## Copyright (C) 2008-2015 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, qmr}
## @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 (isnumeric (A) && 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 (isnumeric (M1) && 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 (isnumeric (M2) && 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);

