Reference documentation for deal.II version Git bed997f895 2020-09-22 11:49:20 -0400
ArpackSolver Class Reference

#include <deal.II/lac/arpack_solver.h>

Inheritance diagram for ArpackSolver:
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## Public Types

enum  WhichEigenvalues {
algebraically_largest, algebraically_smallest, largest_magnitude, smallest_magnitude,
largest_real_part, smallest_real_part, largest_imaginary_part, smallest_imaginary_part,
both_ends
}

using size_type = types::global_dof_index

## Public Member Functions

SolverControlcontrol () const

template<typename VectorType >
void set_initial_vector (const VectorType &vec)

void set_shift (const std::complex< double > sigma)

template<typename VectorType , typename MatrixType1 , typename MatrixType2 , typename INVERSE >
void solve (const MatrixType1 &A, const MatrixType2 &B, const INVERSE &inverse, std::vector< std::complex< double >> &eigenvalues, std::vector< VectorType > &eigenvectors, const unsigned int n_eigenvalues=0)

Public Member Functions inherited from Subscriptor
Subscriptor ()

Subscriptor (const Subscriptor &)

Subscriptor (Subscriptor &&) noexcept

virtual ~Subscriptor ()

Subscriptoroperator= (const Subscriptor &)

Subscriptoroperator= (Subscriptor &&) noexcept

void subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int n_subscriptions () const

template<typename StreamType >
void list_subscribers (StreamType &stream) const

void list_subscribers () const

template<class Archive >
void serialize (Archive &ar, const unsigned int version)

## Protected Attributes

SolverControlsolver_control

bool initial_vector_provided

std::vector< doubleresid

double sigmar

double sigmai

## Static Private Member Functions

static ::ExceptionBaseArpackExcInvalidNumberofEigenvalues (int arg1, int arg2)

static ::ExceptionBaseArpackExcInvalidEigenvectorSize (int arg1, int arg2)

static ::ExceptionBaseArpackExcInvalidEigenvectorSizeNonsymmetric (int arg1, int arg2)

static ::ExceptionBaseArpackExcInvalidEigenvalueSize (int arg1, int arg2)

static ::ExceptionBaseArpackExcInvalidNumberofArnoldiVectors (int arg1, int arg2)

static ::ExceptionBaseArpackExcSmallNumberofArnoldiVectors (int arg1, int arg2)

static ::ExceptionBaseArpackExcArpackIdo (int arg1)

static ::ExceptionBaseArpackExcArpackMode (int arg1)

static ::ExceptionBaseArpackExcArpackInfodsaupd (int arg1)

static ::ExceptionBaseArpackExcArpackInfodnaupd (int arg1)

static ::ExceptionBaseArpackExcArpackInfodseupd (int arg1)

static ::ExceptionBaseArpackExcArpackInfodneupd (int arg1)

static ::ExceptionBaseArpackExcArpackInfoMaxIt (int arg1)

static ::ExceptionBaseArpackExcArpackNoShifts ()

Static Public Member Functions inherited from Subscriptor
static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

## Detailed Description

Interface for using ARPACK. ARPACK is a collection of Fortran77 subroutines designed to solve large scale eigenvalue problems. Here we interface to the routines dnaupd and dneupd of ARPACK. If the operator is specified to be symmetric we use the symmetric interface dsaupd and dseupd of ARPACK instead. The package is designed to compute a few eigenvalues and corresponding eigenvectors of a general n by n matrix A. It is most appropriate for large sparse matrices A.

In this class we make use of the method applied to the generalized eigenspectrum problem $$(A-\lambda B)x=0$$, for $$x\neq0$$; where $$A$$ is a system matrix, $$B$$ is a mass matrix, and $$\lambda, x$$ are a set of eigenvalues and eigenvectors respectively.

The ArpackSolver can be used in application codes with serial objects in the following way:

solver.solve(A, B, OP, lambda, x, size_of_spectrum);

for the generalized eigenvalue problem $$Ax=B\lambda x$$, where the variable size_of_spectrum tells ARPACK the number of eigenvector/eigenvalue pairs to solve for. Here, lambda is a vector that will contain the eigenvalues computed, x a vector that will contain the eigenvectors computed, and OP is an inverse operation for the matrix A. Shift and invert transformation around zero is applied.

Through the AdditionalData the user can specify some of the parameters to be set.

For further information on how the ARPACK routines dsaupd, dseupd, dnaupd and dneupd work and also how to set the parameters appropriately please take a look into the ARPACK manual.

Note
Whenever you eliminate degrees of freedom using AffineConstraints, you generate spurious eigenvalues and eigenvectors. If you make sure that the diagonals of eliminated matrix rows are all equal to one, you get a single additional eigenvalue. But beware that some functions in deal.II set these diagonals to rather arbitrary (from the point of view of eigenvalue problems) values. See also step-36 for an example.

Definition at line 167 of file arpack_solver.h.

## ◆ size_type

Declare the type for container size.

Definition at line 173 of file arpack_solver.h.

## ◆ WhichEigenvalues

An enum that lists the possible choices for which eigenvalues to compute in the solve() function.

Enumerator
algebraically_largest

The algebraically largest eigenvalues.

algebraically_smallest

The algebraically smallest eigenvalues.

largest_magnitude

The eigenvalue with the largest magnitudes.

smallest_magnitude

The eigenvalue with the smallest magnitudes.

largest_real_part

The eigenvalues with the largest real parts.

smallest_real_part

The eigenvalues with the smallest real parts.

largest_imaginary_part

The eigenvalues with the largest imaginary parts.

smallest_imaginary_part

The eigenvalues with the smallest imaginary parts.

both_ends

Compute half of the eigenvalues from the high end of the spectrum and the other half from the low end. If the number of requested eigenvectors is odd, then the extra eigenvector comes from the high end of the spectrum.

Definition at line 180 of file arpack_solver.h.

## ◆ ArpackSolver()

 ArpackSolver::ArpackSolver ( SolverControl & control, const AdditionalData & data = AdditionalData() )
inline

Constructor.

Definition at line 502 of file arpack_solver.h.

## ◆ control()

 SolverControl & ArpackSolver::control ( ) const
inline

Definition at line 895 of file arpack_solver.h.

## ◆ set_initial_vector()

template<typename VectorType >
 void ArpackSolver::set_initial_vector ( const VectorType & vec )
inline

Set initial vector for building Krylov space.

Definition at line 524 of file arpack_solver.h.

## ◆ set_shift()

 void ArpackSolver::set_shift ( const std::complex< double > sigma )
inline

Set shift sigma for shift-and-invert spectral transformation.

If this function is not called, the shift is assumed to be zero.

Definition at line 514 of file arpack_solver.h.

## ◆ solve()

template<typename VectorType , typename MatrixType1 , typename MatrixType2 , typename INVERSE >
 void ArpackSolver::solve ( const MatrixType1 & A, const MatrixType2 & B, const INVERSE & inverse, std::vector< std::complex< double >> & eigenvalues, std::vector< VectorType > & eigenvectors, const unsigned int n_eigenvalues = 0 )
inline

Solve the generalized eigensprectrum problem $$A x=\lambda B x$$ by calling the dsaupd and dseupd or dnaupd and dneupd functions of ARPACK.

The function returns a vector of eigenvalues of length n and a vector of eigenvectors of length n in the symmetric case and of length n+1 in the non-symmetric case. In the symmetric case all eigenvectors are real. In the non-symmetric case complex eigenvalues always occur as complex conjugate pairs. Therefore the eigenvector for an eigenvalue with nonzero complex part is stored by putting the real and the imaginary parts in consecutive real-valued vectors. The eigenvector of the complex conjugate eigenvalue does not need to be stored, since it is just the complex conjugate of the stored eigenvector. Thus, if the last n-th eigenvalue has a nonzero imaginary part, Arpack needs in total n+1 real-valued vectors to store real and imaginary parts of the eigenvectors.

Parameters
 A The operator for which we want to compute eigenvalues. Actually, this parameter is entirely unused. B The inner product of the underlying space, typically the mass matrix. For constrained problems, it can be a partial mass matrix, like for instance the velocity mass matrix of a Stokes problem. Only its function vmult() is used. inverse This is the possibly shifted inverse that is actually used instead of A. Only its function vmult() is used. eigenvalues is a vector of complex numbers in which the eigenvalues are returned. eigenvectors is a real vector of eigenvectors, containing the real parts of all eigenvectors and the imaginary parts of the eigenvectors corresponding to complex conjugate eigenvalue pairs. Therefore, its length should be n in the symmetric case and n+1 in the non-symmetric case. In the non-symmetric case the storage scheme leads for example to the following pattern. Suppose that the first two eigenvalues are real and the third and fourth are a complex conjugate pair. Asking for three eigenpairs results in [real(v1),real(v2), real(v3),imag(v3)]. Note that we get the same pattern if we ask for four eigenpairs in this example, since the fourth eigenvector is simply the complex conjugate of the third one. n_eigenvalues The purpose of this parameter is not clear, but it is safe to set it to the size of eigenvalues or greater. Leave it at its default zero, which will be reset to the size of eigenvalues internally.

Definition at line 538 of file arpack_solver.h.

## ◆ solver_control

 SolverControl& ArpackSolver::solver_control
protected

Reference to the object that controls convergence of the iterative solver.

Definition at line 349 of file arpack_solver.h.

protected

Store a copy of the flags for this particular solver.

Definition at line 354 of file arpack_solver.h.

## ◆ initial_vector_provided

 bool ArpackSolver::initial_vector_provided
protected

Store an initial vector

Definition at line 359 of file arpack_solver.h.

## ◆ resid

 std::vector ArpackSolver::resid
protected

Definition at line 360 of file arpack_solver.h.

## ◆ sigmar

 double ArpackSolver::sigmar
protected

Real part of the shift

Definition at line 365 of file arpack_solver.h.

## ◆ sigmai

 double ArpackSolver::sigmai
protected

Imaginary part of the shift

Definition at line 370 of file arpack_solver.h.

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