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+## Copyright (C) 2008 Radek Salac
+##
+## 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} {} cgs (@var{A}, @var{b})
+## @deftypefnx {Function File} {} cgs (@var{A}, @var{b}, @var{tol}, @var{maxit}, @var{M1}, @var{M2}, @var{x0})
+## This procedure attempts to solve a system of linear equations A*x = b for x.
+## The @var{A} must be square, symmetric and positive definite real matrix N*N.
+## The @var{b} must be a one column vector with a length of N.
+## The @var{tol} specifies the tolerance of the method, default value is 1e-6.
+## The @var{maxit} specifies the maximum number of iteration, default value is MIN(20,N).
+## The @var{M1} specifies a preconditioner, can also be a function handler which returns M\X.
+## The @var{M2} combined with @var{M1} defines preconditioner as preconditioner=M1*M2.
+## The @var{x0} is initial guess, default value is zeros(N,1).
+##
+## @end deftypefn
+
+function [x, flag, relres, iter, resvec] = cgs (A, b, tol, maxit, M1, M2, x0)
+
+  if (nargin < 2 || nargin > 7 || nargout > 5)
+    print_usage ();
+  elseif (!isnumeric (A) || rows (A) != columns (A))
+    error ("cgs: first argument must be a n-by-n matrix");
+  elseif (!isvector (b))
+    error ("cgs: b must be a vector");
+  elseif (rows (A) != rows (b))
+    error ("cgs: first and second argument must have the same number of rows");
+  elseif (nargin > 2 && !isscalar (tol))
+    error ("cgs: tol must be a scalar");
+  elseif (nargin > 3 && !isscalar (maxit))
+    error ("cgs: maxit must be a scalar");
+  elseif (nargin > 4 && ismatrix (M1) && (rows (M1) != rows (A) || columns (M1) != columns (A)))
+    error ("cgs: M1 must have the same number of rows and columns as A");
+  elseif (nargin > 5 && (!ismatrix (M2) || rows (M2) != rows (A) || columns (M2) != columns (A)))
+    error ("cgs: M2 must have the same number of rows and columns as A");
+  elseif (nargin > 6 && !isvector (x0))
+    error ("cgs: x0 must be a vector");
+  elseif (nargin > 6 && rows (x0) != rows (b))
+    error ("cgs: xO must have the same number of rows as b");
+  endif
+
+  ## default toleration
+  if (nargin < 3)
+    tol = 1e-6;
+  endif
+
+  ## default maximum number of iteration
+  if (nargin < 4)
+    maxit = min (rows (b),20);
+  endif
+
+
+  ## left preconditioner
+  precon = [];
+  if (nargin == 5)
+    precon = M1;
+  elseif (nargin > 5)
+    if (isparse(M1) && issparse(M2))
+      precon = @(x) M1 * (M2 * x);
+    else
+      precon = M1 * M2;
+    endif
+  endif
+
+  ## precon can by also function
+  if (nargin > 4 && isnumeric (precon))
+
+    ## we can compute inverse preconditioner and use quicker algorithm
+    if (det (precon) != 0)
+     precon=inv (precon);
+    else
+     error ("cgs: preconditioner is ill conditioned");
+    endif
+
+    ## we must make test if preconditioner isn't ill conditioned
+    if (isinf (cond (precon))); 
+      error ("cgs: preconditioner is ill conditioned");
+    endif
+  endif
+
+  ## specifies initial estimate x0
+  if (nargin < 7)
+    x = zeros (rows (b), 1);
+  else
+    x = x0;
+  endif
+
+  relres = b - A * x;
+  ## vector of the residual norms for each iteration
+  resvec = [norm(relres)];
+  ro = 0;
+  norm_b = norm (b);
+  ## default behaviour we don't reach tolerance tol within maxit iterations
+  flag = 1;
+  for iter = 1 : maxit
+
+    if (nargin > 4 && isnumeric (precon))
+      ## we have computed inverse matrix so we can use quick algorithm
+      z = precon * relres;
+    elseif (nargin > 4)
+      ## our preconditioner is a function
+      z = feval (precon, relres);
+    else
+      ## we don't use preconditioning
+      z = relres;
+    endif
+
+    ## cache
+    ro_old = ro;
+    ro = relres' * z;
+    if (iter == 1)
+      p = z;
+    else
+      beta = ro / ro_old;
+      p = z + beta * p;
+    endif
+    q = A * p; #cache
+    alpha = ro / (p' * q);
+    x = x + alpha * p;
+
+    relres = relres - alpha * q;
+    resvec = [resvec; norm(relres)];
+
+    relres_distance = resvec (end) / norm_b;
+    if (relres_distance <= 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;
+
+  relres = relres_distance;
+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)