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+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2011 Gael Guennebaud <gael.guennebaud@inria.fr>
+// Copyright (C) 2012, 2014 Kolja Brix <brix@igpm.rwth-aaachen.de>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#ifndef EIGEN_GMRES_H
+#define EIGEN_GMRES_H
+
+namespace Eigen { 
+
+namespace internal {
+
+/**
+ * Generalized Minimal Residual Algorithm based on the
+ * Arnoldi algorithm implemented with Householder reflections.
+ *
+ * Parameters:
+ *  \param mat       matrix of linear system of equations
+ *  \param Rhs       right hand side vector of linear system of equations
+ *  \param x         on input: initial guess, on output: solution
+ *  \param precond   preconditioner used
+ *  \param iters     on input: maximum number of iterations to perform
+ *                   on output: number of iterations performed
+ *  \param restart   number of iterations for a restart
+ *  \param tol_error on input: residual tolerance
+ *                   on output: residuum achieved
+ *
+ * \sa IterativeMethods::bicgstab() 
+ *  
+ *
+ * For references, please see:
+ *
+ * Saad, Y. and Schultz, M. H.
+ * GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems.
+ * SIAM J.Sci.Stat.Comp. 7, 1986, pp. 856 - 869.
+ *
+ * Saad, Y.
+ * Iterative Methods for Sparse Linear Systems.
+ * Society for Industrial and Applied Mathematics, Philadelphia, 2003.
+ *
+ * Walker, H. F.
+ * Implementations of the GMRES method.
+ * Comput.Phys.Comm. 53, 1989, pp. 311 - 320.
+ *
+ * Walker, H. F.
+ * Implementation of the GMRES Method using Householder Transformations.
+ * SIAM J.Sci.Stat.Comp. 9, 1988, pp. 152 - 163.
+ *
+ */
+template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
+bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Preconditioner & precond,
+		int &iters, const int &restart, typename Dest::RealScalar & tol_error) {
+
+	using std::sqrt;
+	using std::abs;
+
+	typedef typename Dest::RealScalar RealScalar;
+	typedef typename Dest::Scalar Scalar;
+	typedef Matrix < Scalar, Dynamic, 1 > VectorType;
+	typedef Matrix < Scalar, Dynamic, Dynamic > FMatrixType;
+
+	RealScalar tol = tol_error;
+	const int maxIters = iters;
+	iters = 0;
+
+	const int m = mat.rows();
+
+	VectorType p0 = rhs - mat*x;
+	VectorType r0 = precond.solve(p0);
+ 
+	// is initial guess already good enough?
+	if(abs(r0.norm()) < tol) {
+		return true; 
+	}
+
+	VectorType w = VectorType::Zero(restart + 1);
+
+	FMatrixType H = FMatrixType::Zero(m, restart + 1); // Hessenberg matrix
+	VectorType tau = VectorType::Zero(restart + 1);
+	std::vector < JacobiRotation < Scalar > > G(restart);
+
+	// generate first Householder vector
+	VectorType e(m-1);
+	RealScalar beta;
+	r0.makeHouseholder(e, tau.coeffRef(0), beta);
+	w(0)=(Scalar) beta;
+	H.bottomLeftCorner(m - 1, 1) = e;
+
+	for (int k = 1; k <= restart; ++k) {
+
+		++iters;
+
+		VectorType v = VectorType::Unit(m, k - 1), workspace(m);
+
+		// apply Householder reflections H_{1} ... H_{k-1} to v
+		for (int i = k - 1; i >= 0; --i) {
+			v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
+		}
+
+		// apply matrix M to v:  v = mat * v;
+		VectorType t=mat*v;
+		v=precond.solve(t);
+
+		// apply Householder reflections H_{k-1} ... H_{1} to v
+		for (int i = 0; i < k; ++i) {
+			v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
+		}
+
+		if (v.tail(m - k).norm() != 0.0) {
+
+			if (k <= restart) {
+
+				// generate new Householder vector
+                                  VectorType e(m - k - 1);
+				RealScalar beta;
+				v.tail(m - k).makeHouseholder(e, tau.coeffRef(k), beta);
+				H.col(k).tail(m - k - 1) = e;
+
+				// apply Householder reflection H_{k} to v
+				v.tail(m - k).applyHouseholderOnTheLeft(H.col(k).tail(m - k - 1), tau.coeffRef(k), workspace.data());
+
+			}
+                }
+
+                if (k > 1) {
+                        for (int i = 0; i < k - 1; ++i) {
+                                // apply old Givens rotations to v
+                                v.applyOnTheLeft(i, i + 1, G[i].adjoint());
+                        }
+                }
+
+                if (k<m && v(k) != (Scalar) 0) {
+                        // determine next Givens rotation
+                        G[k - 1].makeGivens(v(k - 1), v(k));
+
+                        // apply Givens rotation to v and w
+                        v.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
+                        w.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
+
+                }
+
+                // insert coefficients into upper matrix triangle
+                H.col(k - 1).head(k) = v.head(k);
+
+                bool stop=(k==m || abs(w(k)) < tol || iters == maxIters);
+
+                if (stop || k == restart) {
+
+                        // solve upper triangular system
+                        VectorType y = w.head(k);
+                        H.topLeftCorner(k, k).template triangularView < Eigen::Upper > ().solveInPlace(y);
+
+                        // use Horner-like scheme to calculate solution vector
+                        VectorType x_new = y(k - 1) * VectorType::Unit(m, k - 1);
+
+                        // apply Householder reflection H_{k} to x_new
+                        x_new.tail(m - k + 1).applyHouseholderOnTheLeft(H.col(k - 1).tail(m - k), tau.coeffRef(k - 1), workspace.data());
+
+                        for (int i = k - 2; i >= 0; --i) {
+                                x_new += y(i) * VectorType::Unit(m, i);
+                                // apply Householder reflection H_{i} to x_new
+                                x_new.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
+                        }
+
+                        x += x_new;
+
+                        if (stop) {
+                                return true;
+                        } else {
+                                k=0;
+
+                                // reset data for a restart  r0 = rhs - mat * x;
+                                VectorType p0=mat*x;
+                                VectorType p1=precond.solve(p0);
+                                r0 = rhs - p1;
+//                                 r0_sqnorm = r0.squaredNorm();
+                                w = VectorType::Zero(restart + 1);
+                                H = FMatrixType::Zero(m, restart + 1);
+                                tau = VectorType::Zero(restart + 1);
+
+                                // generate first Householder vector
+                                RealScalar beta;
+                                r0.makeHouseholder(e, tau.coeffRef(0), beta);
+                                w(0)=(Scalar) beta;
+                                H.bottomLeftCorner(m - 1, 1) = e;
+
+                        }
+
+                }
+
+
+
+	}
+	
+	return false;
+
+}
+
+}
+
+template< typename _MatrixType,
+          typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
+class GMRES;
+
+namespace internal {
+
+template< typename _MatrixType, typename _Preconditioner>
+struct traits<GMRES<_MatrixType,_Preconditioner> >
+{
+  typedef _MatrixType MatrixType;
+  typedef _Preconditioner Preconditioner;
+};
+
+}
+
+/** \ingroup IterativeLinearSolvers_Module
+  * \brief A GMRES solver for sparse square problems
+  *
+  * This class allows to solve for A.x = b sparse linear problems using a generalized minimal
+  * residual method. The vectors x and b can be either dense or sparse.
+  *
+  * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
+  * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
+  *
+  * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
+  * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
+  * and NumTraits<Scalar>::epsilon() for the tolerance.
+  * 
+  * This class can be used as the direct solver classes. Here is a typical usage example:
+  * \code
+  * int n = 10000;
+  * VectorXd x(n), b(n);
+  * SparseMatrix<double> A(n,n);
+  * // fill A and b
+  * GMRES<SparseMatrix<double> > solver(A);
+  * x = solver.solve(b);
+  * std::cout << "#iterations:     " << solver.iterations() << std::endl;
+  * std::cout << "estimated error: " << solver.error()      << std::endl;
+  * // update b, and solve again
+  * x = solver.solve(b);
+  * \endcode
+  * 
+  * By default the iterations start with x=0 as an initial guess of the solution.
+  * One can control the start using the solveWithGuess() method.
+  * 
+  * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
+  */
+template< typename _MatrixType, typename _Preconditioner>
+class GMRES : public IterativeSolverBase<GMRES<_MatrixType,_Preconditioner> >
+{
+  typedef IterativeSolverBase<GMRES> Base;
+  using Base::mp_matrix;
+  using Base::m_error;
+  using Base::m_iterations;
+  using Base::m_info;
+  using Base::m_isInitialized;
+ 
+private:
+  int m_restart;
+  
+public:
+  typedef _MatrixType MatrixType;
+  typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::Index Index;
+  typedef typename MatrixType::RealScalar RealScalar;
+  typedef _Preconditioner Preconditioner;
+
+public:
+
+  /** Default constructor. */
+  GMRES() : Base(), m_restart(30) {}
+
+  /** Initialize the solver with matrix \a A for further \c Ax=b solving.
+    * 
+    * This constructor is a shortcut for the default constructor followed
+    * by a call to compute().
+    * 
+    * \warning this class stores a reference to the matrix A as well as some
+    * precomputed values that depend on it. Therefore, if \a A is changed
+    * this class becomes invalid. Call compute() to update it with the new
+    * matrix A, or modify a copy of A.
+    */
+  template<typename MatrixDerived>
+  explicit GMRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()), m_restart(30) {}
+
+  ~GMRES() {}
+  
+  /** Get the number of iterations after that a restart is performed.
+    */
+  int get_restart() { return m_restart; }
+  
+  /** Set the number of iterations after that a restart is performed.
+    *  \param restart   number of iterations for a restarti, default is 30.
+    */
+  void set_restart(const int restart) { m_restart=restart; }
+  
+  /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A
+    * \a x0 as an initial solution.
+    *
+    * \sa compute()
+    */
+  template<typename Rhs,typename Guess>
+  inline const internal::solve_retval_with_guess<GMRES, Rhs, Guess>
+  solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const
+  {
+    eigen_assert(m_isInitialized && "GMRES is not initialized.");
+    eigen_assert(Base::rows()==b.rows()
+              && "GMRES::solve(): invalid number of rows of the right hand side matrix b");
+    return internal::solve_retval_with_guess
+            <GMRES, Rhs, Guess>(*this, b.derived(), x0);
+  }
+  
+  /** \internal */
+  template<typename Rhs,typename Dest>
+  void _solveWithGuess(const Rhs& b, Dest& x) const
+  {    
+    bool failed = false;
+    for(int j=0; j<b.cols(); ++j)
+    {
+      m_iterations = Base::maxIterations();
+      m_error = Base::m_tolerance;
+      
+      typename Dest::ColXpr xj(x,j);
+      if(!internal::gmres(*mp_matrix, b.col(j), xj, Base::m_preconditioner, m_iterations, m_restart, m_error))
+        failed = true;
+    }
+    m_info = failed ? NumericalIssue
+           : m_error <= Base::m_tolerance ? Success
+           : NoConvergence;
+    m_isInitialized = true;
+  }
+
+  /** \internal */
+  template<typename Rhs,typename Dest>
+  void _solve(const Rhs& b, Dest& x) const
+  {
+    x = b;
+    if(x.squaredNorm() == 0) return; // Check Zero right hand side
+    _solveWithGuess(b,x);
+  }
+
+protected:
+
+};
+
+
+namespace internal {
+
+  template<typename _MatrixType, typename _Preconditioner, typename Rhs>
+struct solve_retval<GMRES<_MatrixType, _Preconditioner>, Rhs>
+  : solve_retval_base<GMRES<_MatrixType, _Preconditioner>, Rhs>
+{
+  typedef GMRES<_MatrixType, _Preconditioner> Dec;
+  EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
+
+  template<typename Dest> void evalTo(Dest& dst) const
+  {
+    dec()._solve(rhs(),dst);
+  }
+};
+
+} // end namespace internal
+
+} // end namespace Eigen
+
+#endif // EIGEN_GMRES_H