// *****************************************************************
// Iterative template routine -- BiCG
//
// BiCG solves the unsymmetric linear system Ax = b
// using the Preconditioned BiConjugate Gradient method
//
// BiCG follows the algorithm described on p. 22 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
BiCG(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 z, ztilde, p, ptilde, q, qtilde;

  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++ )
    {
    z = M.solve(r);
    ztilde = M.trans_solve(rtilde);
    rho_1(0) = dot(z, rtilde);
    if( rho_1(0) == 0 )
      {
      tol = norm(r) / normb;
      max_iter = i;
      return 2;
      }
    if( i == 1 )
      {
      p = z;
      ptilde = ztilde;
      }
    else
      {
      beta(0) = rho_1(0) / rho_2(0);
      p = z + beta(0) * p;
      ptilde = ztilde + beta(0) * ptilde;
      }
    q = A * p;
    qtilde = A.trans_mult(ptilde);
    alpha(0) = rho_1(0) / dot(ptilde, q);
    x += alpha(0) * p;
    r -= alpha(0) * q;
    rtilde -= alpha(0) * qtilde;

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

  tol = resid;
  return 1;
}
