// *****************************************************************
// Iterative template routine -- BiCGSTAB
//
// BiCGSTAB solves the unsymmetric linear system Ax = b
// using the Preconditioned BiConjugate Gradient Stabilized method
//
// BiCGSTAB follows the algorithm described on p. 27 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
BiCGSTAB(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), omega(1);
  Vector p, phat, s, shat, t, v;

  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 )
      {
      p = r;
      }
    else
      {
      beta(0) = (rho_1(0) / rho_2(0) ) * (alpha(0) / omega(0) );
      p = r + beta(0) * (p - omega(0) * v);
      }
    phat = M.solve(p);
    v = A * phat;
    alpha(0) = rho_1(0) / dot(rtilde, v);
    s = r - alpha(0) * v;
    if( (resid = norm(s) / normb) < tol )
      {
      x += alpha(0) * phat;
      tol = resid;
      return 0;
      }
    shat = M.solve(s);
    t = A * shat;
    omega = dot(t, s) / dot(t, t);
    x += alpha(0) * phat + omega(0) * shat;
    r = s - omega(0) * t;

    rho_2(0) = rho_1(0);
    if( (resid = norm(r) / normb) < tol )
      {
      tol = resid;
      max_iter = i;
      return 0;
      }
    if( omega(0) == 0 )
      {
      tol = norm(r) / normb;
      return 3;
      }
    }

  tol = resid;
  return 1;
}
