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1 ## Copyright (C) 2004 Piotr Krzyzanowski <piotr.krzyzanowski@mimuw.edu.pl> |
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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 2, or (at your option) |
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8 ## 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, write to the Free |
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17 ## Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
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18 ## 02110-1301, USA. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @deftypefn {Function File} {@var{x} =} pcg (@var{A}, @var{b}, @var{tol}, @var{maxit}, @var{M}, @var{x0}, @dots{}) |
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22 ## @deftypefnx {Function File} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}, @var{eigest}] =} pcg (@dots{}) |
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23 ## |
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24 ## Solves the linear system of equations @code{@var{A} * @var{x} = |
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25 ## @var{b}} by means of the Preconditioned Conjugate Gradient iterative |
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26 ## method. The input arguments are |
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27 ## |
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28 ## @itemize |
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29 ## @item |
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30 ## @var{A} can be either a square (preferably sparse) matrix or a |
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31 ## function handle, inline function or string containing the name |
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32 ## of a function which computes @code{@var{A} * @var{x}}. In principle |
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33 ## @var{A} should be symmetric and positive definite; if @code{pcg} |
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34 ## finds @var{A} to not be positive definite, you will get a warning |
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35 ## message and the @var{flag} output parameter will be set. |
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36 ## |
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37 ## @item |
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38 ## @var{b} is the right hand side vector. |
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39 ## |
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40 ## @item |
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41 ## @var{tol} is the required relative tolerance for the residual error, |
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42 ## @code{@var{b} - @var{A} * @var{x}}. The iteration stops if @code{norm |
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43 ## (@var{b} - @var{A} * @var{x}) <= @var{tol} * norm (@var{b} - @var{A} * |
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44 ## @var{x0})}. If @var{tol} is empty or is omitted, the function sets |
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45 ## @code{@var{tol} = 1e-6} by default. |
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46 ## |
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47 ## @item |
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48 ## @var{maxit} is the maximum allowable number of iterations; if |
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49 ## @code{[]} is supplied for @code{maxit}, or @code{pcg} has less |
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50 ## arguments, a default value equal to 20 is used. |
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51 ## |
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52 ## @item |
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53 ## @var{M} is the (left) preconditioning matrix, so that the iteration is |
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54 ## (theoretically) equivalent to solving by @code{pcg} @code{@var{P} * |
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55 ## @var{x} = @var{M} \ @var{b}}, with @code{@var{P} = @var{M} \ @var{A}}. |
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56 ## Note that a proper choice of the preconditioner may dramatically |
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57 ## improve the overall performance of the method. Instead of matrix |
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58 ## @var{M}, the user may pass a function which returns the results of |
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59 ## applying the inverse of @var{M} to a vector (usually this is the |
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60 ## preferred way of using the preconditioner). If @code{[]} is supplied |
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61 ## for @var{M}, or @var{M} is omitted, no preconditioning is applied. |
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62 ## |
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63 ## @item |
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64 ## @var{x0} is the initial guess. If @var{x0} is empty or omitted, the |
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65 ## function sets @var{x0} to a zero vector by default. |
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66 ## @end itemize |
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67 ## |
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68 ## The arguments which follow @var{x0} are treated as parameters, and |
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69 ## passed in a proper way to any of the functions (@var{A} or @var{M}) |
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70 ## which are passed to @code{pcg}. See the examples below for further |
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71 ## details. The output arguments are |
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72 ## |
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73 ## @itemize |
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74 ## @item |
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75 ## @var{x} is the computed approximation to the solution of |
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76 ## @code{@var{A} * @var{x} = @var{b}}. |
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77 ## |
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78 ## @item |
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79 ## @var{flag} reports on the convergence. @code{@var{flag} = 0} means |
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80 ## the solution converged and the tolerance criterion given by @var{tol} |
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81 ## is satisfied. @code{@var{flag} = 1} means that the @var{maxit} limit |
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82 ## for the iteration count was reached. @code{@var{flag} = 3} reports that |
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83 ## the (preconditioned) matrix was found not positive definite. |
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84 ## |
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85 ## @item |
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86 ## @var{relres} is the ratio of the final residual to its initial value, |
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87 ## measured in the Euclidean norm. |
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88 ## |
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89 ## @item |
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90 ## @var{iter} is the actual number of iterations performed. |
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91 ## |
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92 ## @item |
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93 ## @var{resvec} describes the convergence history of the method. |
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94 ## @code{@var{resvec} (i,1)} is the Euclidean norm of the residual, and |
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95 ## @code{@var{resvec} (i,2)} is the preconditioned residual norm, |
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96 ## after the (@var{i}-1)-th iteration, @code{@var{i} = |
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97 ## 1,2,...@var{iter}+1}. The preconditioned residual norm is defined as |
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98 ## @code{norm (@var{r}) ^ 2 = @var{r}' * (@var{M} \ @var{r})} where |
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99 ## @code{@var{r} = @var{b} - @var{A} * @var{x}}, see also the |
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100 ## description of @var{M}. If @var{eigest} is not required, only |
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101 ## @code{@var{resvec} (:,1)} is returned. |
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102 ## |
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103 ## @item |
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104 ## @var{eigest} returns the estimate for the smallest @code{@var{eigest} |
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105 ## (1)} and largest @code{@var{eigest} (2)} eigenvalues of the |
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106 ## preconditioned matrix @code{@var{P} = @var{M} \ @var{A}}. In |
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107 ## particular, if no preconditioning is used, the extimates for the |
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108 ## extreme eigenvalues of @var{A} are returned. @code{@var{eigest} (1)} |
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109 ## is an overestimate and @code{@var{eigest} (2)} is an underestimate, |
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110 ## so that @code{@var{eigest} (2) / @var{eigest} (1)} is a lower bound |
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111 ## for @code{cond (@var{P}, 2)}, which nevertheless in the limit should |
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112 ## theoretically be equal to the actual value of the condition number. |
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113 ## The method which computes @var{eigest} works only for symmetric positive |
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114 ## definite @var{A} and @var{M}, and the user is responsible for |
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115 ## verifying this assumption. |
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116 ## @end itemize |
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117 ## |
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118 ## Let us consider a trivial problem with a diagonal matrix (we exploit the |
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119 ## sparsity of A) |
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120 ## |
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121 ## @example |
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122 ## @group |
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123 ## N = 10; |
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124 ## A = diag([1:N]); A = sparse(A); |
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125 ## b = rand(N,1); |
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126 ## @end group |
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127 ## @end example |
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128 ## |
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129 ## @sc{Example 1:} Simplest use of @code{pcg} |
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130 ## |
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131 ## @example |
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132 ## x = pcg(A,b) |
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133 ## @end example |
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134 ## |
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135 ## @sc{Example 2:} @code{pcg} with a function which computes |
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136 ## @code{@var{A} * @var{x}} |
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137 ## |
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138 ## @example |
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139 ## @group |
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140 ## function y = applyA(x) |
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141 ## y = [1:N]'.*x; |
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142 ## endfunction |
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143 ## |
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144 ## x = pcg('applyA',b) |
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145 ## @end group |
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146 ## @end example |
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147 ## |
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148 ## @sc{Example 3:} Preconditioned iteration, with full diagnostics. The |
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149 ## preconditioner (quite strange, because even the original matrix |
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150 ## @var{A} is trivial) is defined as a function |
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151 ## |
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152 ## @example |
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153 ## @group |
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154 ## function y = applyM(x) |
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155 ## K = floor(length(x)-2); |
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156 ## y = x; |
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157 ## y(1:K) = x(1:K)./[1:K]'; |
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158 ## endfunction |
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159 ## |
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160 ## [x, flag, relres, iter, resvec, eigest] = pcg(A,b,[],[],'applyM') |
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161 ## semilogy([1:iter+1], resvec); |
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162 ## @end group |
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163 ## @end example |
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164 ## |
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165 ## @sc{Example 4:} Finally, a preconditioner which depends on a |
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166 ## parameter @var{k}. |
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167 ## |
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168 ## @example |
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169 ## @group |
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170 ## function y = applyM(x, varargin) |
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171 ## K = varargin@{1@}; |
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172 ## y = x; y(1:K) = x(1:K)./[1:K]'; |
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173 ## endfuntion |
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174 ## |
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175 ## [x, flag, relres, iter, resvec, eigest] = ... |
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176 ## pcg(A,b,[],[],'applyM',[],3) |
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177 ## @end group |
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178 ## @end example |
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179 ## |
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180 ## @sc{References} |
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181 ## |
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182 ## [1] C.T.Kelley, 'Iterative methods for linear and nonlinear equations', |
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183 ## SIAM, 1995 (the base PCG algorithm) |
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184 ## |
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185 ## [2] Y.Saad, 'Iterative methods for sparse linear systems', PWS 1996 |
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186 ## (condition number estimate from PCG) Revised version of this book is |
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187 ## available online at http://www-users.cs.umn.edu/~saad/books.html |
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188 ## |
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189 ## |
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190 ## @seealso{sparse, pcr} |
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191 ## @end deftypefn |
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192 |
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193 ## REVISION HISTORY |
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194 ## |
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195 ## 2004-05-21, Piotr Krzyzanowski: |
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196 ## Added 4 demos and 4 tests |
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197 ## |
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198 ## 2004-05-18, Piotr Krzyzanowski: |
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199 ## Warnings use warning() function now |
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200 ## |
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201 ## 2004-04-29, Piotr Krzyzanowski: |
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202 ## Added more warning messages when FLAG is not required |
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203 ## |
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204 ## 2004-04-28, Piotr Krzyzanowski: |
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205 ## When eigest is required, resvec returns both the Euclidean and the |
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206 ## preconditioned residual norm convergence history |
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207 ## |
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208 ## 2004-04-20, Piotr Krzyzanowski: |
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209 ## Corrected eigenvalue estimator. Changed the tridiagonal matrix |
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210 ## eigenvalue solver to regular eig |
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211 ## |
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212 |
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213 function [x, flag, relres, iter, resvec, eigest] = ... |
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214 pcg( A, b, tol, maxit, M, x0, varargin ) |
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215 |
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216 if ((nargin < 6) || isempty(x0)) |
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217 x = zeros(size(b)); |
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218 else |
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219 x = x0; |
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220 endif |
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221 |
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222 if (nargin < 5) |
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223 M = []; |
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224 endif |
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225 |
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226 if ((nargin < 4) || isempty(maxit)) |
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227 maxit = min(size(b,1),20); |
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228 endif |
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229 |
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230 maxit = maxit + 2; |
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231 |
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232 if ((nargin < 3) || isempty(tol)) |
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233 tol = 1e-6; |
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234 endif |
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235 |
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236 preconditioned_residual_out = false; |
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237 if (nargout > 5) |
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238 T = zeros(maxit,maxit); |
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239 preconditioned_residual_out = true; |
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240 endif |
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241 |
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242 matrix_positive_definite = true; # assume A is positive definite |
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243 |
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244 p = zeros(size(b)); |
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245 oldtau = 1; |
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246 if (isnumeric(A)) # is A a matrix? |
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247 r = b - A*x; |
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248 else # then A should be a function! |
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249 r = b - feval(A,x,varargin{:}); |
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250 endif |
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251 |
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252 resvec(1,1) = norm(r); |
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253 alpha = 1; |
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254 iter = 2; |
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255 |
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256 while ((resvec(iter-1,1) > tol*resvec(1,1)) && (iter < maxit)) |
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257 if (isnumeric(M)) # is M a matrix? |
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258 if isempty(M) # if M is empty, use no precond |
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259 z = r; |
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260 else # otherwise, apply the precond |
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261 z = M \ r; |
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262 endif |
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263 else # then M should be a function! |
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264 z = feval(M,r,varargin{:}); |
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265 endif |
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266 tau = z'*r; |
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267 resvec(iter-1,2) = sqrt(tau); |
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268 beta = tau/oldtau; |
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269 oldtau = tau; |
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270 p = z + beta*p; |
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271 if (isnumeric(A)) # is A a matrix? |
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272 w = A*p; |
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273 else # then A should be a function! |
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274 w = feval(A,p,varargin{:}); |
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275 endif |
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276 oldalpha = alpha; # needed only for eigest |
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277 alpha = tau/(p'*w); |
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278 if (alpha <= 0.0) # negative matrix? |
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279 matrix_positive_definite = false; |
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280 endif |
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281 x = x + alpha*p; |
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282 r = r - alpha*w; |
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283 if ((nargout > 5) && (iter > 2)) |
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284 T(iter-1:iter, iter-1:iter) = T(iter-1:iter, iter-1:iter) + ... |
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285 [1 sqrt(beta); sqrt(beta) beta]./oldalpha; |
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286 ## EVS = eig(T(2:iter-1,2:iter-1)); |
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287 ## fprintf(stderr,"PCG condest: %g (iteration: %d)\n", max(EVS)/min(EVS),iter); |
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288 endif |
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289 resvec(iter,1) = norm(r); |
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290 iter = iter + 1; |
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291 endwhile |
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292 |
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293 if (nargout > 5) |
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294 if (matrix_positive_definite ) |
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295 if (iter > 3) |
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296 T = T(2:iter-2,2:iter-2); |
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297 l = eig(T); |
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298 eigest = [min(l) max(l)]; |
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299 ## fprintf(stderr, "PCG condest: %g\n",eigest(2)/eigest(1)); |
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300 else |
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301 eigest = [NaN NaN]; |
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302 warning("PCG: eigenvalue estimate failed: iteration converged too fast."); |
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303 endif |
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304 else |
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305 eigest = [NaN NaN]; |
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306 endif |
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307 |
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308 ## apply the preconditioner once more and finish with the precond |
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309 ## residual |
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310 if (isnumeric(M)) # is M a matrix? |
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311 if isempty(M) # if M is empty, use no precond |
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312 z = r; |
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313 else # otherwise, apply the precond |
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314 z = M\r; |
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315 endif |
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316 else # then M should be a function! |
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317 z = feval(M,r,varargin{:}); |
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318 endif |
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319 resvec(iter-1,2) = sqrt(r'*z); |
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320 else |
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321 resvec = resvec(:,1); |
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322 endif |
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323 |
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324 flag = 0; |
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325 relres = resvec(iter-1,1)./resvec(1,1); |
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326 iter = iter - 2; |
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327 if (iter >= (maxit-2)) |
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328 flag = 1; |
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329 if (nargout < 2) |
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330 warning("PCG: maximum number of iterations (%d) reached\n", iter); |
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331 warning("The initial residual norm was reduced %g times.\n", 1.0/relres); |
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332 endif |
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333 else |
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334 if (nargout < 2) |
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335 fprintf(stderr, "PCG: converged in %d iterations. ", iter); |
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336 fprintf(stderr, "The initial residual norm was reduced %g times.\n",... |
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337 1.0/relres); |
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338 endif |
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339 endif |
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340 |
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341 if (!matrix_positive_definite) |
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342 flag = 3; |
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343 if (nargout < 2) |
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344 warning("PCG: matrix not positive definite?\n"); |
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345 endif |
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346 endif |
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347 endfunction |
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348 |
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349 %!demo |
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350 %! |
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351 %! # Simplest usage of pcg (see also 'help pcg') |
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352 %! |
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353 %! N = 10; |
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354 %! A = diag([1:N]); b = rand(N,1); y = A\b; #y is the true solution |
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355 %! x = pcg(A,b); |
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356 %! printf('The solution relative error is %g\n', norm(x-y)/norm(y)); |
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357 %! |
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358 %! # You shouldn't be afraid if pcg issues some warning messages in this |
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359 %! # example: watch out in the second example, why it takes N iterations |
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360 %! # of pcg to converge to (a very accurate, by the way) solution |
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361 %!demo |
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362 %! |
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363 %! # Full output from pcg, except for the eigenvalue estimates |
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364 %! # We use this output to plot the convergence history |
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365 %! |
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366 %! N = 10; |
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367 %! A = diag([1:N]); b = rand(N,1); X = A\b; #X is the true solution |
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368 %! [x, flag, relres, iter, resvec] = pcg(A,b); |
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369 %! printf('The solution relative error is %g\n', norm(x-X)/norm(X)); |
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370 %! title('Convergence history'); xlabel('Iteration'); ylabel('log(||b-Ax||/||b||)'); |
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371 %! semilogy([0:iter],resvec/resvec(1),'o-g;relative residual;'); |
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372 %!demo |
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373 %! |
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374 %! # Full output from pcg, including the eigenvalue estimates |
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375 %! # Hilbert matrix is extremely ill conditioned, so pcg WILL have problems |
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376 %! |
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377 %! N = 10; |
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378 %! A = hilb(N); b = rand(N,1); X = A\b; #X is the true solution |
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379 %! [x, flag, relres, iter, resvec, eigest] = pcg(A,b,[],200); |
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380 %! printf('The solution relative error is %g\n', norm(x-X)/norm(X)); |
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381 %! printf('Condition number estimate is %g\n', eigest(2)/eigest(1)); |
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382 %! printf('Actual condition number is %g\n', cond(A)); |
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383 %! title('Convergence history'); xlabel('Iteration'); ylabel('log(||b-Ax||)'); |
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384 %! semilogy([0:iter],resvec,['o-g;absolute residual;';'+-r;absolute preconditioned residual;']); |
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385 %!demo |
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386 %! |
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387 %! # Full output from pcg, including the eigenvalue estimates |
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388 %! # We use the 1-D Laplacian matrix for A, and cond(A) = O(N^2) |
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389 %! # and that's the reasone we need some preconditioner; here we take |
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390 %! # a very simple and not powerful Jacobi preconditioner, |
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391 %! # which is the diagonal of A |
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392 %! |
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393 %! N = 100; |
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394 %! A = zeros(N,N); |
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395 %! for i=1:N-1 # form 1-D Laplacian matrix |
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396 %! A(i:i+1,i:i+1) = [2 -1; -1 2]; |
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397 %! endfor |
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398 %! b = rand(N,1); X = A\b; #X is the true solution |
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399 %! maxit = 80; |
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400 %! printf('System condition number is %g\n',cond(A)); |
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401 %! # No preconditioner: the convergence is very slow! |
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402 %! |
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403 %! [x, flag, relres, iter, resvec, eigest] = pcg(A,b,[],maxit); |
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404 %! printf('System condition number estimate is %g\n',eigest(2)/eigest(1)); |
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405 %! title('Convergence history'); xlabel('Iteration'); ylabel('log(||b-Ax||)'); |
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406 %! semilogy([0:iter],resvec(:,1),'o-g;NO preconditioning: absolute residual;'); |
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407 %! |
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408 %! pause(1); |
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409 %! # Test Jacobi preconditioner: it will not help much!!! |
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410 %! |
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411 %! M = diag(diag(A)); # Jacobi preconditioner |
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412 %! [x, flag, relres, iter, resvec, eigest] = pcg(A,b,[],maxit,M); |
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413 %! printf('JACOBI preconditioned system condition number estimate is %g\n',eigest(2)/eigest(1)); |
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414 %! hold on; |
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415 %! semilogy([0:iter],resvec(:,1),'o-r;JACOBI preconditioner: absolute residual;'); |
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416 %! |
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417 %! pause(1); |
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418 %! # Test nonoverlapping block Jacobi preconditioner: it will help much! |
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419 %! |
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420 %! M = zeros(N,N);k=4 |
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421 %! for i=1:k:N # form 1-D Laplacian matrix |
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422 %! M(i:i+k-1,i:i+k-1) = A(i:i+k-1,i:i+k-1); |
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423 %! endfor |
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424 %! [x, flag, relres, iter, resvec, eigest] = pcg(A,b,[],maxit,M); |
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425 %! printf('BLOCK JACOBI preconditioned system condition number estimate is %g\n',eigest(2)/eigest(1)); |
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426 %! semilogy([0:iter],resvec(:,1),'o-b;BLOCK JACOBI preconditioner: absolute residual;'); |
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427 %! hold off; |
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428 %!test |
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429 %! |
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430 %! #solve small diagonal system |
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431 %! |
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432 %! N = 10; |
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433 %! A = diag([1:N]); b = rand(N,1); X = A\b; #X is the true solution |
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434 %! [x, flag] = pcg(A,b,[],N+1); |
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435 %! assert(norm(x-X)/norm(X),0,1e-10); |
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436 %! assert(flag,0); |
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437 %! |
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438 %!test |
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439 %! |
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440 %! #solve small indefinite diagonal system |
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441 %! #despite A is indefinite, the iteration continues and converges |
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442 %! #indefiniteness of A is detected |
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443 %! |
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444 %! N = 10; |
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445 %! A = diag([1:N].*(-ones(1,N).^2)); b = rand(N,1); X = A\b; #X is the true solution |
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446 %! [x, flag] = pcg(A,b,[],N+1); |
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447 %! assert(norm(x-X)/norm(X),0,1e-10); |
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448 %! assert(flag,3); |
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449 %! |
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450 %!test |
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451 %! |
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452 %! #solve tridiagonal system, do not converge in default 20 iterations |
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453 %! |
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454 %! N = 100; |
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455 %! A = zeros(N,N); |
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456 %! for i=1:N-1 # form 1-D Laplacian matrix |
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457 %! A(i:i+1,i:i+1) = [2 -1; -1 2]; |
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458 %! endfor |
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459 %! b = ones(N,1); X = A\b; #X is the true solution |
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460 %! [x, flag, relres, iter, resvec, eigest] = pcg(A,b,1e-12); |
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461 %! assert(flag); |
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462 %! assert(relres>1.0); |
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463 %! assert(iter,20); #should perform max allowable default number of iterations |
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464 %! |
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465 %!test |
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466 %! |
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467 %! #solve tridiagonal system with 'prefect' preconditioner |
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468 %! #converges in one iteration, so the eigest does not work |
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469 %! #and issues a warning |
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470 %! |
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471 %! N = 100; |
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472 %! A = zeros(N,N); |
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473 %! for i=1:N-1 # form 1-D Laplacian matrix |
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474 %! A(i:i+1,i:i+1) = [2 -1; -1 2]; |
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475 %! endfor |
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476 %! b = ones(N,1); X = A\b; #X is the true solution |
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477 %! [x, flag, relres, iter, resvec, eigest] = pcg(A,b,[],[],A,b); |
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478 %! assert(norm(x-X)/norm(X),0,1e-6); |
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479 %! assert(flag,0); |
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480 %! assert(iter,1); #should converge in one iteration |
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481 %! assert(isnan(eigest),isnan([NaN NaN])); |
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482 %! |
