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annotate scripts/linear-algebra/gmres.m @ 12541:dd2c70b30f28

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Add tests for ifftshift.m
author Robert T. Short <octave@phaselockedsystems.com.com>
date Sat, 26 Mar 2011 06:50:12 -0700
parents e742720c5e71
children f96b9b9f141b
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12509
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1 ## Copyright (C) 2009-2011 Carlo de Falco
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2 ##
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3 ## This file is part of Octave.
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4 ##
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5 ## Octave is free software; you can redistribute it and/or modify it
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6 ## under the terms of the GNU General Public License as published by the
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7 ## Free Software Foundation; either version 3 of the License, or (at your
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8 ## option) any later version.
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9 ##
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10 ## Octave is distributed in the hope that it will be useful, but WITHOUT
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11 ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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12 ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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13 ## for more details.
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14 ##
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15 ## You should have received a copy of the GNU General Public License
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16 ## along with Octave; see the file COPYING. If not, see
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17 ## <http://www.gnu.org/licenses/>.
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18
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19 ## -*- texinfo -*-
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20 ## @deftypefn {Function File} {@var{x} =} gmres (@var{A}, @var{b}, @var{m}, @var{rtol}, @var{maxit}, @var{M1}, @var{M2}, @var{x0})
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21 ## @deftypefnx {Function File} {@var{x} =} gmres (@var{A}, @var{b}, @var{m}, @var{rtol}, @var{maxit}, @var{P})
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22 ## @deftypefnx {Function File} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}] =} gmres (@dots{})
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23 ## Solve @code{A x = b} using the Preconditioned GMRES iterative method
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24 ## with restart, a.k.a. PGMRES(m).
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25 ##
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26 ## @itemize @minus
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27 ## @item @var{rtol} is the relative tolerance,
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28 ## if not given or set to [] the default value 1e-6 is used.
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29 ##
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30 ## @item @var{maxit} is the maximum number of outer iterations,
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31 ## if not given or set to [] the default value
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32 ## @code{min (10, numel (b) / restart)} is used.
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33 ##
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34 ## @item @var{x0} is the initial guess,
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35 ## if not given or set to [] the default value @code{zeros(size (b))} is used.
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36 ##
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37 ## @item @var{m} is the restart parameter,
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38 ## if not given or set to [] the default value @code{numel (b)} is used.
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39 ## @end itemize
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40 ##
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41 ## Argument @var{A} can be passed as a matrix, function handle, or
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42 ## inline function @code{f} such that @code{f(x) = A*x}.
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43 ##
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44 ## The preconditioner @var{P} is given as @code{P = M1 * M2}.
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45 ## Both @var{M1} and @var{M2} can be passed as a matrix, function handle, or
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46 ## inline function @code{g} such that @code{g(x) = M1\x} or @code{g(x) = M2\x}.
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47 ##
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48 ## Besides the vector @var{x}, additional outputs are:
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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 ##
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53 ## @table @asis
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54 ## @item 0 : iteration converged to within the specified tolerance
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55 ##
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56 ## @item 1 : maximum number of iterations exceeded
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57 ##
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58 ## @item 2 : unused, but skipped for compatibility
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59 ##
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60 ## @item 3 : algorithm reached stagnation
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61 ## @end table
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62 ##
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63 ## @item @var{relres} is the final value of the relative residual.
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64 ##
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65 ## @item @var{iter} is a vector containing the number of outer iterations and
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66 ## total iterations performed.
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67 ##
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68 ## @item @var{resvec} is a vector containing the relative residual at each
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69 ## iteration.
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70 ## @end itemize
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71 ##
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72 ## @seealso{pcg, cgs, bigcstab}
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73 ## @end deftypefn
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74
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75 function [x, flag, prec_res_norm, itcnt] = gmres (A, b, restart, rtol, maxit, M1, M2, x0)
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76
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77 if (nargin < 2 || nargin > 8)
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78 print_usage ();
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79 end
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80
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81 if (ischar (A))
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82 Ax = str2func (A);
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83 elseif (ismatrix (A))
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84 Ax = @(x) A*x;
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85 elseif (isa (A, "function_handle"))
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86 Ax = A;
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87 else
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88 error ("gmres: A must be a function or matrix");
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89 endif
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90
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91 if (nargin < 3 || isempty (restart))
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92 restart = rows (b);
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93 endif
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94
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95 if (nargin < 4 || isempty (rtol))
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96 rtol = 1e-6;
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97 endif
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98
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99 if (nargin < 5 || isempty (maxit))
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100 maxit = min (rows (b)/restart, 10);
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101 endif
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102
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103 if (nargin < 6 || isempty (M1))
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104 M1m1x = @(x) x;
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105 elseif (ischar (M1))
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106 M1m1x = str2func (M1);
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107 elseif (ismatrix (M1))
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108 M1m1x = @(x) M1 \ x;
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109 elseif (isa (M1, "function_handle"))
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110 M1m1x = M1;
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111 else
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112 error ("gmres: preconditioner M1 must be a function or matrix");
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113 endif
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114
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115 if (nargin < 7 || isempty (M2))
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116 M2m1x = @(x) x;
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117 elseif (ischar (M2))
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118 M2m1x = str2func (M2);
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119 elseif (ismatrix (M2))
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120 M2m1x = @(x) M2 \ x;
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121 elseif (isa (M2, "function_handle"))
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122 M2m1x = M2;
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123 else
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124 error ("gmres: preconditioner M2 must be a function or matrix");
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125 endif
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
126
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
127 Pm1x = @(x) M2m1x (M1m1x (x));
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
128
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
129 if (nargin < 8 || isempty (x0))
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
130 x0 = zeros (size (b));
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
131 endif
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
132
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
133 x_old = x0;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
134 x = x_old;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
135 prec_res = Pm1x (b - Ax (x_old));
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
136 prec_res_norm = norm (prec_res, 2);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
137
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
138 B = zeros (restart + 1, 1);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
139 V = zeros (rows (x), restart);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
140 H = zeros (restart + 1, restart);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
141
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
142 ## begin loop
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
143 iter = 1;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
144 restart_it = restart + 1;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
145 resids = zeros (maxit, 1);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
146 resids(1) = prec_res_norm;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
147 prec_b_norm = norm (Pm1x (b), 2);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
148 flag = 1;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
149
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
150 while ((iter <= maxit * restart) && (prec_res_norm > rtol * prec_b_norm))
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
151
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
152 ## restart
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
153 if (restart_it > restart)
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
154 restart_it = 1;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
155 x_old = x;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
156 prec_res = Pm1x (b - Ax (x_old));
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
157 prec_res_norm = norm (prec_res, 2);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
158 B(1) = prec_res_norm;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
159 H(:) = 0;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
160 V(:, 1) = prec_res / prec_res_norm;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
161 endif
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
162
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
163 ## basic iteration
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
164 tmp = Pm1x (Ax (V(:, restart_it)));
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
165 [V(:,restart_it+1), H(1:restart_it+1, restart_it)] = mgorth (tmp, V(:,1:restart_it));
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
166
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
167 Y = (H(1:restart_it+1, 1:restart_it) \ B (1:restart_it+1));
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
168
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
169 little_res = B(1:restart_it+1) - H(1:restart_it+1, 1:restart_it) * Y(1:restart_it);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
170 prec_res_norm = norm (little_res, 2);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
171
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
172 x = x_old + V(:, 1:restart_it) * Y(1:restart_it);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
173
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
174 resids(iter) = prec_res_norm ;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
175 if (norm (x - x_old, inf) <= eps)
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
176 flag = 3;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
177 break
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
178 endif
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
179
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
180 restart_it++ ; iter++;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
181 endwhile
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
182
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
183 if (prec_res_norm > rtol * prec_b_norm)
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
184 flag = 0;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
185 endif
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
186
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
187 resids = resids(1:iter-1);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
188 itcnt = [floor(maxit/restart), rem(maxit, restart)];
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
189 endfunction
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
190
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
191
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
192 %!shared A, b, dim
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
193 %! dim = 100;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
194 %!test
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
195 %! A = spdiags ([-ones(dim,1) 2*ones(dim,1) ones(dim,1)], [-1:1], dim, dim);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
196 %! b = ones(dim, 1);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
197 %! x = gmres (A, b, 10, 1e-10, dim, @(x) x./diag(A), [], b);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
198 %! assert(x, A\b, 1e-9*norm(x,inf));
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
199 %!
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
200 %!test
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
201 %! x = gmres (A, b, dim, 1e-10, 1e4, @(x) diag(diag(A))\x, [], b);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
202 %! assert(x, A\b, 1e-7*norm(x,inf));
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
203 %!
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
204 %!test
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
205 %! A = spdiags ([[1./(2:2:2*(dim-1)) 0]; 1./(1:2:2*dim-1); [0 1./(2:2:2*(dim-1))]]', -1:1, dim, dim);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
206 %! A = A'*A;
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
207 %! b = rand (dim, 1);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
208 %! [x, resids] = gmres (@(x) A*x, b, dim, 1e-10, dim, @(x) x./diag(A), [], []);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
209 %! assert(x, A\b, 1e-9*norm(x,inf))
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
210 %! x = gmres (@(x) A*x, b, dim, 1e-10, 1e6, @(x) diag(diag(A))\x, [], []);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
211 %! assert(x, A\b, 1e-9*norm(x,inf));
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
212 %!test
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
213 %! x = gmres (@(x) A*x, b, dim, 1e-10, 1e6, @(x) x./diag(A), [], []);
e742720c5e71 Add mgorth and gmres functions from Octave Forge
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
214 %! assert(x, A\b, 1e-7*norm(x,inf));