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

#include <IDRS.h>

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

Public Types
typedef MatrixType_	MatrixType

typedef MatrixType::Scalar	Scalar

typedef MatrixType::RealScalar	RealScalar

typedef Preconditioner_	Preconditioner

Public Types inherited from Eigen::IterativeSolverBase< IDRS< MatrixType_, Preconditioner_ > >
enum

typedef internal::traits< IDRS< MatrixType_, Preconditioner_ > >::MatrixType	MatrixType

typedef internal::traits< IDRS< MatrixType_, Preconditioner_ > >::Preconditioner	Preconditioner

typedef MatrixType::Scalar	Scalar

typedef MatrixType::StorageIndex	StorageIndex

typedef MatrixType::RealScalar	RealScalar

Public Member Functions
	IDRS ()

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

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

void	setS (Index S)

void	setSmoothing (bool smoothing)

void	setAngle (RealScalar angle)

void	setResidualUpdate (bool update)

Public Member Functions inherited from Eigen::IterativeSolverBase< IDRS< MatrixType_, Preconditioner_ > >
	IterativeSolverBase ()

	IterativeSolverBase (const EigenBase< MatrixDerived > &A)

	IterativeSolverBase (IterativeSolverBase &&)=default

	~IterativeSolverBase ()

IDRS< MatrixType_, Preconditioner_ > &	analyzePattern (const EigenBase< MatrixDerived > &A)

IDRS< MatrixType_, Preconditioner_ > &	factorize (const EigenBase< MatrixDerived > &A)

IDRS< MatrixType_, Preconditioner_ > &	compute (const EigenBase< MatrixDerived > &A)

EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

RealScalar	tolerance () const

IDRS< MatrixType_, Preconditioner_ > &	setTolerance (const RealScalar &tolerance)

Preconditioner &	preconditioner ()

const Preconditioner &	preconditioner () const

Index	maxIterations () const

IDRS< MatrixType_, Preconditioner_ > &	setMaxIterations (Index maxIters)

Index	iterations () const

RealScalar	error () const

const SolveWithGuess< IDRS< 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

IDRS< MatrixType_, Preconditioner_ > &	derived ()

const IDRS< 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

Private Types
typedef IterativeSolverBase< IDRS >	Base

Private Member Functions
const ActualMatrixType &	matrix () const

Private Attributes
Index	m_S

bool	m_smoothing

RealScalar	m_angle

bool	m_residual

RealScalar	m_error

ComputationInfo	m_info

bool	m_isInitialized

Index	m_iterations

Additional Inherited Members
Protected Types inherited from Eigen::IterativeSolverBase< IDRS< MatrixType_, Preconditioner_ > >
typedef SparseSolverBase< IDRS< MatrixType_, Preconditioner_ > >	Base

typedef internal::generic_matrix_wrapper< MatrixType >	MatrixWrapper

typedef MatrixWrapper::ActualMatrixType	ActualMatrixType

Protected Member Functions inherited from Eigen::IterativeSolverBase< IDRS< MatrixType_, Preconditioner_ > >
void	init ()

const ActualMatrixType &	matrix () const

void	grab (const InputType &A)

Protected Attributes inherited from Eigen::IterativeSolverBase< IDRS< 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

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

\implsparsesolverconcept

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

Member Typedef Documentation

◆ Base

template<typename MatrixType_ , typename Preconditioner_ >

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

private

◆ MatrixType

template<typename MatrixType_ , typename Preconditioner_ >

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

◆ Preconditioner

template<typename MatrixType_ , typename Preconditioner_ >

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

◆ RealScalar

template<typename MatrixType_ , typename Preconditioner_ >

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

◆ Scalar

template<typename MatrixType_ , typename Preconditioner_ >

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

Constructor & Destructor Documentation

◆ IDRS() [1/2]

template<typename MatrixType_ , typename Preconditioner_ >

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

inline

Default constructor.

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

Eigen::IDRS::m_residual

bool m_residual

Definition: IDRS.h:322

Eigen::IDRS::RealScalar

MatrixType::RealScalar RealScalar

Definition: IDRS.h:309

Eigen::IDRS::m_smoothing

bool m_smoothing

Definition: IDRS.h:320

Eigen::IDRS::m_S

Index m_S

Definition: IDRS.h:319

Eigen::IDRS::m_angle

RealScalar m_angle

Definition: IDRS.h:321

◆ 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.

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

A

Matrix< SCALARA, Dynamic, Dynamic, opt_A > A

Definition: bench_gemm.cpp:47

Eigen::IDRS::Base

IterativeSolverBase< IDRS > Base

Definition: IDRS.h:313

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

                                                                   {
     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< Derived >::matrix

inlineprivate

374 { return m_matrixWrapper.matrix(); }

Eigen::IterativeSolverBase< IDRS< MatrixType_, Preconditioner_ > >::m_matrixWrapper

MatrixWrapper m_matrixWrapper

Definition: IterativeSolverBase.h:381

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

◆ 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.

385 { m_angle = angle; }

Jeffery_Solution::angle

double angle(const double &t)

Angular position as a function of time t.

Definition: jeffery_orbit.cc:98

References Jeffery_Solution::angle(), and Eigen::IDRS< MatrixType_, Preconditioner_ >::m_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.

390 { m_residual = update; }

References Eigen::IDRS< MatrixType_, Preconditioner_ >::m_residual.

◆ 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

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

References Eigen::IDRS< MatrixType_, Preconditioner_ >::m_S, and oomph::QuadTreeNames::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

373 { m_smoothing = smoothing; }

References Eigen::IDRS< MatrixType_, Preconditioner_ >::m_smoothing.