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## Copyright (C) 2008-2013 Radek Salac
## Copyright (C) 2012 Carlo de Falco
##
## This file is part of Octave.
##
## Octave is free software; you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or (at
## your option) any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
## General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {@var{x} =} cgs (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{M1}, @var{M2}, @var{x0})
## @deftypefnx {Function File} {@var{x} =} cgs (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{P})
## @deftypefnx {Function File} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}] =} cgs (@var{A}, @var{b}, @dots{})
## Solve @code{A x = b}, where @var{A} is a square matrix, using the
## Conjugate Gradients Squared method.
##
## @itemize @minus
## @item @var{rtol} is the relative tolerance, if not given or set to []
## the default value 1e-6 is used.
##
## @item @var{maxit} the maximum number of outer iterations, if not
## given or set to [] the default value @code{min (20, numel (b))} is
## used.
##
## @item @var{x0} the initial guess, if not given or set to [] the
## default value @code{zeros (size (b))} is used.
## @end itemize
##
## @var{A} can be passed as a matrix or as a function handle or
## inline function @code{f} such that @code{f(x) = A*x}.
##
## The preconditioner @var{P} is given as @code{P = M1 * M2}.
## Both @var{M1} and @var{M2} can be passed as a matrix or as a function
## handle or inline function @code{g} such that @code{g(x) = M1 \ x} or
## @code{g(x) = M2 \ x}.
##
## If called with more than one output parameter
##
## @itemize @minus
## @item @var{flag} indicates the exit status:
##
## @itemize @minus
## @item 0: iteration converged to the within the chosen tolerance
##
## @item 1: the maximum number of iterations was reached before convergence
##
## @item 3: the algorithm reached stagnation
## @end itemize
##
## (the value 2 is unused but skipped for compatibility).
##
## @item @var{relres} is the final value of the relative residual.
##
## @item @var{iter} is the number of iterations performed.
##
## @item @var{resvec} is a vector containing the relative residual at
## each iteration.
## @end itemize
##
## @seealso{pcg, bicgstab, bicg, gmres}
## @end deftypefn
function [x, flag, relres, iter, resvec] = cgs (A, b, tol, maxit, M1, M2, x0)
if (nargin >= 2 && nargin <= 7 && isvector (full (b)))
if (ischar (A))
A = str2func (A);
elseif (ismatrix (A))
Ax = @(x) A * x;
elseif (isa (A, "function_handle"))
Ax = @(x) feval (A, x);
else
error (["cgs: first argument is expected to "...
"be a function or a square matrix"]);
endif
if (nargin < 3 || isempty (tol))
tol = 1e-6;
endif
if (nargin < 4 || isempty (maxit))
maxit = min (rows (b), 20);
endif
if (nargin < 5 || isempty (M1))
M1m1x = @(x) x;
elseif (ischar (M1))
M1m1x = str2func (M1);
elseif (ismatrix (M1))
M1m1x = @(x) M1 \ x;
elseif (isa (M1, "function_handle"))
M1m1x = @(x) feval (M1, x);
else
error ("cgs: preconditioner is expected to be a function or matrix");
endif
if (nargin < 6 || isempty (M2))
M2m1x = @(x) x;
elseif (ischar (M2))
M2m1x = str2func (M2);
elseif (ismatrix (M2))
M2m1x = @(x) M2 \ x;
elseif (isa (M2, "function_handle"))
M2m1x = @(x) feval (M2, x);
else
error ("cgs: preconditioner is expected to be a function or matrix");
endif
precon = @(x) M2m1x (M1m1x (x));
if (nargin < 7 || isempty (x0))
x0 = zeros (size (b));
endif
x = x0;
res = b - Ax (x);
norm_b = norm (b);
## Vector of the residual norms for each iteration.
resvec = norm (res) / norm_b;
ro = 0;
## Default behavior we don't reach tolerance tol within maxit iterations.
flag = 1;
for iter = 1:maxit
z = precon (res);
## Cache.
ro_old = ro;
ro = res' * z;
if (iter == 1)
p = z;
else
beta = ro / ro_old;
p = z + beta * p;
endif
## Cache.
q = Ax (p);
alpha = ro / (p' * q);
x = x + alpha * p;
res = res - alpha * q;
relres = norm (res) / norm_b;
resvec = [resvec; relres];
if (relres <= tol)
## We reach tolerance tol within maxit iterations.
flag = 0;
break
elseif (resvec (end) == resvec (end - 1))
## The method stagnates.
flag = 3;
break
endif
endfor
if (nargout < 1)
if (flag == 0)
printf ("cgs converged at iteration %i to a solution with relative residual %e\n",
iter, relres);
elseif (flag == 3)
printf (["cgs stopped at iteration %i without converging to the desired tolerance %e\n",
"because the method stagnated.\n",
"The iterate returned (number %i) has relative residual %e\n"],
iter, tol, iter, relres);
else
printf (["cgs stopped at iteration %i without converging to the desired tolerance %e\n",
"because the maximum number of iterations was reached.\n",
"The iterate returned (number %i) has relative residual %e\n"],
iter, tol, iter, relres);
endif
endif
else
print_usage ();
endif
endfunction
%!demo
%! % Solve system of A*x=b
%! A = [5 -1 3;-1 2 -2;3 -2 3];
%! b = [7;-1;4];
%! [a,b,c,d,e] = cgs (A,b)
%!shared A, b, n, M
%!
%!test
%! n = 100;
%! A = spdiags ([-ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n);
%! b = sum (A, 2);
%! tol = 1e-8;
%! maxit = 1000;
%! M = 4*eye (n);
%! [x, flag, relres, iter, resvec] = cgs (A, b, tol, maxit, M);
%! assert (x, ones (size (b)), 1e-7);
%!
%!test
%! tol = 1e-8;
%! maxit = 15;
%!
%! [x, flag, relres, iter, resvec] = cgs (@(x) A * x, b, tol, maxit, M);
%! assert (x, ones (size (b)), 1e-7);
%!test
%! n = 100;
%! tol = 1e-8;
%! a = sprand (n, n, .1);
%! A = a'*a + 100 * eye (n);
%! b = sum (A, 2);
%! [x, flag, relres, iter, resvec] = cgs (A, b, tol, [], diag (diag (A)));
%! assert (x, ones (size (b)), 1e-7);
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