// *****************************************************************
// Iterative template routine -- CGS
//
// CGS solves the unsymmetric linear system Ax = b
// using the Conjugate Gradient Squared method
//
// CGS follows the algorithm described on p. 26 of the
// SIAM Templates book.
//
// The return value indicates convergence within max_iter (input)
// iterations (0), or no convergence within max_iter iterations (1).
//
// Upon successful return, output arguments have the following values:
//
//        x  --  approximate solution to Ax = b
// max_iter  --  the number of iterations performed before the
//               tolerance was reached
//      tol  --  the residual after the final iteration
//
// *****************************************************************

template <class Matrix, class Vector, class Preconditioner, class Real>
int
CGS(const Matrix & A, Vector & x, const Vector & b,
    const Preconditioner & M, int & max_iter, Real & tol)
{
  Real   resid;
  Vector rho_1(1), rho_2(1), alpha(1), beta(1);
  Vector p, phat, q, qhat, vhat, u, uhat;

  Real   normb = norm(b);
  Vector r = b - A * x;
  Vector rtilde = r;

  if( normb == 0.0 )
    {
    normb = 1;
    }

  if( (resid = norm(r) / normb) <= tol )
    {
    tol = resid;
    max_iter = 0;
    return 0;
    }
  for( int i = 1; i <= max_iter; i++ )
    {
    rho_1(0) = dot(rtilde, r);
    if( rho_1(0) == 0 )
      {
      tol = norm(r) / normb;
      return 2;
      }
    if( i == 1 )
      {
      u = r;
      p = u;
      }
    else
      {
      beta(0) = rho_1(0) / rho_2(0);
      u = r + beta(0) * q;
      p = u + beta(0) * (q + beta(0) * p);
      }
    phat = M.solve(p);
    vhat = A * phat;
    alpha(0) = rho_1(0) / dot(rtilde, vhat);
    q = u - alpha(0) * vhat;
    uhat = M.solve(u + q);
    x += alpha(0) * uhat;
    qhat = A * uhat;
    r -= alpha(0) * qhat;
    rho_2(0) = rho_1(0);
    if( (resid = norm(r) / normb) < tol )
      {
      tol = resid;
      max_iter = i;
      return 0;
      }
    }

  tol = resid;
  return 1;
}
