
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30 ## Solve the linear system of equations @code{@var{A} * @var{x} = @var{b}} by
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31 ## means of the Preconditioned Conjugate Residuals iterative method.
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32 ##
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33 ## The input arguments are
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34 ##
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35 ## @itemize
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36 ## @item
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37 ## @var{A} can be either a square (preferably sparse) matrix or a function
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38 ## handle, inline function or string containing the name of a function which
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39 ## computes @code{@var{A} * @var{x}}. In principle @var{A} should be
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40 ## symmetric and non-singular; if @code{pcr} finds @var{A} to be numerically
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41 ## singular, you will get a warning message and the @var{flag} output
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42 ## parameter will be set.
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43 ##
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44 ## @item
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45 ## @var{b} is the right hand side vector.
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46 ##
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47 ## @item
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48 ## @var{tol} is the required relative tolerance for the residual error,
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49 ## @code{@var{b} - @var{A} * @var{x}}. The iteration stops if
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50 ## @code{norm (@var{b} - @var{A} * @var{x}) <=
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51 ## @var{tol} * norm (@var{b} - @var{A} * @var{x0})}.
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52 ## If @var{tol} is empty or is omitted, the function sets
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53 ## @code{@var{tol} = 1e-6} by default.
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54 ##
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55 ## @item
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56 ## @var{maxit} is the maximum allowable number of iterations; if @code{[]} is
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57 ## supplied for @var{maxit}, or @code{pcr} has less arguments, a default
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58 ## value equal to 20 is used.
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59 ##
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60 ## @item
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61 ## @var{m} is the (left) preconditioning matrix, so that the iteration is
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62 ## (theoretically) equivalent to solving by
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63 ## @code{pcr} @code{@var{P} * @var{x} = @var{m} \ @var{b}}, with
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64 ## @code{@var{P} = @var{m} \ @var{A}}. Note that a proper choice of the
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65 ## preconditioner may dramatically improve the overall performance of the
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66 ## method. Instead of matrix @var{m}, the user may pass a function which
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67 ## returns the results of applying the inverse of @var{m} to a vector
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68 ## (usually this is the preferred way of using the preconditioner). If
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69 ## @code{[]} is supplied for @var{m}, or @var{m} is omitted, no
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70 ## preconditioning is applied.
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71 ##
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72 ## @item
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73 ## @var{x0} is the initial guess. If @var{x0} is empty or omitted, the
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74 ## function sets @var{x0} to a zero vector by default.
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75 ## @end itemize
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76 ##
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77 ## The arguments which follow @var{x0} are treated as parameters, and passed
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78 ## in a proper way to any of the functions (@var{A} or @var{m}) which are
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79 ## passed to @code{pcr}. See the examples below for further details.
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80 ##
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81 ## The output arguments are
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82 ##
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83 ## @itemize
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84 ## @item
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85 ## @var{x} is the computed approximation to the solution of
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86 ## @code{@var{A} * @var{x} = @var{b}}.
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87 ##
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88 ## @item
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89 ## @var{flag} reports on the convergence. @code{@var{flag} = 0} means the
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90 ## solution converged and the tolerance criterion given by @var{tol} is
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91 ## satisfied. @code{@var{flag} = 1} means that the @var{maxit} limit for the
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92 ## iteration count was reached. @code{@var{flag} = 3} reports a @code{pcr}
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93 ## breakdown, see [1] for details.
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94 ##
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95 ## @item
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96 ## @var{relres} is the ratio of the final residual to its initial value,
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97 ## measured in the Euclidean norm.
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98 ##
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99 ## @item
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100 ## @var{iter} is the actual number of iterations performed.
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101 ##
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102 ## @item
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103 ## @var{resvec} describes the convergence history of the method, so that
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104 ## @code{@var{resvec} (i)} contains the Euclidean norms of the residual after
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105 ## the (@var{i}-1)-th iteration, @code{@var{i} = 1,2, @dots{}, @var{iter}+1}.
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106 ## @end itemize
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107 ##
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108 ## Let us consider a trivial problem with a diagonal matrix (we exploit the
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109 ## sparsity of A)
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110 ##
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111 ## @example
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112 ## @group
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113 ## n = 10;
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114 ## A = sparse (diag (1:n));
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115 ## b = rand (N, 1);
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116 ## @end group
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117 ## @end example
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118 ##
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119 ## @sc{Example 1:} Simplest use of @code{pcr}
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120 ##
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121 ## @example
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122 ## x = pcr (A, b)
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123 ## @end example
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124 ##
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125 ## @sc{Example 2:} @code{pcr} with a function which computes
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126 ## @code{@var{A} * @var{x}}.
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127 ##
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128 ## @example
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129 ## @group
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130 ## function y = apply_a (x)
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131 ## y = [1:10]' .* x;
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132 ## endfunction
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133 ##
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134 ## x = pcr ("apply_a", b)
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135 ## @end group
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136 ## @end example
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137 ##
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138 ## @sc{Example 3:} Preconditioned iteration, with full diagnostics. The
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139 ## preconditioner (quite strange, because even the original matrix
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140 ## @var{A} is trivial) is defined as a function
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141 ##
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142 ## @example
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143 ## @group
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144 ## function y = apply_m (x)
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145 ## k = floor (length (x) - 2);
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146 ## y = x;
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147 ## y(1:k) = x(1:k) ./ [1:k]';
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148 ## endfunction
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149 ##
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150 ## [x, flag, relres, iter, resvec] = ...
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151 ## pcr (A, b, [], [], "apply_m")
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152 ## semilogy ([1:iter+1], resvec);
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153 ## @end group
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154 ## @end example
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155 ##
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156 ## @sc{Example 4:} Finally, a preconditioner which depends on a
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157 ## parameter @var{k}.
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158 ##
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159 ## @example
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160 ## @group
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161 ## function y = apply_m (x, varargin)
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162 ## k = varargin@{1@};
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163 ## y = x;
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164 ## y(1:k) = x(1:k) ./ [1:k]';
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165 ## endfunction
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166 ##
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167 ## [x, flag, relres, iter, resvec] = ...
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168 ## pcr (A, b, [], [], "apply_m"', [], 3)
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169 ## @end group
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170 ## @end example
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171 ##
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172 ## Reference:
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173 ##
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174 ## @nospell{W. Hackbusch}, @cite{Iterative Solution of Large Sparse
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175 ## Systems of Equations}, section 9.5.4; @nospell{Springer}, 1994
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176 ##
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177 ## @seealso{sparse, pcg}
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178 ## @end deftypefn
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179
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180 function [x, flag, relres, iter, resvec] = pcr (A, b, tol, maxit, m, x0, varargin)
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181
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182 breakdown = false;
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183
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184 if (nargin < 6 || isempty (x0))
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185 x = zeros (size (b));
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186 else
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187 x = x0;
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188 endif
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189
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190 if (nargin < 5)
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191 m = [];
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192 endif
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193
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194 if (nargin < 4 || isempty (maxit))
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195 maxit = 20;
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196 endif
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197
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198 maxit += 2;
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199
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200 if (nargin < 3 || isempty (tol))
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201 tol = 1e-6;
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202 endif
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203
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204 if (nargin < 2)
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205 print_usage ();
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206 endif
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207
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208 ## init
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209 if (isnumeric (A)) # is A a matrix?
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210 r = b - A*x;
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211 else # then A should be a function!
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212 r = b - feval (A, x, varargin{:});
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213 endif
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214
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215 if (isnumeric (m)) # is M a matrix?
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216 if (isempty (m)) # if M is empty, use no precond
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217 p = r;
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218 else # otherwise, apply the precond
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219 p = m \ r;
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220 endif
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221 else # then M should be a function!
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222 p = feval (m, r, varargin{:});
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223 endif
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224
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225 iter = 2;
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226
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227 b_bot_old = 1;
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228 q_old = p_old = s_old = zeros (size (x));
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229
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230 if (isnumeric (A)) # is A a matrix?
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231 q = A * p;
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232 else # then A should be a function!
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233 q = feval (A, p, varargin{:});
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234 endif
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235
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236 resvec(1) = abs (norm (r));
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237
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238 ## iteration
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239 while (resvec(iter-1) > tol*resvec(1) && iter < maxit)
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240
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241 if (isnumeric (m)) # is M a matrix?
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242 if (isempty (m)) # if M is empty, use no precond
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243 s = q;
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244 else # otherwise, apply the precond
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245 s = m \ q;
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246 endif
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247 else # then M should be a function!
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248 s = feval (m, q, varargin{:});
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249 endif
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250 b_top = r' * s;
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251 b_bot = q' * s;
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252
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253 if (b_bot == 0.0)
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254 breakdown = true;
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255 break;
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256 endif
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257 lambda = b_top / b_bot;
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258
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259 x += lambda*p;
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260 r -= lambda*q;
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261
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262 if (isnumeric (A)) # is A a matrix?
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263 t = A*s;
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264 else # then A should be a function!
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265 t = feval (A, s, varargin{:});
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266 endif
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267
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268 alpha0 = (t'*s) / b_bot;
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269 alpha1 = (t'*s_old) / b_bot_old;
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270
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271 p_temp = p;
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272 q_temp = q;
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273
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274 p = s - alpha0*p - alpha1*p_old;
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275 q = t - alpha0*q - alpha1*q_old;
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276
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277 s_old = s;
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278 p_old = p_temp;
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279 q_old = q_temp;
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280 b_bot_old = b_bot;
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281
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282 resvec(iter) = abs (norm (r));
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283 iter += 1;
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284 endwhile
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285
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286 flag = 0;
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287 relres = resvec(iter-1) ./ resvec(1);
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288 iter -= 2;
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289 if (iter >= maxit-2)
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290 flag = 1;
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291 if (nargout < 2)
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292 warning ("pcr: maximum number of iterations (%d) reached\n", iter);
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293 warning ("pcr: the initial residual norm was reduced %g times\n",
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294 1.0/relres);
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295 endif
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296 elseif (nargout < 2 && ! breakdown)
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297 fprintf (stderr, "pcr: converged in %d iterations. \n", iter);
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298 fprintf (stderr, "pcr: the initial residual norm was reduced %g times\n",
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299 1.0 / relres);
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300 endif
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301
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302 if (breakdown)
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303 flag = 3;
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304 if (nargout < 2)
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305 warning ("pcr: breakdown occurred:\n");
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306 warning ("system matrix singular or preconditioner indefinite?\n");
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307 endif
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308 endif
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309
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310 endfunction
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311
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312
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313 %!demo
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314 %! ## Simplest usage of PCR (see also 'help pcr')
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315 %!
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316 %! N = 20;
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317 %! A = diag (linspace (-3.1,3,N)); b = rand (N,1);
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318 %! y = A \ b; # y is the true solution
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319 %! x = pcr (A,b);
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320 %! printf ("The solution relative error is %g\n", norm (x-y) / norm (y));
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321 %!
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322 %! ## You shouldn't be afraid if PCR issues some warning messages in this
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323 %! ## example: watch out in the second example, why it takes N iterations
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324 %! ## of PCR to converge to (a very accurate, by the way) solution.
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325
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326 %!demo
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327 %! ## Full output from PCR
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328 %! ## We use this output to plot the convergence history
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329 %!
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330 %! N = 20;
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331 %! A = diag (linspace (-3.1,30,N)); b = rand (N,1);
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332 %! X = A \ b; # X is the true solution
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333 %! [x, flag, relres, iter, resvec] = pcr (A,b);
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334 %! printf ("The solution relative error is %g\n", norm (x-X) / norm (X));
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335 %! clf;
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336 %! title ("Convergence history");
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337 %! xlabel ("Iteration"); ylabel ("log (||b-Ax||/||b||)");
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338 %! semilogy ([0:iter], resvec/resvec(1), "o-g;relative residual;");
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339
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340 %!demo
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341 %! ## Full output from PCR
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342 %! ## We use indefinite matrix based on the Hilbert matrix, with one
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343 %! ## strongly negative eigenvalue
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344 %! ## Hilbert matrix is extremely ill conditioned, so is ours,
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345 %! ## and that's why PCR WILL have problems
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346 %!
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347 %! N = 10;
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348 %! A = hilb (N); A(1,1) = -A(1,1); b = rand (N,1);
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349 %! X = A \ b; # X is the true solution
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350 %! printf ("Condition number of A is %g\n", cond (A));
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351 %! [x, flag, relres, iter, resvec] = pcr (A,b,[],200);
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352 %! if (flag == 3)
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353 %! printf ("PCR breakdown. System matrix is [close to] singular\n");
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354 %! endif
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355 %! clf;
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356 %! title ("Convergence history");
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357 %! xlabel ("Iteration"); ylabel ("log (||b-Ax||)");
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358 %! semilogy ([0:iter], resvec, "o-g;absolute residual;");
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359
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360 %!demo
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361 %! ## Full output from PCR
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362 %! ## We use an indefinite matrix based on the 1-D Laplacian matrix for A,
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363 %! ## and here we have cond (A) = O(N^2)
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364 %! ## That's the reason we need some preconditioner; here we take
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diff changeset
365 %! ## a very simple and not powerful Jacobi preconditioner,
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366 %! ## which is the diagonal of A.
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367 %!
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368 %! ## Note that we use here indefinite preconditioners!
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369 %!
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370 %! N = 100;
25755
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371 %! ## Form 1-D Laplacian matrix
ddf8a03beffb Calculate Laplacian matrix in BIST tests without for loop.
Rik <rik@octave.org>
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372 %! A = 2 * eye (N,N);
ddf8a03beffb Calculate Laplacian matrix in BIST tests without for loop.
Rik <rik@octave.org>
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373 %! A(2:(N+1):end) = -1;
ddf8a03beffb Calculate Laplacian matrix in BIST tests without for loop.
Rik <rik@octave.org>
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374 %! A((N+1):(N+1):end) = -1;
ddf8a03beffb Calculate Laplacian matrix in BIST tests without for loop.
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375 %!
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376 %! A = [A, zeros(size(A)); zeros(size(A)), -A];
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377 %! b = rand (2*N,1);
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378 %! X = A \ b; # X is the true solution
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379 %! maxit = 80;
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380 %! printf ("System condition number is %g\n", cond (A));
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381 %! ## No preconditioner: the convergence is very slow!
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382 %!
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383 %! [x, flag, relres, iter, resvec] = pcr (A,b,[],maxit);
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diff changeset
384 %! clf;
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385 %! title ("Convergence history");
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386 %! xlabel ("Iteration"); ylabel ("log (||b-Ax||)");
14363
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diff changeset
387 %! semilogy ([0:iter], resvec, "o-g;NO preconditioning: absolute residual;");
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388 %!
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diff changeset
389 %! pause (1);
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390 %! ## Test Jacobi preconditioner: it will not help much!!!
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391 %!
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392 %! M = diag (diag (A)); # Jacobi preconditioner
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diff changeset
393 %! [x, flag, relres, iter, resvec] = pcr (A,b,[],maxit,M);
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diff changeset
394 %! hold on;
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diff changeset
395 %! semilogy ([0:iter],resvec,"o-r;JACOBI preconditioner: absolute residual;");
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diff changeset
396 %!
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397 %! pause (1);
17336
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398 %! ## Test nonoverlapping block Jacobi preconditioner: this one should give
b81b9d079515 Use '##' for comments which stand alone on a line.
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diff changeset
399 %! ## some convergence speedup!
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400 %!
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401 %! M = zeros (N,N); k = 4;
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diff changeset
402 %! for i=1:k:N # get k x k diagonal blocks of A
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diff changeset
403 %! M(i:i+k-1,i:i+k-1) = A(i:i+k-1,i:i+k-1);
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
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diff changeset
404 %! endfor
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
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diff changeset
405 %! M = [M, zeros(size (M)); zeros(size(M)), -M];
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406 %! [x, flag, relres, iter, resvec] = pcr (A,b,[],maxit,M);
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
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diff changeset
407 %! semilogy ([0:iter], resvec, "o-b;BLOCK JACOBI preconditioner: absolute residual;");
f3d52523cde1 Use Octave coding conventions in all m-file %!test blocks
Rik <octave@nomad.inbox5.com>
parents: 14335
diff changeset
408 %! hold off;
14237
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
409
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55404f3b0da1 [project @ 2006-06-01 19:05:31 by jwe]
jwe
parents:
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410 %!test
17336
b81b9d079515 Use '##' for comments which stand alone on a line.
Rik <rik@octave.org>
parents: 14868
diff changeset
411 %! ## solve small indefinite diagonal system
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55404f3b0da1 [project @ 2006-06-01 19:05:31 by jwe]
jwe
parents:
diff changeset
412 %!
14237
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
413 %! N = 10;
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
414 %! A = diag (linspace (-10.1,10,N)); b = ones (N,1);
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
415 %! X = A \ b; # X is the true solution
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
416 %! [x, flag] = pcr (A,b,[],N+1);
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
417 %! assert (norm (x-X) / norm (X) < 1e-10);
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
418 %! assert (flag, 0);
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
419
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55404f3b0da1 [project @ 2006-06-01 19:05:31 by jwe]
jwe
parents:
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420 %!test
17336
b81b9d079515 Use '##' for comments which stand alone on a line.
Rik <rik@octave.org>
parents: 14868
diff changeset
421 %! ## solve tridiagonal system, do not converge in default 20 iterations
b81b9d079515 Use '##' for comments which stand alone on a line.
Rik <rik@octave.org>
parents: 14868
diff changeset
422 %! ## should perform max allowable default number of iterations
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jwe
parents:
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423 %!
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Rik <octave@nomad.inbox5.com>
parents: 14138
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424 %! N = 100;
25755
ddf8a03beffb Calculate Laplacian matrix in BIST tests without for loop.
Rik <rik@octave.org>
parents: 25472
diff changeset
425 %! ## Form 1-D Laplacian matrix
ddf8a03beffb Calculate Laplacian matrix in BIST tests without for loop.
Rik <rik@octave.org>
parents: 25472
diff changeset
426 %! A = 2 * eye (N,N);
ddf8a03beffb Calculate Laplacian matrix in BIST tests without for loop.
Rik <rik@octave.org>
parents: 25472
diff changeset
427 %! A(2:(N+1):end) = -1;
ddf8a03beffb Calculate Laplacian matrix in BIST tests without for loop.
Rik <rik@octave.org>
parents: 25472
diff changeset
428 %! A((N+1):(N+1):end) = -1;
14237
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
429 %! b = ones (N,1);
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
430 %! X = A \ b; # X is the true solution
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
431 %! [x, flag, relres, iter, resvec] = pcr (A,b,1e-12);
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
432 %! assert (flag, 1);
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
433 %! assert (relres > 0.6);
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
434 %! assert (iter, 20);
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
435
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
436 %!test
17336
b81b9d079515 Use '##' for comments which stand alone on a line.
Rik <rik@octave.org>
parents: 14868
diff changeset
437 %! ## solve tridiagonal system with "perfect" preconditioner
b81b9d079515 Use '##' for comments which stand alone on a line.
Rik <rik@octave.org>
parents: 14868
diff changeset
438 %! ## converges in one iteration
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jwe
parents:
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439 %!
14237
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Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
440 %! N = 100;
25755
ddf8a03beffb Calculate Laplacian matrix in BIST tests without for loop.
Rik <rik@octave.org>
parents: 25472
diff changeset
441 %! ## Form 1-D Laplacian matrix
ddf8a03beffb Calculate Laplacian matrix in BIST tests without for loop.
Rik <rik@octave.org>
parents: 25472
diff changeset
442 %! A = 2 * eye (N,N);
ddf8a03beffb Calculate Laplacian matrix in BIST tests without for loop.
Rik <rik@octave.org>
parents: 25472
diff changeset
443 %! A(2:(N+1):end) = -1;
ddf8a03beffb Calculate Laplacian matrix in BIST tests without for loop.
Rik <rik@octave.org>
parents: 25472
diff changeset
444 %! A((N+1):(N+1):end) = -1;
14237
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
445 %! b = ones (N,1);
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
446 %! X = A \ b; # X is the true solution
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
447 %! [x, flag, relres, iter] = pcr (A,b,[],[],A,b);
28912
0de38a6ef693 maint: Use Octave convention of space after function name in scripts dir.
Rik <rik@octave.org>
parents: 27984
diff changeset
448 %! assert (norm (x-X) / norm (X) < 1e-6);
14237
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
449 %! assert (relres < 1e-6);
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
450 %! assert (flag, 0);
11949c9795a0 Revamp %!demos in m-files to use Octave coding conventions on spacing, etc.
Rik <octave@nomad.inbox5.com>
parents: 14138
diff changeset
451 %! assert (iter, 1); # should converge in one iteration