src/IterativeLinearSolvers/ConjugateGradient.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2011 Gael Guennebaud <gael.guennebaud (at) inria.fr>
//
// 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_CONJUGATE_GRADIENT_H
#define EIGEN_CONJUGATE_GRADIENT_H

namespace Eigen {

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>
EIGEN_DONT_INLINE
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

  RealScalar rhsNorm2 = rhs.squaredNorm();
  if(rhsNorm2 == 0)
  {
    x.setZero();
    iters = 0;
    tol_error = 0;
    return;
  }
  RealScalar threshold = tol*tol*rhsNorm2;
  RealScalar residualNorm2 = residual.squaredNorm();
  if (residualNorm2 < threshold)
  {
    iters = 0;
    tol_error = sqrt(residualNorm2 / rhsNorm2);
    return;
  }

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

  VectorType z(n), tmp(n);
  RealScalar absNew = numext::real(residual.dot(p));  // the square of the absolute value of r scaled by invM
  int i = 0;
  while(i < maxIters)
  {
    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

    residualNorm2 = residual.squaredNorm();
    if(residualNorm2 < threshold)
      break;

    z = precond.solve(residual);          // approximately solve for "A z = residual"

    RealScalar absOld = absNew;
    absNew = numext::real(residual.dot(z));     // update the absolute value of r
    RealScalar beta = absNew / absOld;            // calculate the Gram-Schmidt value used to create the new search direction
    p = z + beta * p;                             // update search direction
    i++;
  }
  tol_error = sqrt(residualNorm2 / rhsNorm2);
  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>
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 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,
  *               Upper, or Lower|Upper in which the full matrix entries will be considered. 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 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
  * 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.
  *
  * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
  */
template< typename _MatrixType, int _UpLo, typename _Preconditioner>
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;
public:
  typedef _MatrixType MatrixType;
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::Index Index;
  typedef typename MatrixType::RealScalar RealScalar;
  typedef _Preconditioner Preconditioner;

  enum {
    UpLo = _UpLo
  };

public:

  /** Default constructor. */
  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.
    */
  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>
  solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const
  {
    eigen_assert(m_isInitialized && "ConjugateGradient is not initialized.");
    eigen_assert(Base::rows()==b.rows()
              && "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>
  void _solveWithGuess(const Rhs& b, Dest& x) const
  {
    typedef typename internal::conditional<UpLo==(Lower|Upper),
                                           const MatrixType&,
                                           SparseSelfAdjointView<const MatrixType, UpLo>
                                          >::type MatrixWrapperType;
    m_iterations = Base::maxIterations();
    m_error = Base::m_tolerance;

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

      typename Dest::ColXpr xj(x,j);
      internal::conjugate_gradient(MatrixWrapperType(*mp_matrix), 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>
  void _solve(const Rhs& b, Dest& x) const
  {
    x.setZero();
    _solveWithGuess(b,x);
  }

protected:

};


namespace internal {

template<typename _MatrixType, int _UpLo, typename _Preconditioner, typename Rhs>
struct solve_retval<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner>, Rhs>
  : solve_retval_base<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner>, Rhs>
{
  typedef ConjugateGradient<_MatrixType,_UpLo,_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_CONJUGATE_GRADIENT_H