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maint: style fixes for cgs * cgs.m: Style fixes.
author John W. Eaton <jwe@octave.org>
date Sat, 03 Sep 2011 16:32:44 -0400
parents 59def8890992
children e81ddf9cacd5
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1 ## Copyright (C) 2008-2011 Radek Salac
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2 ## Copyright (C) 2011 Carlo de Falco
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3 ##
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4 ## This file is part of Octave.
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5 ##
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6 ## Octave is free software; you can redistribute it and/or modify it
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7 ## under the terms of the GNU General Public License as published by
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8 ## the Free Software Foundation; either version 3 of the License, or (at
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9 ## your option) any later version.
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10 ##
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11 ## Octave is distributed in the hope that it will be useful, but
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12 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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13 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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14 ## General Public License for more details.
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15 ##
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16 ## You should have received a copy of the GNU General Public License
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17 ## along with Octave; see the file COPYING. If not, see
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18 ## <http://www.gnu.org/licenses/>.
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19
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20 ## -*- texinfo -*-
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21 ##
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22 ## @deftypefn {Function File} {@var{x} =} cgs (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{M1}, @var{M2}, @var{x0})
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23 ## @deftypefnx {Function File} {@var{x} =} cgs (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{P})
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24 ## @deftypefnx {Function File} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}] =} cgs (@var{A}, @var{b}, ...)
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25 ## Solve @code{A x = b}, where @var{A} is a square matrix, using the
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26 ## Conjugate Gradients Squared method.
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27 ##
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28 ## @itemize @minus
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29 ## @item @var{rtol} is the relative tolerance, if not given or set to []
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30 ## the default value 1e-6 is used.
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31 ## @item @var{maxit} the maximum number of outer iterations, if not
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32 ## given or set to [] the default value @code{min (20, numel (b))} is
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33 ## used.
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34 ## @item @var{x0} the initial guess, if not given or set to [] the
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35 ## default value @code{zeros (size (b))} is used.
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36 ## @end itemize
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37 ##
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38 ## @var{A} can be passed as a matrix or as a function handle or
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39 ## inline function @code{f} such that @code{f(x) = A*x}.
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40 ##
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41 ## The preconditioner @var{P} is given as @code{P = M1 * M2}.
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42 ## Both @var{M1} and @var{M2} can be passed as a matrix or as a function
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43 ## handle or inline function @code{g} such that @code{g(x) = M1 \ x} or
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44 ## @code{g(x) = M2 \ x}.
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45 ##
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46 ## If called with more than one output parameter
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47 ##
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48 ## @itemize @minus
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49 ## @item @var{flag} indicates the exit status:
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50 ## @itemize @minus
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51 ## @item 0: iteration converged to the within the chosen tolerance
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52 ## @item 1: the maximum number of iterations was reached before convergence
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53 ## @item 3: the algorithm reached stagnation
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54 ## @end itemize
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55 ## (the value 2 is unused but skipped for compatibility).
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56 ## @item @var{relres} is the final value of the relative residual.
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57 ## @item @var{iter} is the number of iterations performed.
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58 ## @item @var{resvec} is a vector containing the relative residual at
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59 ## each iteration.
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60 ## @end itemize
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61 ##
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62 ## @seealso{pcg,bicgstab,bicg,gmres}
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63 ## @end deftypefn
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64
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65 function [x, flag, relres, iter, resvec] = cgs (A, b, tol, maxit, M1, M2, x0)
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66
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67 if (nargin >= 2 && nargin <= 7 && isvector (full (b)))
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68
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69 if (ischar (A))
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70 A = str2func (A);
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71 elseif (ismatrix (A))
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72 Ax = @(x) A * x;
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73 elseif (isa (A, "function_handle"))
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74 Ax = @(x) feval (A, x);
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75 else
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76 error (["cgs: first argument is expected to "...
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77 "be a function or a square matrix"]);
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78 endif
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79
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80 if (nargin < 3 || isempty (tol))
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81 tol = 1e-6;
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82 endif
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83
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84 if (nargin < 4 || isempty (maxit))
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85 maxit = min (rows (b), 20);
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86 endif
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87
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88 if (nargin < 5 || isempty (M1))
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89 M1m1x = @(x) x;
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90 elseif (ischar (M1))
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91 M1m1x = str2func (M1);
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92 elseif (ismatrix (M1))
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93 M1m1x = @(x) M1 \ x;
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94 elseif (isa (M1, "function_handle"))
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95 M1m1x = @(x) feval (M1, x);
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96 else
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97 error ("cgs: preconditioner is expected to be a function or matrix");
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98 endif
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99
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100 if (nargin < 6 || isempty (M2))
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101 M2m1x = @(x) x;
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102 elseif (ischar (M2))
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103 M2m1x = str2func (M2);
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104 elseif (ismatrix (M2))
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105 M2m1x = @(x) M2 \ x;
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106 elseif (isa (M2, "function_handle"))
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107 M2m1x = @(x) feval (M2, x);
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108 else
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109 error ("cgs: preconditioner is expected to be a function or matrix");
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110 endif
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111
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112 precon = @(x) M2m1x (M1m1x (x));
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113
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114 if (nargin < 7 || isempty (x0))
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115 x0 = zeros (size (b));
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116 endif
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117
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118
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119 x = x0;
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120
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121 res = b - Ax (x);
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122 norm_b = norm (b);
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123 ## Vector of the residual norms for each iteration.
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124 resvec = norm (res) / norm_b;
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125 ro = 0;
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126 ## Default behavior we don't reach tolerance tol within maxit iterations.
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127 flag = 1;
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128 for iter = 1:maxit
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129
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130 z = precon (res);
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131
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132 ## Cache.
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133 ro_old = ro;
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134 ro = res' * z;
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135 if (iter == 1)
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136 p = z;
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137 else
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138 beta = ro / ro_old;
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139 p = z + beta * p;
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140 endif
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141 ## Cache.
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142 q = Ax (p);
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143 alpha = ro / (p' * q);
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144 x = x + alpha * p;
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145
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146 res = res - alpha * q;
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147 relres = norm (res) / norm_b;
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148 resvec = [resvec; relres];
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149
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150 if (relres <= tol)
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151 ## We reach tolerance tol within maxit iterations.
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152 flag = 0;
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153 break
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154 elseif (resvec (end) == resvec (end - 1))
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155 ## The method stagnates.
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156 flag = 3;
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157 break
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158 endif
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159 endfor
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160
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161 if (nargout < 1)
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162 if (flag == 0)
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163 printf ("cgs converged at iteration %i to a solution with relative residual %e\n",
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164 iter, relres);
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165 elseif (flag == 3)
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166 printf (["cgs stopped at iteration %i without converging to the desired tolerance %e\n",
13063
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167 "because the method stagnated.\n",
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168 "The iterate returned (number %i) has relative residual %e\n"],
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169 iter, tol, iter, relres);
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170 else
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171 printf (["cgs stopped at iteration %i without converging to the desired tolerance %e\n",
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172 "because the maximum number of iterations was reached.\n",
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173 "The iterate returned (number %i) has relative residual %e\n"],
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174 iter, tol, iter, relres);
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175 endif
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176 endif
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177
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178 else
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179 print_usage ();
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180 endif
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181
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182 endfunction
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183
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184
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185
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186 %!demo
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187 %! % Solve system of A*x=b
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188 %! A=[5 -1 3;-1 2 -2;3 -2 3]
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189 %! b=[7;-1;4]
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190 %! [a,b,c,d,e]=cgs(A,b)
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191
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192 %!shared A, b, n, M
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193 %!
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194 %!test
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195 %! n = 100;
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196 %! A = spdiags ([-ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n);
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197 %! b = sum (A, 2);
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198 %! tol = 1e-8;
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199 %! maxit = 1000;
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200 %! M = 4*eye (n);
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201 %! [x, flag, relres, iter, resvec] = cgs (A, b, tol, maxit, M);
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202 %! assert (x, ones (size (b)), 1e-7);
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203 %!
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204 %!test
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205 %! tol = 1e-8;
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206 %! maxit = 15;
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207 %!
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208 %! [x, flag, relres, iter, resvec] = cgs (@(x) A * x, b, tol, maxit, M);
13063
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209 %! assert (x, ones (size (b)), 1e-7);
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210
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diff changeset
211 %!test
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212 %! n = 100;
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213 %! tol = 1e-8;
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214 %! a = sprand (n, n, .1);
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215 %! A = a'*a + 100 * eye (n);
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diff changeset
216 %! b = sum (A, 2);
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diff changeset
217 %! [x, flag, relres, iter, resvec] = cgs (A, b, tol, [], diag (diag (A)));
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diff changeset
218 %! assert (x, ones (size (b)), 1e-7);