The Induced Dimension Reduction method (IDR(s)) is a short-recurrences Krylov method for sparse square problems. More...

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

Public Types
typedef MatrixType_	MatrixType

typedef Preconditioner_	Preconditioner

typedef MatrixType::RealScalar	RealScalar

typedef MatrixType::Scalar	Scalar

Public Types inherited from Eigen::IterativeSolverBase< IDRS< MatrixType_, Preconditioner_ > >
typedef internal::traits< Derived >::MatrixType	MatrixType

typedef internal::traits< Derived >::Preconditioner	Preconditioner

typedef MatrixType::RealScalar	RealScalar

typedef MatrixType::Scalar	Scalar

typedef MatrixType::StorageIndex	StorageIndex

Public Member Functions
template<typename Rhs , typename Dest >
void	_solve_vector_with_guess_impl (const Rhs &b, Dest &x) const

	IDRS ()

template<typename MatrixDerived >
	IDRS (const EigenBase< MatrixDerived > &A)

void	setAngle (RealScalar angle)

void	setResidualUpdate (bool update)

void	setS (Index S)

void	setSmoothing (bool smoothing)

Public Member Functions inherited from Eigen::IterativeSolverBase< IDRS< MatrixType_, Preconditioner_ > >
void	_solve_impl (const Rhs &b, Dest &x) 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_with_guess_impl (const Rhs &b, SparseMatrixBase< DestDerived > &aDest) const

Derived &	analyzePattern (const EigenBase< MatrixDerived > &A)

EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

Derived &	compute (const EigenBase< MatrixDerived > &A)

Derived &	derived ()

const Derived &	derived () const

RealScalar	error () const

Derived &	factorize (const EigenBase< MatrixDerived > &A)

ComputationInfo	info () const

Index	iterations () const

	IterativeSolverBase ()

	IterativeSolverBase (const EigenBase< MatrixDerived > &A)

	IterativeSolverBase (IterativeSolverBase &&)=default

Index	maxIterations () const

Preconditioner &	preconditioner ()

const Preconditioner &	preconditioner () const

EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

Derived &	setMaxIterations (Index maxIters)

Derived &	setTolerance (const RealScalar &tolerance)

const SolveWithGuess< Derived, Rhs, Guess >	solveWithGuess (const MatrixBase< Rhs > &b, const Guess &x0) const

RealScalar	tolerance () const

	~IterativeSolverBase ()

Public Member Functions inherited from Eigen::SparseSolverBase< class >
Derived &	derived ()

const Derived &	derived () const

const Solve< Derived, Rhs >	solve (const MatrixBase< Rhs > &b) const

const Solve< Derived, Rhs >	solve (const SparseMatrixBase< Rhs > &b) const

	SparseSolverBase ()

	SparseSolverBase (SparseSolverBase &&other)

	~SparseSolverBase ()

Private Types
typedef IterativeSolverBase< IDRS >	Base

Private Member Functions
const ActualMatrixType &	matrix () const

Private Attributes
RealScalar	m_angle

RealScalar	m_error

ComputationInfo	m_info

bool	m_isInitialized

Index	m_iterations

bool	m_residual

Index	m_S

bool	m_smoothing

Additional Inherited Members
Public Attributes inherited from Eigen::IterativeSolverBase< IDRS< MatrixType_, Preconditioner_ > >
	ColsAtCompileTime

	MaxColsAtCompileTime

Protected Types inherited from Eigen::IterativeSolverBase< IDRS< MatrixType_, Preconditioner_ > >
typedef MatrixWrapper::ActualMatrixType	ActualMatrixType

typedef SparseSolverBase< Derived >	Base

typedef internal::generic_matrix_wrapper< MatrixType >	MatrixWrapper

Protected Member Functions inherited from Eigen::IterativeSolverBase< IDRS< MatrixType_, Preconditioner_ > >
void	grab (const InputType &A)

void	init ()

const ActualMatrixType &	matrix () const

Protected Attributes inherited from Eigen::IterativeSolverBase< IDRS< MatrixType_, Preconditioner_ > >
bool	m_analysisIsOk

RealScalar	m_error

bool	m_factorizationIsOk

ComputationInfo	m_info

bool	m_isInitialized

Index	m_iterations

MatrixWrapper	m_matrixWrapper

Index	m_maxIterations

Preconditioner	m_preconditioner

RealScalar	m_tolerance

Protected Attributes inherited from Eigen::SparseSolverBase< class >
bool	m_isInitialized

Detailed Description

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

The Induced Dimension Reduction method (IDR(s)) is a short-recurrences Krylov method for sparse square problems.

This class allows to solve for A.x = b sparse linear problems. The vectors x and b can be either dense or sparse. he Induced Dimension Reduction method, IDR(), is a robust and efficient short-recurrence Krylov subspace method for solving large nonsymmetric systems of linear equations.

For indefinite systems IDR(S) outperforms both BiCGStab and BiCGStab(L). Additionally, IDR(S) can handle matrices with complex eigenvalues more efficiently than BiCGStab.

Many problems that do not converge for BiCGSTAB converge for IDR(s) (for larger values of s). And if both methods converge the convergence for IDR(s) is typically much faster for difficult systems (for example indefinite problems).

IDR(s) is a limited memory finite termination method. In exact arithmetic it converges in at most N+N/s iterations, with N the system size. It uses a fixed number of 4+3s vector. In comparison, BiCGSTAB terminates in 2N iterations and uses 7 vectors. GMRES terminates in at most N iterations, and uses I+3 vectors, with I the number of iterations. Restarting GMRES limits the memory consumption, but destroys the finite termination property.

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

This class follows the sparse solver concept .

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.

The tolerance corresponds to the relative residual error: |Ax-b|/|b|

Performance: when using sparse matrices, best performance is achied for a row-major sparse matrix format. Moreover, in this case multi-threading can be exploited if the user code is compiled with OpenMP enabled. See Eigen and multi-threading for details.

By default the iterations start with x=0 as an initial guess of the solution. One can control the start using the solveWithGuess() method.

IDR(s) can also be used in a matrix-free context, see the following example .

See also: class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner

Definition at line 304 of file IDRS.h.

Member Typedef Documentation

◆ Base

template<typename MatrixType_ , typename Preconditioner_ >

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

private

Definition at line 312 of file IDRS.h.

◆ MatrixType

template<typename MatrixType_ , typename Preconditioner_ >

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

Definition at line 306 of file IDRS.h.

◆ Preconditioner

template<typename MatrixType_ , typename Preconditioner_ >

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

Definition at line 309 of file IDRS.h.

◆ RealScalar

template<typename MatrixType_ , typename Preconditioner_ >

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

Definition at line 308 of file IDRS.h.

◆ Scalar

template<typename MatrixType_ , typename Preconditioner_ >

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

Definition at line 307 of file IDRS.h.

Constructor & Destructor Documentation

◆ IDRS() [1/2]

template<typename MatrixType_ , typename Preconditioner_ >

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

inline

Default constructor.

Definition at line 325 of file IDRS.h.

325 : m_S(4), m_smoothing(false), m_angle(RealScalar(0.7)), m_residual(false) {}

Eigen::IDRS::m_residual

bool m_residual

Definition: IDRS.h:321

Eigen::IDRS::RealScalar

MatrixType::RealScalar RealScalar

Definition: IDRS.h:308

Eigen::IDRS::m_smoothing

bool m_smoothing

Definition: IDRS.h:319

Eigen::IDRS::m_S

Index m_S

Definition: IDRS.h:318

Eigen::IDRS::m_angle

RealScalar m_angle

Definition: IDRS.h:320

◆ IDRS() [2/2]

template<typename MatrixType_ , typename Preconditioner_ >

template<typename MatrixDerived >

Eigen::IDRS< MatrixType_, Preconditioner_ >::IDRS ( 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.

Definition at line 338 of file IDRS.h.

339 : Base(A.derived()), m_S(4), m_smoothing(false), m_angle(RealScalar(0.7)), m_residual(false) {}

A

SparseMatrix< double > A(n, n)

Eigen::IDRS::Base

IterativeSolverBase< IDRS > Base

Definition: IDRS.h:312

EigenBase< SparseMatrix< double, Options_, StorageIndex_ > >::derived

SparseMatrix< double, Options_, StorageIndex_ > & derived()

Member Function Documentation

◆ _solve_vector_with_guess_impl()

template<typename MatrixType_ , typename Preconditioner_ >

template<typename Rhs , typename Dest >

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

inline

Loops over the number of columns of b and does the following:

sets the tolerance and maxIterations
Calls the function that has the core solver routine

Definition at line 347 of file IDRS.h.

                                                                   {
     m_iterations = Base::maxIterations();
     m_error = Base::m_tolerance;
  
     bool ret = internal::idrs(matrix(), b, x, Base::m_preconditioner, m_iterations, m_error, m_S, m_smoothing, m_angle,
                               m_residual);
  
     m_info = (!ret) ? NumericalIssue : m_error <= Base::m_tolerance ? Success : NoConvergence;
   }

◆ matrix()

template<typename MatrixType_ , typename Preconditioner_ >

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

private

◆ setAngle()

template<typename MatrixType_ , typename Preconditioner_ >

void Eigen::IDRS< MatrixType_, Preconditioner_ >::setAngle ( RealScalar angle )

inline

The angle must be a real scalar. In IDR(s), a value for the iteration parameter omega must be chosen in every s+1th step. The most natural choice is to select a value to minimize the norm of the next residual. This corresponds to the parameter omega = 0. In practice, this may lead to values of omega that are so small that the other iteration parameters cannot be computed with sufficient accuracy. In such cases it is better to increase the value of omega sufficiently such that a compromise is reached between accurate computations and reduction of the residual norm. The parameter angle =0.7 (”maintaining the convergence strategy”) results in such a compromise.

Definition at line 384 of file IDRS.h.

384 { m_angle = angle; }

◆ setResidualUpdate()

template<typename MatrixType_ , typename Preconditioner_ >

void Eigen::IDRS< MatrixType_, Preconditioner_ >::setResidualUpdate ( bool update )

inline

The parameter replace is a logical that determines whether a residual replacement strategy is employed to increase the accuracy of the solution.

Definition at line 389 of file IDRS.h.

389 { m_residual = update; }

◆ setS()

template<typename MatrixType_ , typename Preconditioner_ >

void Eigen::IDRS< MatrixType_, Preconditioner_ >::setS ( Index S )

inline

Sets the parameter S, indicating the dimension of the shadow space. Default is 4

Definition at line 358 of file IDRS.h.

                      {
     if (S < 1) {
       S = 4;
     }
  
     m_S = S;
   }

◆ setSmoothing()

template<typename MatrixType_ , typename Preconditioner_ >

void Eigen::IDRS< MatrixType_, Preconditioner_ >::setSmoothing ( bool smoothing )

inline

Switches off and on smoothing. Residual smoothing results in monotonically decreasing residual norms at the expense of two extra vectors of storage and a few extra vector operations. Although monotonic decrease of the residual norms is a desirable property, the rate of convergence of the unsmoothed process and the smoothed process is basically the same. Default is off

Definition at line 372 of file IDRS.h.

372 { m_smoothing = smoothing; }

Member Data Documentation

◆ m_angle

template<typename MatrixType_ , typename Preconditioner_ >

RealScalar Eigen::IDRS< MatrixType_, Preconditioner_ >::m_angle

private

Definition at line 320 of file IDRS.h.

◆ m_error

template<typename MatrixType_ , typename Preconditioner_ >

RealScalar Eigen::IterativeSolverBase< class >::m_error

private

◆ m_info

template<typename MatrixType_ , typename Preconditioner_ >

ComputationInfo Eigen::IterativeSolverBase< class >::m_info

private

◆ m_isInitialized

template<typename MatrixType_ , typename Preconditioner_ >

bool Eigen::IterativeSolverBase< class >::m_isInitialized

private

◆ m_iterations

template<typename MatrixType_ , typename Preconditioner_ >

Index Eigen::IterativeSolverBase< class >::m_iterations

private

◆ m_residual

template<typename MatrixType_ , typename Preconditioner_ >

bool Eigen::IDRS< MatrixType_, Preconditioner_ >::m_residual

private

Definition at line 321 of file IDRS.h.

◆ m_S

template<typename MatrixType_ , typename Preconditioner_ >

Index Eigen::IDRS< MatrixType_, Preconditioner_ >::m_S

private

Definition at line 318 of file IDRS.h.

◆ m_smoothing

template<typename MatrixType_ , typename Preconditioner_ >

bool Eigen::IDRS< MatrixType_, Preconditioner_ >::m_smoothing

private

Definition at line 319 of file IDRS.h.

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

IDRS.h

Public Types

Public Member Functions

Private Types

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

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

Member Typedef Documentation

◆ Base

◆ MatrixType

◆ Preconditioner

◆ RealScalar

◆ Scalar

Constructor & Destructor Documentation

◆ IDRS() [1/2]

◆ IDRS() [2/2]

Member Function Documentation

◆ _solve_vector_with_guess_impl()

◆ matrix()

◆ setAngle()

◆ setResidualUpdate()

◆ setS()

◆ setSmoothing()

Member Data Documentation

◆ m_angle

◆ m_error

◆ m_info

◆ m_isInitialized

◆ m_iterations

◆ m_residual

◆ m_S

◆ m_smoothing

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