// *****************************************************************
// Iterative template routine -- CG
//
// CG solves the symmetric positive definite linear
// system Ax=b using the Conjugate Gradient method.
//
// CG follows the algorithm described on p. 15 in 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
CG(const Matrix & A, Vector & x, const Vector & b,
   const Preconditioner & M, int & max_iter, Real & tol)
{
  Real   resid;
  Vector p, z, q;
  Vector alpha(1), beta(1), rho(1), rho_1(1);

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

  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++ )
    {
    z = M.solve(r);
    rho(0) = dot(r, z);

    if( i == 1 )
      {
      p = z;
      }
    else
      {
      beta(0) = rho(0) / rho_1(0);
      p = z + beta(0) * p;
      }

    q = A * p;
    alpha(0) = rho(0) / dot(p, q);

    x += alpha(0) * p;
    r -= alpha(0) * q;

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

    rho_1(0) = rho(0);
    }

  tol = resid;
  return 1;
}
