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author John W. Eaton <jwe@octave.org>
date Thu, 20 Jan 2011 17:35:29 -0500
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1 ## Copyright (C) 2008-2011 Radek Salac
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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
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7 ## the Free Software Foundation; either version 3 of the License, or (at
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8 ## your 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
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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13 ## General Public License 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} {} cgs (@var{A}, @var{b})
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21 ## @deftypefnx {Function File} {} cgs (@var{A}, @var{b}, @var{tol}, @var{maxit}, @var{M1}, @var{M2}, @var{x0})
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22 ## This procedure attempts to solve a system of linear equations A*x = b for x.
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23 ## The @var{A} must be square, symmetric and positive definite real matrix N*N.
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24 ## The @var{b} must be a one column vector with a length of N.
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25 ## The @var{tol} specifies the tolerance of the method, default value is 1e-6.
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26 ## The @var{maxit} specifies the maximum number of iteration, default value is
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27 ## MIN(20,N).
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28 ## The @var{M1} specifies a preconditioner, can also be a function handler which
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29 ## returns M\X.
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30 ## The @var{M2} combined with @var{M1} defines preconditioner as
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31 ## preconditioner=M1*M2.
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32 ## The @var{x0} is initial guess, default value is zeros(N,1).
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33 ##
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34 ## @end deftypefn
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35
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36 function [x, flag, relres, iter, resvec] = cgs (A, b, tol, maxit, M1, M2, x0)
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37
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38 if (nargin < 2 || nargin > 7 || nargout > 5)
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39 print_usage ();
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40 elseif (!(isnumeric (A) && issquare (A)))
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41 error ("cgs: A must be a square numeric matrix");
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42 elseif (!isvector (b))
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43 error ("cgs: B must be a vector");
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44 elseif (rows (A) != rows (b))
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45 error ("cgs: A and B must have the same number of rows");
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46 elseif (nargin > 2 && !isscalar (tol))
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47 error ("cgs: TOL must be a scalar");
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48 elseif (nargin > 3 && !isscalar (maxit))
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49 error ("cgs: MAXIT must be a scalar");
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50 elseif (nargin > 4 && ismatrix (M1) && (rows (M1) != rows (A) || columns (M1) != columns (A)))
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51 error ("cgs: M1 must have the same number of rows and columns as A");
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52 elseif (nargin > 5 && (!ismatrix (M2) || rows (M2) != rows (A) || columns (M2) != columns (A)))
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53 error ("cgs: M2 must have the same number of rows and columns as A");
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54 elseif (nargin > 6 && !isvector (x0))
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55 error ("cgs: X0 must be a vector");
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56 elseif (nargin > 6 && rows (x0) != rows (b))
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57 error ("cgs: X0 must have the same number of rows as B");
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58 endif
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59
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60 ## Default tolerance.
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61 if (nargin < 3)
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62 tol = 1e-6;
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63 endif
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64
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65 ## Default maximum number of iteration.
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66 if (nargin < 4)
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67 maxit = min (rows (b),20);
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68 endif
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69
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70 ## Left preconditioner.
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71 if (nargin == 5)
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72 if (isnumeric (M1))
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73 precon = @(x) M1 \ x;
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74 endif
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75 elseif (nargin > 5)
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76 if (issparse (M1) && issparse (M2))
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77 precon = @(x) M2 \ (M1 \ x);
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78 else
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79 M = M1*M2;
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80 precon = @(x) M \ x;
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81 endif
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82 else
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83 precon = @(x) x;
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84 endif
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85
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86 ## Specifies initial estimate x0.
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87 if (nargin < 7)
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88 x = zeros (rows (b), 1);
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89 else
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90 x = x0;
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91 endif
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92
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93 res = b - A * x;
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94 norm_b = norm (b);
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95 ## Vector of the residual norms for each iteration.
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96 resvec = [ norm(res)/norm_b ];
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97 ro = 0;
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98 ## Default behavior we don't reach tolerance tol within maxit iterations.
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99 flag = 1;
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100 for iter = 1 : maxit
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101
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102 z = precon (res);
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103
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104 ## Cache.
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105 ro_old = ro;
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106 ro = res' * z;
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107 if (iter == 1)
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108 p = z;
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109 else
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110 beta = ro / ro_old;
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111 p = z + beta * p;
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112 endif
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113 ## Cache.
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114 q = A * p;
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115 alpha = ro / (p' * q);
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116 x = x + alpha * p;
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117
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118 res = res - alpha * q;
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119 relres = norm(res) / norm_b;
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120 resvec = [resvec;relres];
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121
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122 if (relres <= tol)
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123 ## We reach tolerance tol within maxit iterations.
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124 flag = 0;
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125 break;
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126 elseif (resvec (end) == resvec (end - 1))
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127 ## The method stagnates.
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128 flag = 3;
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129 break;
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130 endif
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131 endfor;
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132
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133 if (nargout < 1)
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134 if ( flag == 0 )
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135 printf (["cgs converged at iteration %i ",
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136 "to a solution with relative residual %e\n"],iter,relres);
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137 elseif (flag == 3)
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138 printf (["cgs stopped at iteration %i ",
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139 "without converging to the desired tolerance %e\n",
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140 "because the method stagnated.\n",
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141 "The iterate returned (number %i) has relative residual %e\n"],iter,tol,iter,relres);
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142 else
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143 printf (["cgs stopped at iteration %i ",
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144 "without converging to the desired tolerance %e\n",
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145 "because the maximum number of iterations was reached.\n",
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146 "The iterate returned (number %i) has relative residual %e\n"],iter,tol,iter,relres);
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147 endif
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148 endif
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149
8338
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150 endfunction
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151
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152
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153
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154 %!demo
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155 %! % Solve system of A*x=b
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156 %! A=[5 -1 3;-1 2 -2;3 -2 3]
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157 %! b=[7;-1;4]
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158 %! [a,b,c,d,e]=cgs(A,b)