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annotate scripts/sparse/bicgstab.m @ 19854:0a3ca546d7fc

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doc: Add qmr to documentation. * linalg.txi: Add DOCSTRING entry for qmr in specialized solvers. bicg.m, bicgstab.m, cgs.m, gmres.m, qmr.m: Add qmr to seealso links.
author Rik <rik@octave.org>
date Tue, 24 Feb 2015 20:50:59 -0800
parents 9fc020886ae9
children df437a52bcaf
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1 ## Copyright (C) 2008-2015 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 ##
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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 ##
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60 ## (the value 2 is unused but skipped for compatibility).
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61 ##
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62 ## @item @var{relres} is the final value of the relative residual.
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63 ##
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64 ## @item @var{iter} is the number of iterations performed.
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65 ##
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66 ## @item @var{resvec} is a vector containing the relative residual at each
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67 ## iteration.
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68 ## @end itemize
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69 ##
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70 ## @seealso{bicg, cgs, gmres, pcg, qmr}
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71 ##
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72 ## @end deftypefn
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73
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74 function [x, flag, relres, iter, resvec] = bicgstab (A, b, tol, maxit,
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75 M1, M2, x0)
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76
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77 if (nargin >= 2 && nargin <= 7 && isvector (full (b)))
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78
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79 if (ischar (A))
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80 A = str2func (A);
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81 elseif (isnumeric(A) && ismatrix (A))
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82 Ax = @(x) A * x;
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83 elseif (isa (A, "function_handle"))
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84 Ax = @(x) feval (A, x);
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85 else
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86 error (["bicgstab: first argument is expected " ...
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87 "to be a function or a square matrix"]);
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88 endif
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89
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90 if (nargin < 3 || isempty (tol))
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91 tol = 1e-6;
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92 endif
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93
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94 if (nargin < 4 || isempty (maxit))
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95 maxit = min (rows (b), 20);
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96 endif
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97
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98 if (nargin < 5 || isempty (M1))
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99 M1m1x = @(x) x;
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100 elseif (ischar (M1))
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101 M1m1x = str2func (M1);
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102 elseif (isnumeric(M1) && ismatrix (M1))
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103 M1m1x = @(x) M1 \ x;
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104 elseif (isa (M1, "function_handle"))
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105 M1m1x = @(x) feval (M1, x);
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106 else
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107 error (["bicgstab: preconditioner is " ...
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108 "expected to be a function or matrix"]);
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109 endif
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110
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111 if (nargin < 6 || isempty (M2))
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112 M2m1x = @(x) x;
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113 elseif (ischar (M2))
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114 M2m1x = str2func (M2);
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115 elseif (isnumeric(M2) && ismatrix (M2))
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116 M2m1x = @(x) M2 \ x;
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117 elseif (isa (M2, "function_handle"))
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118 M2m1x = @(x) feval (M2, x);
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119 else
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120 error (["bicgstab: preconditioner is "...
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121 "expected to be a function or matrix"]);
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122 endif
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123
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124 precon = @(x) M2m1x (M1m1x (x));
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125
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126 if (nargin < 7 || isempty (x0))
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127 x0 = zeros (size (b));
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128 endif
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129
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130 ## specifies initial estimate x0
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131 if (nargin < 7)
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132 x = zeros (rows (b), 1);
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133 else
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134 x = x0;
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135 endif
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136
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137 norm_b = norm (b);
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138
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139 res = b - Ax (x);
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140 rr = res;
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141
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142 ## Vector of the residual norms for each iteration.
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143 resvec = norm (res) / norm_b;
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144
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145 ## Default behaviour we don't reach tolerance tol within maxit iterations.
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146 flag = 1;
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147
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148 for iter = 1:maxit
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149 rho_1 = rr' * res;
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150
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151 if (iter == 1)
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152 p = res;
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153 else
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154 beta = (rho_1 / rho_2) * (alpha / omega);
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155 p = res + beta * (p - omega * v);
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156 endif
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157
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158 phat = precon (p);
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159
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160 v = Ax (phat);
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161 alpha = rho_1 / (rr' * v);
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162 s = res - alpha * v;
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163
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164 shat = precon (s);
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165
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166 t = Ax (shat);
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167 omega = (s' * t) / (t' * t);
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168 x = x + alpha * phat + omega * shat;
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169 res = s - omega * t;
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170 rho_2 = rho_1;
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171
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172 relres = norm (res) / norm_b;
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173 resvec = [resvec; relres];
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174
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175 if (relres <= tol)
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176 ## We reach tolerance tol within maxit iterations.
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177 flag = 0;
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178 break;
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179 elseif (resvec(end) == resvec(end - 1))
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180 ## The method stagnates.
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181 flag = 3;
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182 break;
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183 endif
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184 endfor
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185
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186 if (nargout < 2)
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187 if (flag == 0)
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188 printf ("bicgstab converged at iteration %i ", iter);
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189 printf ("to a solution with relative residual %e\n", relres);
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190 elseif (flag == 3)
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191 printf ("bicgstab stopped at iteration %i ", iter);
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192 printf ("without converging to the desired tolerance %e\n", tol);
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193 printf ("because the method stagnated.\n");
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194 printf ("The iterate returned (number %i) ", iter);
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195 printf ("has relative residual %e\n", relres);
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196 else
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197 printf ("bicgstab stopped at iteration %i ", iter);
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198 printf ("without converging to the desired toleranc %e\n", tol);
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199 printf ("because the maximum number of iterations was reached.\n");
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200 printf ("The iterate returned (number %i) ", iter);
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201 printf ("has relative residual %e\n", relres);
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202 endif
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203 endif
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204
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205 else
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206 print_usage ();
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207 endif
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diff changeset
208
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209 endfunction
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210
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211
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212 %!demo
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213 %! % Solve system of A*x=b
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214 %! A = [5 -1 3;-1 2 -2;3 -2 3];
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215 %! b = [7;-1;4];
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216 %! [x, flag, relres, iter, resvec] = bicgstab (A, b)
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217
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218 %!shared A, b, n, M1, M2
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219 %!
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220 %!test
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221 %! n = 100;
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222 %! A = spdiags ([-2*ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n);
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223 %! b = sum (A, 2);
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224 %! tol = 1e-8;
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225 %! maxit = 15;
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226 %! M1 = spdiags ([ones(n,1)/(-2) ones(n,1)],-1:0, n, n);
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227 %! M2 = spdiags ([4*ones(n,1) -ones(n,1)], 0:1, n, n);
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228 %! [x, flag, relres, iter, resvec] = bicgstab (A, b, 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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diff changeset
231 %!test
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232 %!function y = afun (x, a)
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233 %! y = a * x;
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234 %!endfunction
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235 %!
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236 %! tol = 1e-8;
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237 %! maxit = 15;
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238 %!
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239 %! [x, flag, relres, iter, resvec] = bicgstab (@(x) afun (x, A), b,
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240 %! tol, maxit, M1, M2);
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diff changeset
241 %! assert (x, ones (size (b)), 1e-7);
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242
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diff changeset
243 %!test
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244 %! n = 100;
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245 %! tol = 1e-8;
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246 %! a = sprand (n, n, .1);
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247 %! A = a'*a + 100 * eye (n);
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248 %! b = sum (A, 2);
13061
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249 %! [x, flag, relres, iter, resvec] = bicgstab (A, b, tol, [], diag (diag (A)));
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250 %! assert (x, ones (size (b)), 1e-7);
14363
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diff changeset
251
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diff changeset
252 %!test
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253 %! A = [1 + 1i, 1 + 1i; 2 - 1i, 2 + 1i];
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254 %! b = A * [1; 1];
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diff changeset
255 %! [x, flag, relres, iter, resvec] = bicgstab (A, b);
7ad3b9ca66f5 fix behaviour of bicgstab for complex matrices (bug #40017)
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diff changeset
256 %! assert (x, [1; 1], 1e-6);