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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>
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
// Alternatively, you can redistribute it and/or
// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
// the License, or (at your option) any later version.
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
// GNU General Public License for more details.
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.


namespace internal {

/** \internal Low-level conjugate gradient algorithm
  * \param mat The matrix A
  * \param rhs The right hand side vector b
  * \param x On input and initial solution, on output the computed solution.
  * \param precond A preconditioner being able to efficiently solve for an
  *                approximation of Ax=b (regardless of b)
  * \param iters On input the max number of iteration, on output the number of performed iterations.
  * \param tol_error On input the tolerance error, on output an estimation of the relative error.
template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
00041 void conjugate_gradient(const MatrixType& mat, const Rhs& rhs, Dest& x,
                        const Preconditioner& precond, int& iters,
                        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;
  RealScalar tol = tol_error;
  int maxIters = iters;
  int n = mat.cols();
  VectorType residual = rhs - mat * x; //initial residual
  VectorType p(n);

  p = precond.solve(residual);      //initial search direction

  VectorType z(n), tmp(n);
  RealScalar absNew = internal::real(residual.dot(p));  // the square of the absolute value of r scaled by invM
  RealScalar absInit = absNew;          // the initial absolute value

  int i = 0;
  while ((i < maxIters) && (absNew > tol*tol*absInit))
    tmp.noalias() = mat * p;              // the bottleneck of the algorithm

    Scalar alpha = absNew / p.dot(tmp);   // the amount we travel on dir
    x += alpha * p;                       // update solution
    residual -= alpha * tmp;              // update residue
    z = precond.solve(residual);          // approximately solve for "A z = residual"

    RealScalar absOld = absNew;
    absNew = internal::real(residual.dot(z));     // update the absolute value of r
    RealScalar beta = absNew / absOld;            // calculate the Gram-Schmidit value used to create the new search direction
    p = z + beta * p;                             // update search direction

  tol_error = sqrt(abs(absNew / absInit));
  iters = i;


template< typename _MatrixType, int _UpLo=Lower,
          typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
class ConjugateGradient;

namespace internal {

template< typename _MatrixType, int _UpLo, typename _Preconditioner>
00094 struct traits<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> >
  typedef _MatrixType MatrixType;
  typedef _Preconditioner Preconditioner;


/** \ingroup IterativeLinearSolvers_Module
  * \brief A conjugate gradient solver for sparse self-adjoint problems
  * This class allows to solve for A.x = b sparse linear problems using a conjugate gradient algorithm.
  * The sparse matrix A must be selfadjoint. 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 _UpLo the triangular part that will be used for the computations. It can be Lower
  *               or Upper. Default is Lower.
  * \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 default are 1000 max 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
  * ConjugateGradient<SparseMatrix<double> > cg;
  * cg.compute(A);
  * x = cg.solve(b);
  * std::cout << "#iterations:     " << cg.iterations() << std::endl;
  * std::cout << "estimated error: " << cg.error()      << std::endl;
  * // update b, and solve again
  * x = cg.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. Here is a step by
  * step execution example starting with a random guess and printing the evolution
  * of the estimated error:
  * * \code
  * x = VectorXd::Random(n);
  * cg.setMaxIterations(1);
  * int i = 0;
  * do {
  *   x = cg.solveWithGuess(b,x);
  *   std::cout << i << " : " << cg.error() << std::endl;
  *   ++i;
  * } while (cg.info()!=Success && i<100);
  * \endcode
  * Note that such a step by step excution is slightly slower.
  * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
template< typename _MatrixType, int _UpLo, typename _Preconditioner>
00151 class ConjugateGradient : public IterativeSolverBase<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> >
  typedef IterativeSolverBase<ConjugateGradient> Base;
  using Base::mp_matrix;
  using Base::m_error;
  using Base::m_iterations;
  using Base::m_info;
  using Base::m_isInitialized;
  typedef _MatrixType MatrixType;
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::Index Index;
  typedef typename MatrixType::RealScalar RealScalar;
  typedef _Preconditioner Preconditioner;

  enum {
    UpLo = _UpLo


  /** Default constructor. */
00173   ConjugateGradient() : Base() {}

  /** 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.
00185   ConjugateGradient(const MatrixType& A) : Base(A) {}

  ~ConjugateGradient() {}
  /** \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<ConjugateGradient, Rhs, Guess>
00196   solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const
    eigen_assert(m_isInitialized && "ConjugateGradient is not initialized.");
              && "ConjugateGradient::solve(): invalid number of rows of the right hand side matrix b");
    return internal::solve_retval_with_guess
            <ConjugateGradient, Rhs, Guess>(*this, b.derived(), x0);

  /** \internal */
  template<typename Rhs,typename Dest>
00207   void _solveWithGuess(const Rhs& b, Dest& x) const
    m_iterations = Base::m_maxIterations;
    m_error = Base::m_tolerance;

    for(int j=0; j<b.cols(); ++j)
      m_iterations = Base::m_maxIterations;
      m_error = Base::m_tolerance;

      typename Dest::ColXpr xj(x,j);
      internal::conjugate_gradient(mp_matrix->template selfadjointView<UpLo>(), b.col(j), xj,
                                   Base::m_preconditioner, m_iterations, m_error);

    m_isInitialized = true;
    m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
  /** \internal */
  template<typename Rhs,typename Dest>
00228   void _solve(const Rhs& b, Dest& x) const



namespace internal {

template<typename _MatrixType, int _UpLo, typename _Preconditioner, typename Rhs>
00242 struct solve_retval<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner>, Rhs>
  : solve_retval_base<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner>, Rhs>
  typedef ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> Dec;

  template<typename Dest> void evalTo(Dest& dst) const



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