--- /dev/null
+## Copyright (C) 2008-2012 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} =} bicgstab (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{M1}, @var{M2}, @var{x0})
+## @deftypefnx {Function File} {@var{x} =} bicgstab (@var{A}, @var{b}, @var{rtol}, @var{maxit}, @var{P})
+## @deftypefnx {Function File} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}] =} bicgstab (@var{A}, @var{b}, @dots{})
+## Solve @code{A x = b} using the stabilizied Bi-conjugate gradient iterative
+## 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{bicg, cgs, gmres, pcg}
+##
+## @end deftypefn
+
+function [x, flag, relres, iter, resvec] = bicgstab (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 (["bicgstab: 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 (["bicgstab: 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 (["bicgstab: 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
+
+ ## specifies initial estimate x0
+ if (nargin < 7)
+ x = zeros (rows (b), 1);
+ else
+ x = x0;
+ endif
+
+ norm_b = norm (b);
+
+ res = b - Ax (x);
+ rr = res;
+
+ ## Vector of the residual norms for each iteration.
+ resvec = norm(res) / norm_b;
+
+ ## Default behaviour we don't reach tolerance tol within maxit iterations.
+ flag = 1;
+
+ for iter = 1:maxit
+ rho_1 = res' * rr;
+
+ if (iter == 1)
+ p = res;
+ else
+ beta = (rho_1 / rho_2) * (alpha / omega);
+ p = res + beta * (p - omega * v);
+ endif
+
+ phat = precon (p);
+
+ v = Ax (phat);
+ alpha = rho_1 / (rr' * v);
+ s = res - alpha * v;
+
+ shat = precon (s);
+
+ t = Ax (shat);
+ omega = (t' * s) / (t' * t);
+ x = x + alpha * phat + omega * shat;
+ res = s - omega * t;
+ rho_2 = rho_1;
+
+ 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 < 2)
+ if (flag == 0)
+ printf ("bicgstab converged at iteration %i ", iter);
+ printf ("to a solution with relative residual %e\n", relres);
+ elseif (flag == 3)
+ printf ("bicgstab stopped at iteration %i ", iter);
+ printf ("without converging to the desired tolerance %e\n", tol);
+ printf ("because the method stagnated.\n");
+ printf ("The iterate returned (number %i) ", iter);
+ printf ("has relative residual %e\n", relres);
+ else
+ printf ("bicgstab stopped at iteration %i ", iter);
+ printf ("without converging to the desired toleranc %e\n", tol);
+ printf ("because the maximum number of iterations was reached.\n");
+ printf ("The iterate returned (number %i) ", iter);
+ printf ("has relative residual %e\n", 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]
+%! [x, flag, relres, iter, resvec] = bicgstab(A, b)
+
+%!shared A, b, n, M1, M2
+%!
+%!test
+%! n = 100;
+%! A = spdiags ([-2*ones(n,1) 4*ones(n,1) -ones(n,1)], -1:1, n, n);
+%! b = sum (A, 2);
+%! tol = 1e-8;
+%! maxit = 15;
+%! M1 = spdiags ([ones(n,1)/(-2) ones(n,1)],-1:0, n, n);
+%! M2 = spdiags ([4*ones(n,1) -ones(n,1)], 0:1, n, n);
+%! [x, flag, relres, iter, resvec] = bicgstab (A, b, tol, maxit, M1, M2);
+%! assert (x, ones (size (b)), 1e-7);
+%!
+%!test
+%!function y = afun (x, a)
+%! y = a * x;
+%!endfunction
+%!
+%! tol = 1e-8;
+%! maxit = 15;
+%!
+%! [x, flag, relres, iter, resvec] = bicgstab (@(x) afun (x, A), b,
+%! tol, maxit, M1, M2);
+%! 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] = bicgstab (A, b, tol, [], diag (diag (A)));
+%! assert (x, ones (size (b)), 1e-7);
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