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annotate scripts/sparse/cgs.m @ 8338:a35bf360b919

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Add the cgs and treelayout functions
author Radek Salac <salac.r@gmail.com>
date Fri, 21 Nov 2008 15:03:03 +0100
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children cadc73247d65
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8338
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1 ## Copyright (C) 2008 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 MIN(20,N).
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27 ## The @var{M1} specifies a preconditioner, can also be a function handler which returns M\X.
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28 ## The @var{M2} combined with @var{M1} defines preconditioner as preconditioner=M1*M2.
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29 ## The @var{x0} is initial guess, default value is zeros(N,1).
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30 ##
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31 ## @end deftypefn
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32
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33 function [x, flag, relres, iter, resvec] = cgs (A, b, tol, maxit, M1, M2, x0)
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34
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35 if (nargin < 2 || nargin > 7 || nargout > 5)
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36 print_usage ();
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37 elseif (!isnumeric (A) || rows (A) != columns (A))
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38 error ("cgs: first argument must be a n-by-n matrix");
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39 elseif (!isvector (b))
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40 error ("cgs: b must be a vector");
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41 elseif (rows (A) != rows (b))
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42 error ("cgs: first and second argument must have the same number of rows");
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43 elseif (nargin > 2 && !isscalar (tol))
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44 error ("cgs: tol must be a scalar");
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45 elseif (nargin > 3 && !isscalar (maxit))
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46 error ("cgs: maxit must be a scalar");
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47 elseif (nargin > 4 && ismatrix (M1) && (rows (M1) != rows (A) || columns (M1) != columns (A)))
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48 error ("cgs: M1 must have the same number of rows and columns as A");
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49 elseif (nargin > 5 && (!ismatrix (M2) || rows (M2) != rows (A) || columns (M2) != columns (A)))
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50 error ("cgs: M2 must have the same number of rows and columns as A");
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51 elseif (nargin > 6 && !isvector (x0))
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52 error ("cgs: x0 must be a vector");
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53 elseif (nargin > 6 && rows (x0) != rows (b))
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54 error ("cgs: xO must have the same number of rows as b");
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55 endif
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56
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57 ## default toleration
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58 if (nargin < 3)
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59 tol = 1e-6;
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60 endif
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61
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62 ## default maximum number of iteration
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63 if (nargin < 4)
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64 maxit = min (rows (b),20);
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65 endif
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66
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67
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68 ## left preconditioner
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69 precon = [];
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70 if (nargin == 5)
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71 precon = M1;
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72 elseif (nargin > 5)
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73 if (isparse(M1) && issparse(M2))
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74 precon = @(x) M1 * (M2 * x);
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75 else
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76 precon = M1 * M2;
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77 endif
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78 endif
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79
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80 ## precon can by also function
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81 if (nargin > 4 && isnumeric (precon))
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82
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83 ## we can compute inverse preconditioner and use quicker algorithm
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84 if (det (precon) != 0)
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85 precon=inv (precon);
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86 else
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87 error ("cgs: preconditioner is ill conditioned");
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88 endif
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89
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90 ## we must make test if preconditioner isn't ill conditioned
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91 if (isinf (cond (precon)));
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92 error ("cgs: preconditioner is ill conditioned");
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93 endif
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94 endif
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95
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96 ## specifies initial estimate x0
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97 if (nargin < 7)
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98 x = zeros (rows (b), 1);
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99 else
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100 x = x0;
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101 endif
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102
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103 relres = b - A * x;
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104 ## vector of the residual norms for each iteration
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105 resvec = [norm(relres)];
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106 ro = 0;
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107 norm_b = norm (b);
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108 ## default behaviour we don't reach tolerance tol within maxit iterations
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109 flag = 1;
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110 for iter = 1 : maxit
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111
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112 if (nargin > 4 && isnumeric (precon))
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113 ## we have computed inverse matrix so we can use quick algorithm
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114 z = precon * relres;
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115 elseif (nargin > 4)
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116 ## our preconditioner is a function
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117 z = feval (precon, relres);
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118 else
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119 ## we don't use preconditioning
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120 z = relres;
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121 endif
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122
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123 ## cache
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124 ro_old = ro;
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125 ro = relres' * z;
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126 if (iter == 1)
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127 p = z;
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128 else
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129 beta = ro / ro_old;
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130 p = z + beta * p;
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131 endif
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132 q = A * p; #cache
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133 alpha = ro / (p' * q);
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134 x = x + alpha * p;
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135
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136 relres = relres - alpha * q;
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137 resvec = [resvec; norm(relres)];
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138
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139 relres_distance = resvec (end) / norm_b;
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140 if (relres_distance <= tol)
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141 ## we reach tolerance tol within maxit iterations
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142 flag = 0;
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143 break;
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144 elseif (resvec (end) == resvec (end - 1) )
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145 ## the method stagnates
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146 flag = 3;
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147 break;
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148 endif
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149 endfor;
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150
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151 relres = relres_distance;
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152 endfunction
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153
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154
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155
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156 %!demo
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157 %! % Solve system of A*x=b
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158 %! A=[5 -1 3;-1 2 -2;3 -2 3]
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159 %! b=[7;-1;4]
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160 %! [a,b,c,d,e]=cgs(A,b)