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annotate scripts/sparse/bicgstab.m @ 13091:e5aaba072d2b

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maint: style fixes in iterative linear solvers * bicg.m, bicgstab.m, cgs.m, gmres.m: Style fixes.
author Carlo de Falco <kingcrimson@tiscali.it>
date Sun, 04 Sep 2011 01:21:43 +0200
parents a0d854f079d2
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} =} bicgstab (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{M1}, @var{M2}, @var{x0})
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23 ## @deftypefnx {Function File} {@var{x} =} bicgstab (@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}] =} bicgstab (@var{A}, @var{b}, ...)
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25 ## Solve @code{A x = b} using the stabilizied Bi-conjugate gradient iterative 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 ## @item @var{maxit} the maximum number of outer iterations, if not
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31 ## given or set to [] the default value @code{min (20, numel (b))} is
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32 ## used.
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33 ## @item @var{x0} the initial guess, if not given or set to [] the
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34 ## default value @code{zeros (size (b))} is used.
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35 ## @end itemize
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36 ##
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37 ## @var{A} can be passed as a matrix or as a function handle or
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38 ## inline function @code{f} such that @code{f(x) = A*x}.
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39 ##
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40 ## The preconditioner @var{P} is given as @code{P = M1 * M2}.
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41 ## Both @var{M1} and @var{M2} can be passed as a matrix or as a function handle or
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42 ## inline function @code{g} such that @code{g(x) = M1 \ x} or @code{g(x) = M2 \ x}.
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43 ##
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44 ## If called with more than one output parameter
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45 ##
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46 ## @itemize @minus
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47 ## @item @var{flag} indicates the exit status:
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48 ## @itemize @minus
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49 ## @item 0: iteration converged to the within the chosen tolerance
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50 ## @item 1: the maximum number of iterations was reached before convergence
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51 ## @item 3: the algorithm reached stagnation
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52 ## @end itemize
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53 ## (the value 2 is unused but skipped for compatibility).
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54 ## @item @var{relres} is the final value of the relative residual.
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55 ## @item @var{iter} is the number of iterations performed.
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56 ## @item @var{resvec} is a vector containing the relative residual at each iteration.
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57 ## @end itemize
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58 ##
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59 ## @seealso{bicg,cgs,gmres,pcg}
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60 ##
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61 ## @end deftypefn
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62
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63 function [x, flag, relres, iter, resvec] = bicgstab (A, b, tol, maxit,
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64 M1, M2, x0)
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65
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66 if (nargin >= 2 && nargin <= 7 && isvector (full (b)))
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67
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68 if (ischar (A))
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69 A = str2func (A);
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70 elseif (ismatrix (A))
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71 Ax = @(x) A * x;
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72 elseif (isa (A, "function_handle"))
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73 Ax = @(x) feval (A, x);
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74 else
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75 error (["bicgstab: first argument is expected " ...
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76 "to be a function or a square matrix"]);
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77 endif
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78
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79 if (nargin < 3 || isempty (tol))
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80 tol = 1e-6;
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81 endif
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82
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83 if (nargin < 4 || isempty (maxit))
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84 maxit = min (rows (b), 20);
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85 endif
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86
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87 if (nargin < 5 || isempty (M1))
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88 M1m1x = @(x) x;
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89 elseif (ischar (M1))
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90 M1m1x = str2func (M1);
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91 elseif (ismatrix (M1))
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92 M1m1x = @(x) M1 \ x;
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93 elseif (isa (M1, "function_handle"))
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94 M1m1x = @(x) feval (M1, x);
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95 else
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96 error (["bicgstab: preconditioner is " ...
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97 "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 (["bicgstab: preconditioner is "...
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110 "expected to be a function or matrix"]);
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111 endif
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112
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113 precon = @(x) M2m1x (M1m1x (x));
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114
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115 if (nargin < 7 || isempty (x0))
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116 x0 = zeros (size (b));
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117 endif
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118
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119 ## specifies initial estimate x0
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120 if (nargin < 7)
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121 x = zeros (rows (b), 1);
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122 else
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123 x = x0;
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124 endif
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125
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126 norm_b = norm (b);
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127
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128 res = b - Ax (x);
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129 rr = res;
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130
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131 ## Vector of the residual norms for each iteration.
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132 resvec = norm(res) / norm_b;
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133
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134 ## Default behaviour we don't reach tolerance tol within maxit iterations.
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135 flag = 1;
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136
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137 for iter = 1:maxit
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138 rho_1 = res' * rr;
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139
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140 if (iter == 1)
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141 p = res;
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142 else
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143 beta = (rho_1 / rho_2) * (alpha / omega);
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144 p = res + beta * (p - omega * v);
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145 endif
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146
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147 phat = precon (p);
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148
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149 v = Ax (phat);
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150 alpha = rho_1 / (rr' * v);
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151 s = res - alpha * v;
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152
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153 shat = precon (s);
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154
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155 t = Ax (shat);
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156 omega = (t' * s) / (t' * t);
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157 x = x + alpha * phat + omega * shat;
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158 res = s - omega * t;
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159 rho_2 = rho_1;
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160
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161 relres = norm (res) / norm_b;
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162 resvec = [resvec; relres];
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163
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164 if (relres <= tol)
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165 ## We reach tolerance tol within maxit iterations.
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166 flag = 0;
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167 break;
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168 elseif (resvec(end) == resvec(end - 1))
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169 ## The method stagnates.
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170 flag = 3;
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171 break;
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172 endif
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173 endfor
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174
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175 if (nargout < 2)
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176 if (flag == 0)
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177 printf ("bicgstab converged at iteration %i ", iter);
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178 printf ("to a solution with relative residual %e\n", relres);
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179 elseif (flag == 3)
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180 printf ("bicgstab stopped at iteration %i ", iter);
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181 printf ("without converging to the desired tolerance %e\n", tol);
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182 printf ("because the method stagnated.\n");
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183 printf ("The iterate returned (number %i) ", iter);
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184 printf ("has relative residual %e\n", relres);
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185 else
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186 printf ("bicgstab stopped at iteration %i ", iter);
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187 printf ("without converging to the desired toleranc %e\n", tol);
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188 printf ("because the maximum number of iterations was reached.\n");
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189 printf ("The iterate returned (number %i) ", iter);
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190 printf ("has relative residual %e\n", relres);
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191 endif
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192 endif
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193
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194 else
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195 print_usage ();
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196 endif
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197
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198 endfunction
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199
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200 %!demo
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201 %! % Solve system of A*x=b
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202 %! A = [5 -1 3;-1 2 -2;3 -2 3]
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203 %! b = [7;-1;4]
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204 %! [x, flag, relres, iter, resvec] = bicgstab(A, b)
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205
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206 %!shared A, b, n, M1, M2
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207 %!
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208 %!test
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209 %! n = 100;
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210 %! A = spdiags ([-2*ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n);
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211 %! b = sum (A, 2);
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212 %! tol = 1e-8;
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213 %! maxit = 15;
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214 %! M1 = spdiags ([ones(n,1)/(-2) ones(n,1)],-1:0, n, n);
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215 %! M2 = spdiags ([4*ones(n,1) -ones(n,1)], 0:1, n, n);
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216 %! [x, flag, relres, iter, resvec] = bicgstab (A, b, tol, maxit, M1, M2);
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217 %! assert (x, ones (size (b)), 1e-7);
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218 %!
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219 %!test
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220 %! tol = 1e-8;
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221 %! maxit = 15;
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222 %!
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223 %! function y = afun (x, a)
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224 %! y = a * x;
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225 %! endfunction
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226 %!
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227 %! [x, flag, relres, iter, resvec] = bicgstab (@(x) afun (x, A), b,
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228 %! tol, maxit, M1, M2);
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229 %! assert (x, ones (size (b)), 1e-7);
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230
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231 %!test
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232 %! n = 100;
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233 %! tol = 1e-8;
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234 %! a = sprand (n, n, .1);
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235 %! A = a'*a + 100 * eye (n);
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236 %! b = sum (A, 2);
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237 %! [x, flag, relres, iter, resvec] = bicgstab (A, b, tol, [], diag (diag (A)));
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238 %! assert (x, ones (size (b)), 1e-7);