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doc: Remove unnecessary blank line after start of Texinfo section. * bicg.m, bicgstab.m, cgs.m: Remove blank line after Texinfo start.
author Rik <octave@nomad.inbox5.com>
date Tue, 08 May 2012 18:02:24 -0700
parents f3d52523cde1
children 72b8b39e12be
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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 ## @deftypefn {Function File} {@var{x} =} bicgstab (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{M1}, @var{M2}, @var{x0})
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22 ## @deftypefnx {Function File} {@var{x} =} bicgstab (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{P})
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23 ## @deftypefnx {Function File} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}] =} bicgstab (@var{A}, @var{b}, @dots{})
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24 ## Solve @code{A x = b} using the stabilizied Bi-conjugate gradient iterative
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25 ## method.
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26 ##
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27 ## @itemize @minus
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28 ## @item @var{rtol} is the relative tolerance, if not given or set to
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29 ## [] the default value 1e-6 is used.
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30 ##
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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 ##
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35 ## @item @var{x0} the initial guess, if not given or set to [] the
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36 ## default value @code{zeros (size (b))} is used.
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37 ## @end itemize
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38 ##
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39 ## @var{A} can be passed as a matrix or as a function handle or
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40 ## inline function @code{f} such that @code{f(x) = A*x}.
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41 ##
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42 ## The preconditioner @var{P} is given as @code{P = M1 * M2}.
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43 ## Both @var{M1} and @var{M2} can be passed as a matrix or as a function
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44 ## handle or inline function @code{g} such that @code{g(x) = M1 \ x} or
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45 ## @code{g(x) = M2 \ x}.
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46 ##
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47 ## If called with more than one output parameter
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48 ##
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49 ## @itemize @minus
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50 ## @item @var{flag} indicates the exit status:
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51 ## @itemize @minus
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52 ## @item 0: iteration converged to the within the chosen tolerance
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53 ##
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54 ## @item 1: the maximum number of iterations was reached before convergence
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55 ##
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56 ## @item 3: the algorithm reached stagnation
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57 ## @end itemize
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58 ## (the value 2 is unused but skipped for compatibility).
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59 ##
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60 ## @item @var{relres} is the final value of the relative residual.
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61 ##
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62 ## @item @var{iter} is the number of iterations performed.
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63 ##
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64 ## @item @var{resvec} is a vector containing the relative residual at each iteration.
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65 ## @end itemize
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66 ##
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67 ## @seealso{bicg, cgs, gmres, pcg}
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68 ##
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69 ## @end deftypefn
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70
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71 function [x, flag, relres, iter, resvec] = bicgstab (A, b, tol, maxit,
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72 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 (["bicgstab: first argument is expected " ...
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84 "to 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 (["bicgstab: preconditioner is " ...
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105 "expected to be a function or matrix"]);
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106 endif
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107
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108 if (nargin < 6 || isempty (M2))
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109 M2m1x = @(x) x;
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110 elseif (ischar (M2))
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111 M2m1x = str2func (M2);
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112 elseif (ismatrix (M2))
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113 M2m1x = @(x) M2 \ x;
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114 elseif (isa (M2, "function_handle"))
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115 M2m1x = @(x) feval (M2, x);
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116 else
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117 error (["bicgstab: preconditioner is "...
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118 "expected to be a function or matrix"]);
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119 endif
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120
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121 precon = @(x) M2m1x (M1m1x (x));
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122
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123 if (nargin < 7 || isempty (x0))
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124 x0 = zeros (size (b));
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125 endif
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126
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127 ## specifies initial estimate x0
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128 if (nargin < 7)
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129 x = zeros (rows (b), 1);
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130 else
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131 x = x0;
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132 endif
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133
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134 norm_b = norm (b);
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135
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136 res = b - Ax (x);
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137 rr = res;
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138
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139 ## Vector of the residual norms for each iteration.
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140 resvec = norm(res) / norm_b;
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141
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142 ## Default behaviour we don't reach tolerance tol within maxit iterations.
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143 flag = 1;
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144
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145 for iter = 1:maxit
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146 rho_1 = res' * rr;
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147
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148 if (iter == 1)
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149 p = res;
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150 else
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151 beta = (rho_1 / rho_2) * (alpha / omega);
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152 p = res + beta * (p - omega * v);
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153 endif
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154
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155 phat = precon (p);
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156
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157 v = Ax (phat);
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158 alpha = rho_1 / (rr' * v);
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159 s = res - alpha * v;
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160
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161 shat = precon (s);
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162
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163 t = Ax (shat);
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164 omega = (t' * s) / (t' * t);
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165 x = x + alpha * phat + omega * shat;
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166 res = s - omega * t;
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167 rho_2 = rho_1;
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168
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169 relres = norm (res) / norm_b;
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170 resvec = [resvec; relres];
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171
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172 if (relres <= tol)
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173 ## We reach tolerance tol within maxit iterations.
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174 flag = 0;
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175 break;
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176 elseif (resvec(end) == resvec(end - 1))
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177 ## The method stagnates.
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178 flag = 3;
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179 break;
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180 endif
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181 endfor
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182
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183 if (nargout < 2)
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184 if (flag == 0)
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185 printf ("bicgstab converged at iteration %i ", iter);
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186 printf ("to a solution with relative residual %e\n", relres);
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187 elseif (flag == 3)
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188 printf ("bicgstab stopped at iteration %i ", iter);
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189 printf ("without converging to the desired tolerance %e\n", tol);
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190 printf ("because the method stagnated.\n");
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191 printf ("The iterate returned (number %i) ", iter);
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192 printf ("has relative residual %e\n", relres);
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193 else
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194 printf ("bicgstab stopped at iteration %i ", iter);
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195 printf ("without converging to the desired toleranc %e\n", tol);
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196 printf ("because the maximum number of iterations was reached.\n");
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197 printf ("The iterate returned (number %i) ", iter);
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198 printf ("has relative residual %e\n", relres);
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199 endif
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200 endif
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201
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202 else
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203 print_usage ();
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204 endif
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205
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206 endfunction
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207
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208
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209 %!demo
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210 %! % Solve system of A*x=b
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211 %! A = [5 -1 3;-1 2 -2;3 -2 3];
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212 %! b = [7;-1;4];
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213 %! [x, flag, relres, iter, resvec] = bicgstab (A, b)
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214
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215 %!shared A, b, n, M1, M2
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216 %!
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217 %!test
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218 %! n = 100;
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219 %! A = spdiags ([-2*ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n);
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220 %! b = sum (A, 2);
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221 %! tol = 1e-8;
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222 %! maxit = 15;
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223 %! M1 = spdiags ([ones(n,1)/(-2) ones(n,1)],-1:0, n, n);
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224 %! M2 = spdiags ([4*ones(n,1) -ones(n,1)], 0:1, n, n);
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225 %! [x, flag, relres, iter, resvec] = bicgstab (A, b, tol, maxit, M1, M2);
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226 %! assert (x, ones (size (b)), 1e-7);
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227 %!
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228 %!test
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229 %!function y = afun (x, a)
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230 %! y = a * x;
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231 %!endfunction
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232 %!
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233 %! tol = 1e-8;
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234 %! maxit = 15;
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235 %!
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236 %! [x, flag, relres, iter, resvec] = bicgstab (@(x) afun (x, A), b,
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237 %! tol, maxit, M1, M2);
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238 %! assert (x, ones (size (b)), 1e-7);
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239
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240 %!test
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241 %! n = 100;
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242 %! tol = 1e-8;
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243 %! a = sprand (n, n, .1);
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244 %! A = a'*a + 100 * eye (n);
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245 %! b = sum (A, 2);
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246 %! [x, flag, relres, iter, resvec] = bicgstab (A, b, tol, [], diag (diag (A)));
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247 %! assert (x, ones (size (b)), 1e-7);
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248