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maint: update copyright notices for 2012
author John W. Eaton <jwe@octave.org>
date Mon, 02 Jan 2012 14:25:41 -0500
parents 9cae456085c2
children 11949c9795a0
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1 ## Copyright (C) 2008-2012 Radek Salac
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2 ## Copyright (C) 2012 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}, @dots{})
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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 ##
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32 ## @item @var{maxit} the maximum number of outer iterations, if not
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33 ## given or set to [] the default value @code{min (20, numel (b))} is
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34 ## used.
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35 ##
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36 ## @item @var{x0} the initial guess, if not given or set to [] the
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37 ## default value @code{zeros (size (b))} is used.
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38 ## @end itemize
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39 ##
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40 ## @var{A} can be passed as a matrix or as a function handle or
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41 ## inline function @code{f} such that @code{f(x) = A*x}.
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42 ##
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43 ## The preconditioner @var{P} is given as @code{P = M1 * M2}.
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44 ## Both @var{M1} and @var{M2} can be passed as a matrix or as a function
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45 ## handle or inline function @code{g} such that @code{g(x) = M1 \ x} or
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46 ## @code{g(x) = M2 \ x}.
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47 ##
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48 ## If called with more than one output parameter
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49 ##
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50 ## @itemize @minus
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51 ## @item @var{flag} indicates the exit status:
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52 ## @itemize @minus
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53 ## @item 0: iteration converged to the within the chosen tolerance
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54 ##
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55 ## @item 1: the maximum number of iterations was reached before convergence
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56 ##
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57 ## @item 3: the algorithm reached stagnation
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58 ## @end itemize
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59 ## (the value 2 is unused but skipped for compatibility).
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60 ##
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61 ## @item @var{relres} is the final value of the relative residual.
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62 ##
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63 ## @item @var{iter} is the number of iterations performed.
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64 ##
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65 ## @item @var{resvec} is a vector containing the relative residual at
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66 ## each iteration.
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67 ## @end itemize
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68 ##
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69 ## @seealso{pcg, bicgstab, bicg, gmres}
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70 ## @end deftypefn
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71
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72 function [x, flag, relres, iter, resvec] = cgs (A, b, tol, maxit, M1, M2, x0)
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73
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74 if (nargin >= 2 && nargin <= 7 && isvector (full (b)))
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75
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76 if (ischar (A))
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77 A = str2func (A);
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78 elseif (ismatrix (A))
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79 Ax = @(x) A * x;
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80 elseif (isa (A, "function_handle"))
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81 Ax = @(x) feval (A, x);
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82 else
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83 error (["cgs: first argument is expected to "...
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84 "be a function or a square matrix"]);
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85 endif
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86
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87 if (nargin < 3 || isempty (tol))
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88 tol = 1e-6;
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89 endif
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90
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91 if (nargin < 4 || isempty (maxit))
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92 maxit = min (rows (b), 20);
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93 endif
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94
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95 if (nargin < 5 || isempty (M1))
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96 M1m1x = @(x) x;
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97 elseif (ischar (M1))
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98 M1m1x = str2func (M1);
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99 elseif (ismatrix (M1))
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100 M1m1x = @(x) M1 \ x;
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101 elseif (isa (M1, "function_handle"))
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102 M1m1x = @(x) feval (M1, x);
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103 else
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104 error ("cgs: preconditioner is expected to be a function or matrix");
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105 endif
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106
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107 if (nargin < 6 || isempty (M2))
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108 M2m1x = @(x) x;
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109 elseif (ischar (M2))
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110 M2m1x = str2func (M2);
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111 elseif (ismatrix (M2))
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112 M2m1x = @(x) M2 \ x;
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113 elseif (isa (M2, "function_handle"))
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114 M2m1x = @(x) feval (M2, x);
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115 else
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116 error ("cgs: preconditioner is expected to be a function or matrix");
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117 endif
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118
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119 precon = @(x) M2m1x (M1m1x (x));
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120
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121 if (nargin < 7 || isempty (x0))
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122 x0 = zeros (size (b));
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123 endif
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124
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125
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126 x = x0;
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127
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128 res = b - Ax (x);
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129 norm_b = norm (b);
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130 ## Vector of the residual norms for each iteration.
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131 resvec = norm (res) / norm_b;
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132 ro = 0;
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133 ## Default behavior we don't reach tolerance tol within maxit iterations.
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134 flag = 1;
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135 for iter = 1:maxit
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136
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137 z = precon (res);
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138
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139 ## Cache.
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140 ro_old = ro;
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141 ro = res' * z;
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142 if (iter == 1)
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143 p = z;
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144 else
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145 beta = ro / ro_old;
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146 p = z + beta * p;
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147 endif
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148 ## Cache.
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149 q = Ax (p);
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150 alpha = ro / (p' * q);
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151 x = x + alpha * p;
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152
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153 res = res - alpha * q;
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154 relres = norm (res) / norm_b;
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155 resvec = [resvec; relres];
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156
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157 if (relres <= tol)
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158 ## We reach tolerance tol within maxit iterations.
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159 flag = 0;
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160 break
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161 elseif (resvec (end) == resvec (end - 1))
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162 ## The method stagnates.
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163 flag = 3;
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164 break
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165 endif
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166 endfor
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167
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168 if (nargout < 1)
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169 if (flag == 0)
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170 printf ("cgs converged at iteration %i to a solution with relative residual %e\n",
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171 iter, relres);
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172 elseif (flag == 3)
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173 printf (["cgs stopped at iteration %i without converging to the desired tolerance %e\n",
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174 "because the method stagnated.\n",
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175 "The iterate returned (number %i) has relative residual %e\n"],
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176 iter, tol, iter, relres);
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177 else
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178 printf (["cgs stopped at iteration %i without converging to the desired tolerance %e\n",
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179 "because the maximum number of iterations was reached.\n",
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180 "The iterate returned (number %i) has relative residual %e\n"],
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181 iter, tol, iter, relres);
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182 endif
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183 endif
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184
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185 else
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186 print_usage ();
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187 endif
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188
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189 endfunction
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190
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191
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192
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193 %!demo
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194 %! % Solve system of A*x=b
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195 %! A=[5 -1 3;-1 2 -2;3 -2 3]
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196 %! b=[7;-1;4]
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197 %! [a,b,c,d,e]=cgs(A,b)
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198
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199 %!shared A, b, n, M
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200 %!
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201 %!test
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202 %! n = 100;
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203 %! A = spdiags ([-ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n);
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204 %! b = sum (A, 2);
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205 %! tol = 1e-8;
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206 %! maxit = 1000;
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207 %! M = 4*eye (n);
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208 %! [x, flag, relres, iter, resvec] = cgs (A, b, tol, maxit, M);
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209 %! assert (x, ones (size (b)), 1e-7);
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210 %!
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211 %!test
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212 %! tol = 1e-8;
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213 %! maxit = 15;
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214 %!
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215 %! [x, flag, relres, iter, resvec] = cgs (@(x) A * x, b, tol, maxit, M);
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216 %! assert (x, ones (size (b)), 1e-7);
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217
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218 %!test
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219 %! n = 100;
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220 %! tol = 1e-8;
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221 %! a = sprand (n, n, .1);
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222 %! A = a'*a + 100 * eye (n);
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223 %! b = sum (A, 2);
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224 %! [x, flag, relres, iter, resvec] = cgs (A, b, tol, [], diag (diag (A)));
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225 %! assert (x, ones (size (b)), 1e-7);