Eigen::DGMRES< MatrixType_, Preconditioner_ > Class Template Reference

A Restarted GMRES with deflation. This class implements a modification of the GMRES solver for sparse linear systems. The basis is built with modified Gram-Schmidt. At each restart, a few approximated eigenvectors corresponding to the smallest eigenvalues are used to build a preconditioner for the next cycle. This preconditioner for deflation can be combined with any other preconditioner, the IncompleteLUT for instance. The preconditioner is applied at right of the matrix and the combination is multiplicative. More...

#include <DGMRES.h>

Inheritance diagram for Eigen::DGMRES< MatrixType_, Preconditioner_ >:

Public Types
typedef MatrixType_	MatrixType
typedef MatrixType::Scalar	Scalar
typedef MatrixType::StorageIndex	StorageIndex
typedef MatrixType::RealScalar	RealScalar
typedef Preconditioner_	Preconditioner
typedef Matrix< Scalar, Dynamic, Dynamic >	DenseMatrix
typedef Matrix< RealScalar, Dynamic, Dynamic >	DenseRealMatrix
typedef Matrix< Scalar, Dynamic, 1 >	DenseVector
typedef Matrix< RealScalar, Dynamic, 1 >	DenseRealVector
typedef Matrix< std::complex< RealScalar >, Dynamic, 1 >	ComplexVector
Public Types inherited from Eigen::IterativeSolverBase< DGMRES< MatrixType_, Preconditioner_ > >
enum
typedef internal::traits< DGMRES< MatrixType_, Preconditioner_ > >::MatrixType	MatrixType
typedef internal::traits< DGMRES< MatrixType_, Preconditioner_ > >::Preconditioner	Preconditioner
typedef MatrixType::Scalar	Scalar
typedef MatrixType::StorageIndex	StorageIndex
typedef MatrixType::RealScalar	RealScalar

Public Member Functions
	DGMRES ()
template<typename MatrixDerived >
	DGMRES (const EigenBase< MatrixDerived > &A)
	~DGMRES ()
template<typename Rhs , typename Dest >
void	_solve_vector_with_guess_impl (const Rhs &b, Dest &x) const
Index	restart ()
void	set_restart (const Index restart)
void	setEigenv (const Index neig)
Index	deflSize ()
void	setMaxEigenv (const Index maxNeig)
template<typename Rhs , typename Dest >
void	_solve_impl (const Rhs &b, Dest &x) const
template<typename Rhs , typename DestDerived >
void	_solve_with_guess_impl (const Rhs &b, SparseMatrixBase< DestDerived > &aDest) const
template<typename Rhs , typename DestDerived >
std::enable_if_t< Rhs::ColsAtCompileTime !=1 &&DestDerived::ColsAtCompileTime !=1 >	_solve_with_guess_impl (const Rhs &b, MatrixBase< DestDerived > &aDest) const
template<typename Rhs , typename DestDerived >
std::enable_if_t< Rhs::ColsAtCompileTime==1||DestDerived::ColsAtCompileTime==1 >	_solve_with_guess_impl (const Rhs &b, MatrixBase< DestDerived > &dest) const
Public Member Functions inherited from Eigen::IterativeSolverBase< DGMRES< MatrixType_, Preconditioner_ > >
	IterativeSolverBase ()
	IterativeSolverBase (const EigenBase< MatrixDerived > &A)
	IterativeSolverBase (IterativeSolverBase &&)=default
	~IterativeSolverBase ()
DGMRES< MatrixType_, Preconditioner_ > &	analyzePattern (const EigenBase< MatrixDerived > &A)
DGMRES< MatrixType_, Preconditioner_ > &	factorize (const EigenBase< MatrixDerived > &A)
DGMRES< MatrixType_, Preconditioner_ > &	compute (const EigenBase< MatrixDerived > &A)
EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT
EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT
RealScalar	tolerance () const
DGMRES< MatrixType_, Preconditioner_ > &	setTolerance (const RealScalar &tolerance)
Preconditioner &	preconditioner ()
const Preconditioner &	preconditioner () const
Index	maxIterations () const
DGMRES< MatrixType_, Preconditioner_ > &	setMaxIterations (Index maxIters)
Index	iterations () const
RealScalar	error () const
const SolveWithGuess< DGMRES< MatrixType_, Preconditioner_ >, Rhs, Guess >	solveWithGuess (const MatrixBase< Rhs > &b, const Guess &x0) const
ComputationInfo	info () const
void	_solve_with_guess_impl (const Rhs &b, SparseMatrixBase< DestDerived > &aDest) const
std::enable_if_t< Rhs::ColsAtCompileTime !=1 &&DestDerived::ColsAtCompileTime !=1 >	_solve_with_guess_impl (const Rhs &b, MatrixBase< DestDerived > &aDest) const
std::enable_if_t< Rhs::ColsAtCompileTime==1||DestDerived::ColsAtCompileTime==1 >	_solve_with_guess_impl (const Rhs &b, MatrixBase< DestDerived > &dest) const
void	_solve_impl (const Rhs &b, Dest &x) const
DGMRES< MatrixType_, Preconditioner_ > &	derived ()
const DGMRES< MatrixType_, Preconditioner_ > &	derived () const
Public Member Functions inherited from Eigen::SparseSolverBase< Derived >
	SparseSolverBase ()
	SparseSolverBase (SparseSolverBase &&other)
	~SparseSolverBase ()
Derived &	derived ()
const Derived &	derived () const
template<typename Rhs >
const Solve< Derived, Rhs >	solve (const MatrixBase< Rhs > &b) const
template<typename Rhs >
const Solve< Derived, Rhs >	solve (const SparseMatrixBase< Rhs > &b) const
template<typename Rhs , typename Dest >
void	_solve_impl (const SparseMatrixBase< Rhs > &b, SparseMatrixBase< Dest > &dest) const

Protected Member Functions
template<typename Rhs , typename Dest >
void	dgmres (const MatrixType &mat, const Rhs &rhs, Dest &x, const Preconditioner &precond) const
	Perform several cycles of restarted GMRES with modified Gram Schmidt,. More...
template<typename Dest >
Index	dgmresCycle (const MatrixType &mat, const Preconditioner &precond, Dest &x, DenseVector &r0, RealScalar &beta, const RealScalar &normRhs, Index &nbIts) const
	Perform one restart cycle of DGMRES. More...
Index	dgmresComputeDeflationData (const MatrixType &mat, const Preconditioner &precond, const Index &it, StorageIndex &neig) const
template<typename RhsType , typename DestType >
Index	dgmresApplyDeflation (const RhsType &In, DestType &Out) const
ComplexVector	schurValues (const ComplexSchur< DenseMatrix > &schurofH) const
ComplexVector	schurValues (const RealSchur< DenseMatrix > &schurofH) const
void	dgmresInitDeflation (Index &rows) const
Protected Member Functions inherited from Eigen::IterativeSolverBase< DGMRES< MatrixType_, Preconditioner_ > >
void	init ()
const ActualMatrixType &	matrix () const
void	grab (const InputType &A)

Protected Attributes
DenseMatrix	m_V
DenseMatrix	m_H
DenseMatrix	m_Hes
Index	m_restart
DenseMatrix	m_U
DenseMatrix	m_MU
DenseMatrix	m_T
PartialPivLU< DenseMatrix >	m_luT
StorageIndex	m_neig
Index	m_r
Index	m_maxNeig
RealScalar	m_lambdaN
bool	m_isDeflAllocated
bool	m_isDeflInitialized
RealScalar	m_smv
bool	m_force
Protected Attributes inherited from Eigen::IterativeSolverBase< DGMRES< MatrixType_, Preconditioner_ > >
MatrixWrapper	m_matrixWrapper
Preconditioner	m_preconditioner
Index	m_maxIterations
RealScalar	m_tolerance
RealScalar	m_error
Index	m_iterations
ComputationInfo	m_info
bool	m_analysisIsOk
bool	m_factorizationIsOk
bool	m_isInitialized
Protected Attributes inherited from Eigen::SparseSolverBase< Derived >
bool	m_isInitialized

Private Types
typedef IterativeSolverBase< DGMRES >	Base

Private Member Functions
const ActualMatrixType &	matrix () const

Private Attributes
RealScalar	m_error
ComputationInfo	m_info
bool	m_isInitialized
Index	m_iterations
RealScalar	m_tolerance

Additional Inherited Members
Protected Types inherited from Eigen::IterativeSolverBase< DGMRES< MatrixType_, Preconditioner_ > >
typedef SparseSolverBase< DGMRES< MatrixType_, Preconditioner_ > >	Base
typedef internal::generic_matrix_wrapper< MatrixType >	MatrixWrapper
typedef MatrixWrapper::ActualMatrixType	ActualMatrixType

Detailed Description

template<typename MatrixType_, typename Preconditioner_>
class Eigen::DGMRES< MatrixType_, Preconditioner_ >

A Restarted GMRES with deflation. This class implements a modification of the GMRES solver for sparse linear systems. The basis is built with modified Gram-Schmidt. At each restart, a few approximated eigenvectors corresponding to the smallest eigenvalues are used to build a preconditioner for the next cycle. This preconditioner for deflation can be combined with any other preconditioner, the IncompleteLUT for instance. The preconditioner is applied at right of the matrix and the combination is multiplicative.

Template Parameters

MatrixType_	the type of the sparse matrix A, can be a dense or a sparse matrix.
Preconditioner_	the type of the preconditioner. Default is DiagonalPreconditioner Typical usage : SparseMatrix<double> A; VectorXd x, b; //Fill A and b ... DGMRES<SparseMatrix<double> > solver; solver.set_restart(30); // Set restarting value solver.setEigenv(1); // Set the number of eigenvalues to deflate solver.compute(A); x = solver.solve(b);

DGMRES can also be used in a matrix-free context, see the following example .

References : [1] D. NUENTSA WAKAM and F. PACULL, Memory Efficient Hybrid Algebraic Solvers for Linear Systems Arising from Compressible Flows, Computers and Fluids, In Press, https://doi.org/10.1016/j.compfluid.2012.03.023 [2] K. Burrage and J. Erhel, On the performance of various adaptive preconditioned GMRES strategies, 5(1998), 101-121. [3] J. Erhel, K. Burrage and B. Pohl, Restarted GMRES preconditioned by deflation,J. Computational and Applied Mathematics, 69(1996), 303-318.

Member Typedef Documentation

Base

template<typename MatrixType_ , typename Preconditioner_ >

typedef IterativeSolverBase<DGMRES> Eigen::DGMRES< MatrixType_, Preconditioner_ >::Base

private

ComplexVector

template<typename MatrixType_ , typename Preconditioner_ >

typedef Matrix<std::complex<RealScalar>, Dynamic, 1> Eigen::DGMRES< MatrixType_, Preconditioner_ >::ComplexVector

DenseMatrix

template<typename MatrixType_ , typename Preconditioner_ >

typedef Matrix<Scalar, Dynamic, Dynamic> Eigen::DGMRES< MatrixType_, Preconditioner_ >::DenseMatrix

DenseRealMatrix

template<typename MatrixType_ , typename Preconditioner_ >

typedef Matrix<RealScalar, Dynamic, Dynamic> Eigen::DGMRES< MatrixType_, Preconditioner_ >::DenseRealMatrix

DenseRealVector

template<typename MatrixType_ , typename Preconditioner_ >

typedef Matrix<RealScalar, Dynamic, 1> Eigen::DGMRES< MatrixType_, Preconditioner_ >::DenseRealVector

DenseVector

template<typename MatrixType_ , typename Preconditioner_ >

typedef Matrix<Scalar, Dynamic, 1> Eigen::DGMRES< MatrixType_, Preconditioner_ >::DenseVector

MatrixType

template<typename MatrixType_ , typename Preconditioner_ >

typedef MatrixType_ Eigen::DGMRES< MatrixType_, Preconditioner_ >::MatrixType

Preconditioner

template<typename MatrixType_ , typename Preconditioner_ >

typedef Preconditioner_ Eigen::DGMRES< MatrixType_, Preconditioner_ >::Preconditioner

RealScalar

template<typename MatrixType_ , typename Preconditioner_ >

typedef MatrixType::RealScalar Eigen::DGMRES< MatrixType_, Preconditioner_ >::RealScalar

Scalar

template<typename MatrixType_ , typename Preconditioner_ >

typedef MatrixType::Scalar Eigen::DGMRES< MatrixType_, Preconditioner_ >::Scalar

StorageIndex

template<typename MatrixType_ , typename Preconditioner_ >

typedef MatrixType::StorageIndex Eigen::DGMRES< MatrixType_, Preconditioner_ >::StorageIndex

Constructor & Destructor Documentation

DGMRES() [1/2]

template<typename MatrixType_ , typename Preconditioner_ >

Eigen::DGMRES< MatrixType_, Preconditioner_ >::DGMRES ( )

inline

Default constructor.

      : Base(), m_restart(30), m_neig(0), m_r(0), m_maxNeig(5), m_isDeflAllocated(false), m_isDeflInitialized(false) {}

DGMRES() [2/2]

template<typename MatrixType_ , typename Preconditioner_ >

template<typename MatrixDerived >

Eigen::DGMRES< MatrixType_, Preconditioner_ >::DGMRES ( const EigenBase< MatrixDerived > &  A )

inlineexplicit

Initialize the solver with matrix A for further 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 is changed this class becomes invalid. Call compute() to update it with the new matrix A, or modify a copy of A.

      : Base(A.derived()),
        m_restart(30),
        m_neig(0),
        m_r(0),
        m_maxNeig(5),
        m_isDeflAllocated(false),
        m_isDeflInitialized(false) {}

~DGMRES()

template<typename MatrixType_ , typename Preconditioner_ >

Eigen::DGMRES< MatrixType_, Preconditioner_ >::~DGMRES ( )

inline

{}

Member Function Documentation

_solve_impl()

template<typename MatrixType_ , typename Preconditioner_ >

template<typename Rhs , typename Dest >

void Eigen::IterativeSolverBase< Derived >::_solve_impl ( typename Rhs  , typename Dest    )

inline

default initial guess = 0

                                                {
    x.setZero();
    derived()._solve_with_guess_impl(b, x);
  }

_solve_vector_with_guess_impl()

template<typename MatrixType_ , typename Preconditioner_ >

template<typename Rhs , typename Dest >

void Eigen::DGMRES< MatrixType_, Preconditioner_ >::_solve_vector_with_guess_impl ( const Rhs &  b, Dest &  x  ) const

inline

                                                                  {
    EIGEN_STATIC_ASSERT(Rhs::ColsAtCompileTime == 1 || Dest::ColsAtCompileTime == 1,
                        YOU_TRIED_CALLING_A_VECTOR_METHOD_ON_A_MATRIX);
 
    m_iterations = Base::maxIterations();
    m_error = Base::m_tolerance;
 
    dgmres(matrix(), b, x, Base::m_preconditioner);
  }

References b, Eigen::DGMRES< MatrixType_, Preconditioner_ >::dgmres(), EIGEN_STATIC_ASSERT, Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_error, Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_iterations, Eigen::IterativeSolverBase< Derived >::m_preconditioner, Eigen::IterativeSolverBase< Derived >::m_tolerance, Eigen::DGMRES< MatrixType_, Preconditioner_ >::matrix(), Eigen::IterativeSolverBase< Derived >::maxIterations(), and plotDoE::x.

_solve_with_guess_impl() [1/3]

template<typename MatrixType_ , typename Preconditioner_ >

template<typename Rhs , typename DestDerived >

std::enable_if_t<Rhs::ColsAtCompileTime != 1 && DestDerived::ColsAtCompileTime != 1> Eigen::IterativeSolverBase< Derived >::_solve_with_guess_impl ( typename Rhs  , typename DestDerived    )

inline

                                                          {
    eigen_assert(rows() == b.rows());
 
    Index rhsCols = b.cols();
    DestDerived& dest(aDest.derived());
    ComputationInfo global_info = Success;
    for (Index k = 0; k < rhsCols; ++k) {
      typename DestDerived::ColXpr xk(dest, k);
      typename Rhs::ConstColXpr bk(b, k);
      derived()._solve_vector_with_guess_impl(bk, xk);
 
      // The call to _solve_vector_with_guess updates m_info, so if it failed for a previous column
      // we need to restore it to the worst value.
      if (m_info == NumericalIssue)
        global_info = NumericalIssue;
      else if (m_info == NoConvergence)
        global_info = NoConvergence;
    }
    m_info = global_info;
  }

_solve_with_guess_impl() [2/3]

template<typename MatrixType_ , typename Preconditioner_ >

template<typename Rhs , typename DestDerived >

std::enable_if_t<Rhs::ColsAtCompileTime == 1 || DestDerived::ColsAtCompileTime == 1> Eigen::IterativeSolverBase< Derived >::_solve_with_guess_impl ( typename Rhs  , typename DestDerived    )

inline

                                                         {
    derived()._solve_vector_with_guess_impl(b, dest.derived());
  }

_solve_with_guess_impl() [3/3]

template<typename MatrixType_ , typename Preconditioner_ >

template<typename Rhs , typename DestDerived >

void Eigen::IterativeSolverBase< Derived >::_solve_with_guess_impl ( typename Rhs  , typename DestDerived    )

inline

                                                                                        {
    eigen_assert(rows() == b.rows());
 
    Index rhsCols = b.cols();
    Index size = b.rows();
    DestDerived& dest(aDest.derived());
    typedef typename DestDerived::Scalar DestScalar;
    Eigen::Matrix<DestScalar, Dynamic, 1> tb(size);
    Eigen::Matrix<DestScalar, Dynamic, 1> tx(cols());
    // We do not directly fill dest because sparse expressions have to be free of aliasing issue.
    // For non square least-square problems, b and dest might not have the same size whereas they might alias
    // each-other.
    typename DestDerived::PlainObject tmp(cols(), rhsCols);
    ComputationInfo global_info = Success;
    for (Index k = 0; k < rhsCols; ++k) {
      tb = b.col(k);
      tx = dest.col(k);
      derived()._solve_vector_with_guess_impl(tb, tx);
      tmp.col(k) = tx.sparseView(0);
 
      // The call to _solve_vector_with_guess_impl updates m_info, so if it failed for a previous column
      // we need to restore it to the worst value.
      if (m_info == NumericalIssue)
        global_info = NumericalIssue;
      else if (m_info == NoConvergence)
        global_info = NoConvergence;
    }
    m_info = global_info;
    dest.swap(tmp);
  }

deflSize()

template<typename MatrixType_ , typename Preconditioner_ >

Index Eigen::DGMRES< MatrixType_, Preconditioner_ >::deflSize ( )

inline

Get the size of the deflation subspace size

{ return m_r; }

References Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_r.

dgmres()

template<typename MatrixType_ , typename Preconditioner_ >

template<typename Rhs , typename Dest >

void Eigen::DGMRES< MatrixType_, Preconditioner_ >::dgmres ( const MatrixType &  mat, const Rhs &  rhs, Dest &  x, const Preconditioner &  precond  ) const

protected

Perform several cycles of restarted GMRES with modified Gram Schmidt,.

A right preconditioner is used combined with deflation.

                                                                                       {
  const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
 
  RealScalar normRhs = rhs.norm();
  if (normRhs <= considerAsZero) {
    x.setZero();
    m_error = 0;
    return;
  }
 
  // Initialization
  m_isDeflInitialized = false;
  Index n = mat.rows();
  DenseVector r0(n);
  Index nbIts = 0;
  m_H.resize(m_restart + 1, m_restart);
  m_Hes.resize(m_restart, m_restart);
  m_V.resize(n, m_restart + 1);
  // Initial residual vector and initial norm
  if (x.squaredNorm() == 0) x = precond.solve(rhs);
  r0 = rhs - mat * x;
  RealScalar beta = r0.norm();
 
  m_error = beta / normRhs;
  if (m_error < m_tolerance)
    m_info = Success;
  else
    m_info = NoConvergence;
 
  // Iterative process
  while (nbIts < m_iterations && m_info == NoConvergence) {
    dgmresCycle(mat, precond, x, r0, beta, normRhs, nbIts);
 
    // Compute the new residual vector for the restart
    if (nbIts < m_iterations && m_info == NoConvergence) {
      r0 = rhs - mat * x;
      beta = r0.norm();
    }
  }
}

References beta, min, n, Eigen::NoConvergence, Eigen::SparseMatrix< Scalar_, Options_, StorageIndex_ >::rows(), Eigen::Success, and plotDoE::x.

Referenced by Eigen::DGMRES< MatrixType_, Preconditioner_ >::_solve_vector_with_guess_impl().

dgmresApplyDeflation()

template<typename MatrixType_ , typename Preconditioner_ >

template<typename RhsType , typename DestType >

Index Eigen::DGMRES< MatrixType_, Preconditioner_ >::dgmresApplyDeflation ( const RhsType &  In, DestType &  Out  ) const

protected

                                                                                                    {
  DenseVector x1 = m_U.leftCols(m_r).transpose() * x;
  y = x + m_U.leftCols(m_r) * (m_lambdaN * m_luT.solve(x1) - x1);
  return 0;
}

References plotDoE::x, Global_parameters::x1(), and y.

dgmresComputeDeflationData()

template<typename MatrixType_ , typename Preconditioner_ >

Index Eigen::DGMRES< MatrixType_, Preconditioner_ >::dgmresComputeDeflationData ( const MatrixType &  mat, const Preconditioner &  precond, const Index &  it, StorageIndex &  neig  ) const

protected

                                                                                                 {
  // First, find the Schur form of the Hessenberg matrix H
  std::conditional_t<NumTraits<Scalar>::IsComplex, ComplexSchur<DenseMatrix>, RealSchur<DenseMatrix> > schurofH;
  bool computeU = true;
  DenseMatrix matrixQ(it, it);
  matrixQ.setIdentity();
  schurofH.computeFromHessenberg(m_Hes.topLeftCorner(it, it), matrixQ, computeU);
 
  ComplexVector eig(it);
  Matrix<StorageIndex, Dynamic, 1> perm(it);
  eig = this->schurValues(schurofH);
 
  // Reorder the absolute values of Schur values
  DenseRealVector modulEig(it);
  for (Index j = 0; j < it; ++j) modulEig(j) = std::abs(eig(j));
  perm.setLinSpaced(it, 0, internal::convert_index<StorageIndex>(it - 1));
  internal::sortWithPermutation(modulEig, perm, neig);
 
  if (!m_lambdaN) {
    m_lambdaN = (std::max)(modulEig.maxCoeff(), m_lambdaN);
  }
  // Count the real number of extracted eigenvalues (with complex conjugates)
  Index nbrEig = 0;
  while (nbrEig < neig) {
    if (eig(perm(it - nbrEig - 1)).imag() == RealScalar(0))
      nbrEig++;
    else
      nbrEig += 2;
  }
  // Extract the  Schur vectors corresponding to the smallest Ritz values
  DenseMatrix Sr(it, nbrEig);
  Sr.setZero();
  for (Index j = 0; j < nbrEig; j++) {
    Sr.col(j) = schurofH.matrixU().col(perm(it - j - 1));
  }
 
  // Form the Schur vectors of the initial matrix using the Krylov basis
  DenseMatrix X;
  X = m_V.leftCols(it) * Sr;
  if (m_r) {
    // Orthogonalize X against m_U using modified Gram-Schmidt
    for (Index j = 0; j < nbrEig; j++)
      for (Index k = 0; k < m_r; k++) X.col(j) = X.col(j) - (m_U.col(k).dot(X.col(j))) * m_U.col(k);
  }
 
  // Compute m_MX = A * M^-1 * X
  Index m = m_V.rows();
  if (!m_isDeflAllocated) dgmresInitDeflation(m);
  DenseMatrix MX(m, nbrEig);
  DenseVector tv1(m);
  for (Index j = 0; j < nbrEig; j++) {
    tv1 = mat * X.col(j);
    MX.col(j) = precond.solve(tv1);
  }
 
  // Update m_T = [U'MU U'MX; X'MU X'MX]
  m_T.block(m_r, m_r, nbrEig, nbrEig) = X.transpose() * MX;
  if (m_r) {
    m_T.block(0, m_r, m_r, nbrEig) = m_U.leftCols(m_r).transpose() * MX;
    m_T.block(m_r, 0, nbrEig, m_r) = X.transpose() * m_MU.leftCols(m_r);
  }
 
  // Save X into m_U and m_MX in m_MU
  for (Index j = 0; j < nbrEig; j++) m_U.col(m_r + j) = X.col(j);
  for (Index j = 0; j < nbrEig; j++) m_MU.col(m_r + j) = MX.col(j);
  // Increase the size of the invariant subspace
  m_r += nbrEig;
 
  // Factorize m_T into m_luT
  m_luT.compute(m_T.topLeftCorner(m_r, m_r));
 
  // FIXME CHeck if the factorization was correctly done (nonsingular matrix)
  m_isDeflInitialized = true;
  return 0;
}

References abs(), IsComplex, j, k, m, max, Eigen::PlainObjectBase< Derived >::setZero(), Eigen::internal::sortWithPermutation(), and X.

dgmresCycle()

template<typename MatrixType_ , typename Preconditioner_ >

template<typename Dest >

Index Eigen::DGMRES< MatrixType_, Preconditioner_ >::dgmresCycle ( const MatrixType &  mat, const Preconditioner &  precond, Dest &  x, DenseVector &  r0, RealScalar &  beta, const RealScalar &  normRhs, Index &  nbIts  ) const

protected

Perform one restart cycle of DGMRES.

Parameters

mat	The coefficient matrix
precond	The preconditioner
x	the new approximated solution
r0	The initial residual vector
beta	The norm of the residual computed so far
normRhs	The norm of the right hand side vector
nbIts	The number of iterations

                                                                            {
  // Initialization
  DenseVector g(m_restart + 1);  // Right hand side of the least square problem
  g.setZero();
  g(0) = Scalar(beta);
  m_V.col(0) = r0 / beta;
  m_info = NoConvergence;
  std::vector<JacobiRotation<Scalar> > gr(m_restart);  // Givens rotations
  Index it = 0;                                        // Number of inner iterations
  Index n = mat.rows();
  DenseVector tv1(n), tv2(n);  // Temporary vectors
  while (m_info == NoConvergence && it < m_restart && nbIts < m_iterations) {
    // Apply preconditioner(s) at right
    if (m_isDeflInitialized) {
      dgmresApplyDeflation(m_V.col(it), tv1);  // Deflation
      tv2 = precond.solve(tv1);
    } else {
      tv2 = precond.solve(m_V.col(it));  // User's selected preconditioner
    }
    tv1 = mat * tv2;
 
    // Orthogonalize it with the previous basis in the basis using modified Gram-Schmidt
    Scalar coef;
    for (Index i = 0; i <= it; ++i) {
      coef = tv1.dot(m_V.col(i));
      tv1 = tv1 - coef * m_V.col(i);
      m_H(i, it) = coef;
      m_Hes(i, it) = coef;
    }
    // Normalize the vector
    coef = tv1.norm();
    m_V.col(it + 1) = tv1 / coef;
    m_H(it + 1, it) = coef;
    //     m_Hes(it+1,it) = coef;
 
    // FIXME Check for happy breakdown
 
    // Update Hessenberg matrix with Givens rotations
    for (Index i = 1; i <= it; ++i) {
      m_H.col(it).applyOnTheLeft(i - 1, i, gr[i - 1].adjoint());
    }
    // Compute the new plane rotation
    gr[it].makeGivens(m_H(it, it), m_H(it + 1, it));
    // Apply the new rotation
    m_H.col(it).applyOnTheLeft(it, it + 1, gr[it].adjoint());
    g.applyOnTheLeft(it, it + 1, gr[it].adjoint());
 
    beta = std::abs(g(it + 1));
    m_error = beta / normRhs;
    // std::cerr << nbIts << " Relative Residual Norm " << m_error << std::endl;
    it++;
    nbIts++;
 
    if (m_error < m_tolerance) {
      // The method has converged
      m_info = Success;
      break;
    }
  }
 
  // Compute the new coefficients by solving the least square problem
  //   it++;
  // FIXME  Check first if the matrix is singular ... zero diagonal
  DenseVector nrs(m_restart);
  nrs = m_H.topLeftCorner(it, it).template triangularView<Upper>().solve(g.head(it));
 
  // Form the new solution
  if (m_isDeflInitialized) {
    tv1 = m_V.leftCols(it) * nrs;
    dgmresApplyDeflation(tv1, tv2);
    x = x + precond.solve(tv2);
  } else
    x = x + precond.solve(m_V.leftCols(it) * nrs);
 
  // Go for a new cycle and compute data for deflation
  if (nbIts < m_iterations && m_info == NoConvergence && m_neig > 0 && (m_r + m_neig) < m_maxNeig)
    dgmresComputeDeflationData(mat, precond, it, m_neig);
  return 0;
}

References abs(), adjoint(), beta, i, n, Eigen::NoConvergence, Eigen::SparseMatrix< Scalar_, Options_, StorageIndex_ >::rows(), Eigen::PlainObjectBase< Derived >::setZero(), Eigen::Success, and plotDoE::x.

dgmresInitDeflation()

template<typename MatrixType_ , typename Preconditioner_ >

void Eigen::DGMRES< MatrixType_, Preconditioner_ >::dgmresInitDeflation ( Index &  rows ) const

protected

                                                                                {
  m_U.resize(rows, m_maxNeig);
  m_MU.resize(rows, m_maxNeig);
  m_T.resize(m_maxNeig, m_maxNeig);
  m_lambdaN = 0.0;
  m_isDeflAllocated = true;
}

References rows.

matrix()

template<typename MatrixType_ , typename Preconditioner_ >

const ActualMatrixType& Eigen::IterativeSolverBase< Derived >::matrix

inlineprivate

{ return m_matrixWrapper.matrix(); }

Referenced by Eigen::DGMRES< MatrixType_, Preconditioner_ >::_solve_vector_with_guess_impl().

restart()

template<typename MatrixType_ , typename Preconditioner_ >

Index Eigen::DGMRES< MatrixType_, Preconditioner_ >::restart ( )

inline

Get the restart value

{ return m_restart; }

References Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_restart.

Referenced by Eigen::DGMRES< MatrixType_, Preconditioner_ >::set_restart().

schurValues() [1/2]

template<typename MatrixType_ , typename Preconditioner_ >

DGMRES< MatrixType_, Preconditioner_ >::ComplexVector Eigen::DGMRES< MatrixType_, Preconditioner_ >::schurValues ( const ComplexSchur< DenseMatrix > &  schurofH ) const

inlineprotected

                                                     {
  return schurofH.matrixT().diagonal();
}

References Eigen::ComplexSchur< MatrixType_ >::matrixT().

schurValues() [2/2]

template<typename MatrixType_ , typename Preconditioner_ >

DGMRES< MatrixType_, Preconditioner_ >::ComplexVector Eigen::DGMRES< MatrixType_, Preconditioner_ >::schurValues ( const RealSchur< DenseMatrix > &  schurofH ) const

inlineprotected

                                                  {
  const DenseMatrix& T = schurofH.matrixT();
  Index it = T.rows();
  ComplexVector eig(it);
  Index j = 0;
  while (j < it - 1) {
    if (T(j + 1, j) == Scalar(0)) {
      eig(j) = std::complex<RealScalar>(T(j, j), RealScalar(0));
      j++;
    } else {
      eig(j) = std::complex<RealScalar>(T(j, j), T(j + 1, j));
      eig(j + 1) = std::complex<RealScalar>(T(j, j + 1), T(j + 1, j + 1));
      j++;
    }
  }
  if (j < it - 1) eig(j) = std::complex<RealScalar>(T(j, j), RealScalar(0));
  return eig;
}

References j, and Eigen::RealSchur< MatrixType_ >::matrixT().

set_restart()

template<typename MatrixType_ , typename Preconditioner_ >

void Eigen::DGMRES< MatrixType_, Preconditioner_ >::set_restart ( const Index  restart )

inline

Set the restart value (default is 30)

{ m_restart = restart; }

References Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_restart, and Eigen::DGMRES< MatrixType_, Preconditioner_ >::restart().

setEigenv()

template<typename MatrixType_ , typename Preconditioner_ >

void Eigen::DGMRES< MatrixType_, Preconditioner_ >::setEigenv ( const Index  neig )

inline

Set the number of eigenvalues to deflate at each restart

                                   {
    m_neig = neig;
    if (neig + 1 > m_maxNeig) m_maxNeig = neig + 1;  // To allow for complex conjugates
  }

References Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_maxNeig, and Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_neig.

setMaxEigenv()

template<typename MatrixType_ , typename Preconditioner_ >

void Eigen::DGMRES< MatrixType_, Preconditioner_ >::setMaxEigenv ( const Index  maxNeig )

inline

Set the maximum size of the deflation subspace

{ m_maxNeig = maxNeig; }

References Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_maxNeig.

Member Data Documentation

m_error

template<typename MatrixType_ , typename Preconditioner_ >

RealScalar Eigen::IterativeSolverBase< Derived >::m_error

mutableprivate

Referenced by Eigen::DGMRES< MatrixType_, Preconditioner_ >::_solve_vector_with_guess_impl().

m_force

template<typename MatrixType_ , typename Preconditioner_ >

bool Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_force

mutableprotected

m_H

template<typename MatrixType_ , typename Preconditioner_ >

DenseMatrix Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_H

mutableprotected

m_Hes

template<typename MatrixType_ , typename Preconditioner_ >

DenseMatrix Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_Hes

mutableprotected

m_info

template<typename MatrixType_ , typename Preconditioner_ >

ComputationInfo Eigen::IterativeSolverBase< Derived >::m_info

mutableprivate

m_isDeflAllocated

template<typename MatrixType_ , typename Preconditioner_ >

bool Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_isDeflAllocated

mutableprotected

m_isDeflInitialized

template<typename MatrixType_ , typename Preconditioner_ >

bool Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_isDeflInitialized

mutableprotected

m_isInitialized

template<typename MatrixType_ , typename Preconditioner_ >

bool Eigen::SparseSolverBase< Derived >::m_isInitialized

mutableprivate

m_iterations

template<typename MatrixType_ , typename Preconditioner_ >

Index Eigen::IterativeSolverBase< Derived >::m_iterations

mutableprivate

Referenced by Eigen::DGMRES< MatrixType_, Preconditioner_ >::_solve_vector_with_guess_impl().

m_lambdaN

template<typename MatrixType_ , typename Preconditioner_ >

RealScalar Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_lambdaN

mutableprotected

m_luT

template<typename MatrixType_ , typename Preconditioner_ >

PartialPivLU<DenseMatrix> Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_luT

mutableprotected

m_maxNeig

template<typename MatrixType_ , typename Preconditioner_ >

Index Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_maxNeig

mutableprotected

Referenced by Eigen::DGMRES< MatrixType_, Preconditioner_ >::setEigenv(), and Eigen::DGMRES< MatrixType_, Preconditioner_ >::setMaxEigenv().

m_MU

template<typename MatrixType_ , typename Preconditioner_ >

DenseMatrix Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_MU

mutableprotected

m_neig

template<typename MatrixType_ , typename Preconditioner_ >

StorageIndex Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_neig

mutableprotected

Referenced by Eigen::DGMRES< MatrixType_, Preconditioner_ >::setEigenv().

m_r

template<typename MatrixType_ , typename Preconditioner_ >

Index Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_r

mutableprotected

Referenced by Eigen::DGMRES< MatrixType_, Preconditioner_ >::deflSize().

m_restart

template<typename MatrixType_ , typename Preconditioner_ >

Index Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_restart

mutableprotected

Referenced by Eigen::DGMRES< MatrixType_, Preconditioner_ >::restart(), and Eigen::DGMRES< MatrixType_, Preconditioner_ >::set_restart().

m_smv

template<typename MatrixType_ , typename Preconditioner_ >

RealScalar Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_smv

mutableprotected

m_T

template<typename MatrixType_ , typename Preconditioner_ >

DenseMatrix Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_T

mutableprotected

m_tolerance

template<typename MatrixType_ , typename Preconditioner_ >

RealScalar Eigen::IterativeSolverBase< Derived >::m_tolerance

private

m_U

template<typename MatrixType_ , typename Preconditioner_ >

DenseMatrix Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_U

mutableprotected

m_V

template<typename MatrixType_ , typename Preconditioner_ >

DenseMatrix Eigen::DGMRES< MatrixType_, Preconditioner_ >::m_V

mutableprotected

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The documentation for this class was generated from the following file:

• DGMRES.h