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annotate scripts/sparse/gmres.m @ 20761:b7ac1e94266e

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maint: Further clean up of functions in ode/private dir. * AbsRel_Norm.m, fuzzy_compare.m, integrate_adaptive.m, integrate_const.m, integrate_n_steps.m, ode_struct_value_check.m, odepkg_event_handle.m, odepkg_structure_check.m, runge_kutta_45_dorpri.m: Place latest copyright first in file. Use two spaces before beginning single-line comment. Use parentheses around variable to be tested in switch stmt. Use space between function name and opening parenthesis.
author Rik <rik@octave.org>
date Mon, 05 Oct 2015 12:03:16 -0700
parents df437a52bcaf
children
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rev   line source
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1 ## Copyright (C) 2009-2015 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 with
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24 ## 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, if not given or
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31 ## set to [] the default value @code{min (10, numel (b) / restart)} is used.
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32 ##
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33 ## @item @var{x0} is the initial guess,
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34 ## if not given or set to [] the default value @code{zeros (size (b))} is used.
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35 ##
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36 ## @item @var{m} is the restart parameter,
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37 ## if not given or set to [] the default value @code{numel (b)} is used.
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38 ## @end itemize
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39 ##
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40 ## Argument @var{A} can be passed as a matrix, function handle, or inline
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41 ## 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}. Both @var{M1}
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44 ## and @var{M2} can be passed as a matrix, function handle, or inline function
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45 ## @code{g} such that @code{g(x) = M1\x} or @code{g(x) = M2\x}.
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46 ##
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47 ## Besides the vector @var{x}, additional outputs are:
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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 ## @table @asis
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53 ## @item 0 : iteration converged to within the specified tolerance
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54 ##
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55 ## @item 1 : maximum number of iterations exceeded
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56 ##
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57 ## @item 2 : unused, but skipped for compatibility
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58 ##
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59 ## @item 3 : algorithm reached stagnation (no change between iterations)
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60 ## @end table
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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 a vector containing the number of outer iterations and
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65 ## total iterations performed.
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66 ##
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67 ## @item @var{resvec} is a vector containing the relative residual at each
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68 ## iteration.
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69 ## @end itemize
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70 ##
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71 ## @seealso{bicg, bicgstab, cgs, pcg, pcr, qmr}
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72 ## @end deftypefn
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73
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74 function [x, flag, relres, it, resvec] = gmres (A, b, restart, rtol, maxit, M1, M2, x0)
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75
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76 if (nargin < 2 || nargin > 8)
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77 print_usage ();
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78 endif
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79
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80 if (ischar (A))
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81 Ax = str2func (A);
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82 elseif (isnumeric (A) && ismatrix (A))
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83 Ax = @(x) A*x;
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84 elseif (isa (A, "function_handle"))
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85 Ax = A;
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86 else
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87 error ("gmres: A must be a function or matrix");
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88 endif
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89
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90 if (nargin < 3 || isempty (restart))
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91 restart = rows (b);
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92 endif
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93
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94 if (nargin < 4 || isempty (rtol))
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95 rtol = 1e-6;
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96 endif
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97
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98 if (nargin < 5 || isempty (maxit))
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99 maxit = min (rows (b)/restart, 10);
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100 endif
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101
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102 if (nargin < 6 || isempty (M1))
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103 M1m1x = @(x) x;
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104 elseif (ischar (M1))
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105 M1m1x = str2func (M1);
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106 elseif (isnumeric (M1) && ismatrix (M1))
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107 M1m1x = @(x) M1 \ x;
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108 elseif (isa (M1, "function_handle"))
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109 M1m1x = M1;
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110 else
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111 error ("gmres: preconditioner M1 must be a function or matrix");
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112 endif
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113
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114 if (nargin < 7 || isempty (M2))
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115 M2m1x = @(x) x;
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116 elseif (ischar (M2))
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117 M2m1x = str2func (M2);
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118 elseif (isnumeric (M2) && ismatrix (M2))
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119 M2m1x = @(x) M2 \ x;
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120 elseif (isa (M2, "function_handle"))
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121 M2m1x = M2;
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122 else
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123 error ("gmres: preconditioner M2 must be a function or matrix");
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124 endif
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parents:
diff changeset
125
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126 Pm1x = @(x) M2m1x (M1m1x (x));
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127
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128 if (nargin < 8 || isempty (x0))
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129 x0 = zeros (size (b));
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130 endif
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131
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132 x_old = x0;
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133 x = x_old;
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134 prec_res = Pm1x (b - Ax (x_old));
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135 presn = norm (prec_res, 2);
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136
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137 B = zeros (restart + 1, 1);
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138 V = zeros (rows (x), restart);
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139 H = zeros (restart + 1, restart);
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140
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141 ## begin loop
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142 iter = 1;
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143 restart_it = restart + 1;
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144 resvec = zeros (maxit, 1);
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145 resvec(1) = presn;
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146 prec_b_norm = norm (Pm1x (b), 2);
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147 flag = 1; # Default flag is maximum # of iterations exceeded
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148
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149 while (iter <= maxit * restart && presn > rtol * prec_b_norm)
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150
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151 ## restart
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152 if (restart_it > restart)
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153 restart_it = 1;
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154 x_old = x;
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155 prec_res = Pm1x (b - Ax (x_old));
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156 presn = norm (prec_res, 2);
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157 B(1) = presn;
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158 H(:) = 0;
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159 V(:, 1) = prec_res / presn;
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160 endif
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161
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162 ## basic iteration
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163 tmp = Pm1x (Ax (V(:, restart_it)));
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164 [V(:,restart_it+1), H(1:restart_it+1, restart_it)] = ...
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165 mgorth (tmp, V(:,1:restart_it));
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166
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9fc020886ae9 maint: Clean up m-files to follow Octave coding conventions.
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diff changeset
167 Y = (H(1:restart_it+1, 1:restart_it) \ B(1:restart_it+1));
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168
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diff changeset
169 little_res = B(1:restart_it+1) - ...
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170 H(1:restart_it+1, 1:restart_it) * Y(1:restart_it);
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diff changeset
171
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172 presn = norm (little_res, 2);
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173
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174 x = x_old + V(:, 1:restart_it) * Y(1:restart_it);
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175
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176 resvec(iter+1) = presn;
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177 if (norm (x - x_old, inf) <= eps)
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178 flag = 3; # Stagnation: no change between iterations
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179 break;
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180 endif
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181
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182 restart_it++ ;
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183 iter++;
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184 endwhile
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185
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186 if (nargout > 1)
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187 ## Calculate extra outputs as requested
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188 relres = presn / prec_b_norm;
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189 if (relres <= rtol)
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190 flag = 0; # Converged to solution within tolerance
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191 endif
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192
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193 it = [floor(iter/restart), restart_it-1];
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194 endif
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195
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196 endfunction
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197
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diff changeset
198
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199 %!demo
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200 %! dim = 20;
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201 %! A = spdiags ([-ones(dim,1) 2*ones(dim,1) ones(dim,1)], [-1:1], dim, dim);
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202 %! b = ones (dim, 1);
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203 %! [x, flag, relres, iter, resvec] = gmres (A, b, 10, 1e-10, dim, @(x) x ./ diag (A), [], b)
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204
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205 %!shared A, b, dim
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206 %! dim = 100;
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diff changeset
207 %!test
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208 %! A = spdiags ([-ones(dim,1) 2*ones(dim,1) ones(dim,1)], [-1:1], dim, dim);
14363
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209 %! b = ones (dim, 1);
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210 %! x = gmres (A, b, 10, 1e-10, dim, @(x) x ./ diag (A), [], b);
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211 %! assert (x, A\b, 1e-9*norm (x, Inf));
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212 %!
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213 %!test
14363
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214 %! x = gmres (A, b, dim, 1e-10, 1e4, @(x) diag (diag (A)) \ x, [], b);
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215 %! assert(x, A\b, 1e-7*norm (x, Inf));
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216 %!
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diff changeset
217 %!test
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218 %! 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);
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219 %! A = A'*A;
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220 %! b = rand (dim, 1);
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221 %! [x, resvec] = gmres (@(x) A*x, b, dim, 1e-10, dim, @(x) x./diag (A), [], []);
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222 %! assert (x, A\b, 1e-9*norm (x, Inf));
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223 %! x = gmres (@(x) A*x, b, dim, 1e-10, 1e6, @(x) diag (diag (A)) \ x, [], []);
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diff changeset
224 %! assert (x, A\b, 1e-9*norm (x, Inf));
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225 %!test
14868
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226 %! x = gmres (@(x) A*x, b, dim, 1e-10, 1e6, @(x) x ./ diag (A), [], []);
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227 %! assert (x, A\b, 1e-7*norm (x, Inf));
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diff changeset
228
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229
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230 %!error gmres (1)
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231 %!error gmres (1,2,3,4,5,6,7,8,9)
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232 %!error <A must be> gmres ({1},2)
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233 %!error <A must be a function or matrix> gmres ({1},2)
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diff changeset
234 %!error <M1 must be a function or matrix> gmres (1,2,3,4,5,{6})
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235 %!error <M2 must be a function or matrix> gmres (1,2,3,4,5,6,{7})
17338
1c89599167a6 maint: End m-files with 1 blank line.
Rik <rik@octave.org>
parents: 14868
diff changeset
236