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doc: Periodic grammarcheck of documentation. * contrib.txi, diagperm.txi, emacs.txi, install.txi, package.txi, plot.txi, poly.txi, vectorize.txi, strread.m, textscan.m, graphics_toolkit.m, bicg.m, bicgstab.m, cgs.m, rand.cc, data.cc: Periodic grammarcheck of documentation.
author Rik <octave@nomad.inbox5.com>
date Mon, 09 Jul 2012 10:34:43 -0700
parents 3edba8b5f430
children e2de3c8882be
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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1 ## Copyright (C) 2006 Sylvain Pelissier <sylvain.pelissier@gmail.com>
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2 ## Copyright (C) 2012 Carlo de Falco
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3 ##
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4 ## This program is free software; you can redistribute it and/or modify
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5 ## it under the terms of the GNU General Public License as published by
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6 ## the Free Software Foundation; either version 2 of the License, or
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7 ## (at your option) any later version.
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8 ##
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9 ## This program is distributed in the hope that it will be useful,
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10 ## but WITHOUT ANY WARRANTY; without even the implied warranty of
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11 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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12 ## GNU General Public License for more details.
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13 ##
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14 ## You should have received a copy of the GNU General Public License
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15 ## along with this program; If not, see <http://www.gnu.org/licenses/>.
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16
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17 ## -*- texinfo -*-
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18 ## @deftypefn {Function File} {@var{x} =} bicg (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{M1}, @var{M2}, @var{x0})
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19 ## @deftypefnx {Function File} {@var{x} =} bicg (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{P})
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20 ## @deftypefnx {Function File} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}] =} bicg (@var{A}, @var{b}, @dots{})
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21 ## Solve @code{A x = b} using the Bi-conjugate gradient iterative method.
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22 ##
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23 ## @itemize @minus
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24 ## @item @var{rtol} is the relative tolerance, if not given
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25 ## or set to [] the default value 1e-6 is used.
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26 ##
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27 ## @item @var{maxit} the maximum number of outer iterations,
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28 ## if not given or set to [] the default value
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29 ## @code{min (20, numel (b))} is used.
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30 ##
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31 ## @item @var{x0} the initial guess, if not given or set to []
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32 ## the default value @code{zeros (size (b))} is used.
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33 ## @end itemize
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34 ##
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35 ## @var{A} can be passed as a matrix or as a function handle or
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36 ## inline function @code{f} such that @code{f(x, "notransp") = A*x}
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37 ## and @code{f(x, "transp") = A'*x}.
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38 ##
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39 ## The preconditioner @var{P} is given as @code{P = M1 * M2}.
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40 ## Both @var{M1} and @var{M2} can be passed as a matrix or as
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41 ## a function handle or inline function @code{g} such that
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42 ## @code{g(x, "notransp") = M1 \ x} or @code{g(x, "notransp") = M2 \ x} and
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43 ## @code{g(x, "transp") = M1' \ x} or @code{g(x, "transp") = M2' \ x}.
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44 ##
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45 ## If called with more than one output parameter
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46 ##
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47 ## @itemize @minus
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48 ## @item @var{flag} indicates the exit status:
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49 ##
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50 ## @itemize @minus
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51 ## @item 0: iteration converged to the within the chosen tolerance
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52 ##
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53 ## @item 1: the maximum number of iterations was reached before convergence
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54 ##
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55 ## @item 3: the algorithm reached stagnation
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56 ## @end itemize
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57 ## (the value 2 is unused but skipped for compatibility).
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58 ##
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59 ## @item @var{relres} is the final value of the relative residual.
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60 ##
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61 ## @item @var{iter} is the number of iterations performed.
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62 ##
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63 ## @item @var{resvec} is a vector containing the relative residual at each iteration.
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64 ## @end itemize
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65 ##
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66 ## @seealso{bicgstab, cgs, gmres, pcg}
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67 ##
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68 ## @end deftypefn
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69
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70
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71 function [x, flag, res1, k, resvec] = bicg (A, b, tol, maxit, M1, M2, x0)
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72
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73 if (nargin >= 2 && isvector (full (b)))
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74
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75 if (ischar (A))
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76 fun = str2func (A);
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77 Ax = @(x) feval (fun, x, "notransp");
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78 Atx = @(x) feval (fun, x, "transp");
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79 elseif (ismatrix (A))
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80 Ax = @(x) A * x;
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81 Atx = @(x) A' * x;
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82 elseif (isa (A, "function_handle"))
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83 Ax = @(x) feval (A, x, "notransp");
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84 Atx = @(x) feval (A, x, "transp");
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85 else
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86 error (["bicg: first argument is expected to " ...
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87 "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, ignore) x;
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100 M1tm1x = M1m1x;
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101 elseif (ischar (M1))
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102 fun = str2func (M1);
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103 M1m1x = @(x) feval (fun, x, "notransp");
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104 M1tm1x = @(x) feval (fun, x, "transp");
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105 elseif (ismatrix (M1))
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106 M1m1x = @(x) M1 \ x;
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107 M1tm1x = @(x) M1' \ x;
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108 elseif (isa (M1, "function_handle"))
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109 M1m1x = @(x) feval (M1, x, "notransp");
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110 M1tm1x = @(x) feval (M1, x, "transp");
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111 else
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112 error (["bicg: preconditioner is expected to " ...
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113 "be a function or matrix"]);
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114 endif
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115
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116 if (nargin < 6 || isempty (M2))
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117 M2m1x = @(x, ignore) x;
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118 M2tm1x = M2m1x;
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119 elseif (ischar (M2))
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parents:
diff changeset
120 fun = str2func (M2);
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parents:
diff changeset
121 M2m1x = @(x) feval (fun, x, "notransp");
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parents:
diff changeset
122 M2tm1x = @(x) feval (fun, x, "transp");
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
123 elseif (ismatrix (M2))
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
124 M2m1x = @(x) M2 \ x;
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
125 M2tm1x = @(x) M2' \ x;
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parents:
diff changeset
126 elseif (isa (M2, "function_handle"))
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parents:
diff changeset
127 M2m1x = @(x) feval (M2, x, "notransp");
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parents:
diff changeset
128 M2tm1x = @(x) feval (M2, x, "transp");
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
129 else
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
130 error (["bicg: preconditioner is expected to " ...
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
131 "be a function or matrix"]);
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
132 endif
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
133
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
134 Pm1x = @(x) M2m1x (M1m1x (x));
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
135 Ptm1x = @(x) M1tm1x (M2tm1x (x));
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
136
13091
e5aaba072d2b maint: style fixes in iterative linear solvers
Carlo de Falco <kingcrimson@tiscali.it>
parents: 13073
diff changeset
137 if (nargin < 7 || isempty (x0))
13021
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
138 x0 = zeros (size (b));
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
139 endif
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
140
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
141 y = x = x0;
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
142 c = b;
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
143
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
144 r0 = b - Ax (x);
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
145 s0 = c - Atx (y);
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
146
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
147 d = Pm1x (r0);
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
148 f = Ptm1x (s0);
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
149
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
150 bnorm = norm (b);
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
151 res0 = Inf;
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parents:
diff changeset
152
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
153 if (any (r0 != 0))
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diff changeset
154
13021
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
155 for k = 1:maxit
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
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parents:
diff changeset
156
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diff changeset
157 a = (s0' * Pm1x (r0)) ./ (f' * Ax (d));
13021
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
158
13141
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diff changeset
159 x += a * d;
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diff changeset
160 y += conj (a) * f;
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
161
13141
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diff changeset
162 r1 = r0 - a * Ax (d);
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diff changeset
163 s1 = s0 - conj (a) * Atx (f);
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
164
13141
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diff changeset
165 beta = (s1' * Pm1x (r1)) ./ (s0' * Pm1x (r0));
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
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parents:
diff changeset
166
13141
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diff changeset
167 d = Pm1x (r1) + beta * d;
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diff changeset
168 f = Ptm1x (s1) + conj (beta) * f;
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
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parents:
diff changeset
169
13141
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diff changeset
170 r0 = r1;
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diff changeset
171 s0 = s1;
13021
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parents:
diff changeset
172
13141
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diff changeset
173 res1 = norm (b - Ax (x)) / bnorm;
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diff changeset
174 if (res1 < tol)
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
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parents:
diff changeset
175 flag = 0;
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parents:
diff changeset
176 if (nargout < 2)
13141
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diff changeset
177 printf ("bicg converged at iteration %i ", k);
13021
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parents:
diff changeset
178 printf ("to a solution with relative residual %e\n", res1);
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parents:
diff changeset
179 endif
13141
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diff changeset
180 break;
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diff changeset
181 endif
e81ddf9cacd5 maint: untabify and remove trailing whitespace from source files
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diff changeset
182
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diff changeset
183 if (res0 <= res1)
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
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parents:
diff changeset
184 flag = 3;
13141
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diff changeset
185 printf ("bicg stopped at iteration %i ", k);
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
186 printf ("without converging to the desired tolerance %e\n", tol);
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
187 printf ("because the method stagnated.\n");
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
188 printf ("The iterate returned (number %i) ", k-1);
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
189 printf ("has relative residual %e\n", res0);
13141
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parents: 13091
diff changeset
190 break
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diff changeset
191 endif
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
192 res0 = res1;
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
193 if (nargout > 4)
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
194 resvec(k) = res0;
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
195 endif
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
196 endfor
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
197
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
198 if (k == maxit)
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
199 flag = 1;
13141
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diff changeset
200 printf ("bicg stopped at iteration %i ", maxit);
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
201 printf ("without converging to the desired tolerance %e\n", tol);
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
202 printf ("because the maximum number of iterations was reached. ");
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
203 printf ("The iterate returned (number %i) has ", maxit);
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
204 printf ("relative residual %e\n", res1);
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
205 endif
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
206
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
207 else
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
208 flag = 0;
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
209 if (nargout < 2)
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
210 printf ("bicg converged after 0 interations\n");
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
211 endif
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
212 endif
13141
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John W. Eaton <jwe@octave.org>
parents: 13091
diff changeset
213
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
214 else
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
215 print_usage ();
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
216 endif
13141
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parents: 13091
diff changeset
217
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
218 endfunction;
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
219
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
220
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
221 %!test
13141
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diff changeset
222 %! n = 100;
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
223 %! A = spdiags ([-2*ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n);
13141
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parents: 13091
diff changeset
224 %! b = sum (A, 2);
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diff changeset
225 %! tol = 1e-8;
13021
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
226 %! maxit = 15;
13141
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diff changeset
227 %! M1 = spdiags ([ones(n,1)/(-2) ones(n,1)],-1:0, n, n);
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diff changeset
228 %! M2 = spdiags ([4*ones(n,1) -ones(n,1)], 0:1, n, n);
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
229 %! [x, flag, relres, iter, resvec] = bicg (A, b, tol, maxit, M1, M2);
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
230 %! assert (x, ones (size (b)), 1e-7);
13796
53674ceb9133 maint: fix function definition in test
John W. Eaton <jwe@octave.org>
parents: 13141
diff changeset
231
53674ceb9133 maint: fix function definition in test
John W. Eaton <jwe@octave.org>
parents: 13141
diff changeset
232 %!function y = afun (x, t, a)
53674ceb9133 maint: fix function definition in test
John W. Eaton <jwe@octave.org>
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diff changeset
233 %! switch t
53674ceb9133 maint: fix function definition in test
John W. Eaton <jwe@octave.org>
parents: 13141
diff changeset
234 %! case "notransp"
53674ceb9133 maint: fix function definition in test
John W. Eaton <jwe@octave.org>
parents: 13141
diff changeset
235 %! y = a * x;
53674ceb9133 maint: fix function definition in test
John W. Eaton <jwe@octave.org>
parents: 13141
diff changeset
236 %! case "transp"
53674ceb9133 maint: fix function definition in test
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parents: 13141
diff changeset
237 %! y = a' * x;
53674ceb9133 maint: fix function definition in test
John W. Eaton <jwe@octave.org>
parents: 13141
diff changeset
238 %! endswitch
53674ceb9133 maint: fix function definition in test
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parents: 13141
diff changeset
239 %!endfunction
53674ceb9133 maint: fix function definition in test
John W. Eaton <jwe@octave.org>
parents: 13141
diff changeset
240 %!
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
241 %!test
13141
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diff changeset
242 %! n = 100;
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
243 %! A = spdiags ([-2*ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n);
13141
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diff changeset
244 %! b = sum (A, 2);
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diff changeset
245 %! tol = 1e-8;
13021
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
diff changeset
246 %! maxit = 15;
13141
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diff changeset
247 %! M1 = spdiags ([ones(n,1)/(-2) ones(n,1)],-1:0, n, n);
e81ddf9cacd5 maint: untabify and remove trailing whitespace from source files
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diff changeset
248 %! M2 = spdiags ([4*ones(n,1) -ones(n,1)], 0:1, n, n);
13021
d55d396a9a55 Add an implementation of the biconjugate gradient iterative method
Carlo de Falco <kingcrimson@tiscali.it>
parents:
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249 %!
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John W. Eaton <jwe@octave.org>
parents: 13091
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250 %! [x, flag, relres, iter, resvec] = bicg (@(x, t) afun (x, t, A),
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
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251 %! b, tol, maxit, M1, M2);
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
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252 %! assert (x, ones (size (b)), 1e-7);
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
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253
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
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254 %!test
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John W. Eaton <jwe@octave.org>
parents: 13091
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255 %! n = 100;
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John W. Eaton <jwe@octave.org>
parents: 13091
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256 %! tol = 1e-8;
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
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257 %! a = sprand (n, n, .1);
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
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258 %! A = a' * a + 100 * eye (n);
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John W. Eaton <jwe@octave.org>
parents: 13091
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259 %! b = sum (A, 2);
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
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260 %! [x, flag, relres, iter, resvec] = bicg (A, b, tol, [], diag (diag (A)));
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Carlo de Falco <kingcrimson@tiscali.it>
parents:
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261 %! assert (x, ones (size (b)), 1e-7);
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Rik <octave@nomad.inbox5.com>
parents: 14359
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262