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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: relative 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,
     Index &iters, const Index &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, ColMajor> FMatrixType;
 
   const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
 
   if(rhs.norm() <= considerAsZero) 
   {
     x.setZero();
     tol_error = 0;
     return true;
   }
 
   RealScalar tol = tol_error;
   const Index maxIters = iters;
   iters = 0;
 
   const Index m = mat.rows();
 
   // residual and preconditioned residual
   VectorType p0 = rhs - mat*x;
   VectorType r0 = precond.solve(p0);
 
   const RealScalar r0Norm = r0.norm();
 
   // is initial guess already good enough?
   if(r0Norm == 0)
   {
     tol_error = 0;
     return true;
   }
 
   // storage for Hessenberg matrix and Householder data
   FMatrixType H   = FMatrixType::Zero(m, restart + 1);
   VectorType w    = VectorType::Zero(restart + 1);
   VectorType tau  = VectorType::Zero(restart + 1);
 
   // storage for Jacobi rotations
   std::vector < JacobiRotation < Scalar > > G(restart);
   
   // storage for temporaries
   VectorType t(m), v(m), workspace(m), x_new(m);
 
   // generate first Householder vector
   Ref<VectorType> H0_tail = H.col(0).tail(m - 1);
   RealScalar beta;
   r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
   w(0) = Scalar(beta);
   
   for (Index k = 1; k <= restart; ++k)
   {
     ++iters;
 
     v = VectorType::Unit(m, k - 1);
 
     // apply Householder reflections H_{1} ... H_{k-1} to v
     // TODO: use a HouseholderSequence
     for (Index 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;
     t.noalias() = mat * v;
     v = precond.solve(t);
 
     // apply Householder reflections H_{k-1} ... H_{1} to v
     // TODO: use a HouseholderSequence
     for (Index 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
         Ref<VectorType> Hk_tail = H.col(k).tail(m - k - 1);
         v.tail(m - k).makeHouseholder(Hk_tail, tau.coeffRef(k), beta);
 
         // apply Householder reflection H_{k} to v
         v.tail(m - k).applyHouseholderOnTheLeft(Hk_tail, tau.coeffRef(k), workspace.data());
       }
     }
 
     if (k > 1)
     {
       for (Index 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);
 
     tol_error = abs(w(k)) / r0Norm;
     bool stop = (k==m || tol_error < tol || iters == maxIters);
 
     if (stop || k == restart)
     {
       // solve upper triangular system
       Ref<VectorType> y = w.head(k);
       H.topLeftCorner(k, k).template triangularView <Upper>().solveInPlace(y);
 
       // use Horner-like scheme to calculate solution vector
       x_new.setZero();
       for (Index i = k - 1; i >= 0; --i)
       {
         x_new(i) += y(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 restart
         p0.noalias() = rhs - mat*x;
         r0 = precond.solve(p0);
 
         // clear Hessenberg matrix and Householder data
         H.setZero();
         w.setZero();
         tau.setZero();
 
         // generate first Householder vector
         r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
         w(0) = Scalar(beta);
       }
     }
   }
 
   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.
   * 
   * GMRES can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
   *
   * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
   */
 template< typename _MatrixType, typename _Preconditioner>
 class GMRES : public IterativeSolverBase<GMRES<_MatrixType,_Preconditioner> >
 {
   typedef IterativeSolverBase<GMRES> Base;
   using Base::matrix;
   using Base::m_error;
   using Base::m_iterations;
   using Base::m_info;
   using Base::m_isInitialized;
 
 private:
   Index m_restart;
 
 public:
   using Base::_solve_impl;
   typedef _MatrixType MatrixType;
   typedef typename MatrixType::Scalar Scalar;
   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.
     */
   Index 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 Index restart) { m_restart=restart; }
 
   /** \internal */
   template<typename Rhs,typename Dest>
   void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const
   {
     m_iterations = Base::maxIterations();
     m_error = Base::m_tolerance;
     bool ret = internal::gmres(matrix(), b, x, Base::m_preconditioner, m_iterations, m_restart, m_error);
     m_info = (!ret) ? NumericalIssue
           : m_error <= Base::m_tolerance ? Success
           : NoConvergence;
   }
 
 protected:
 
 };
 
 } // end namespace Eigen
 
 #endif // EIGEN_GMRES_H

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