Solves a generalized eigenvalue problem subject to linear, homogeneous constraints. More...

#include <constrained.hh>

Inheritance diagram for eigensolver::Constrained:

Collaboration diagram for eigensolver::Constrained:

Public Member Functions
	Constrained (concepts::Operator< Real > &A, concepts::Operator< Real > &M, const concepts::SparseMatrix< Real > &C, SolverFabric< Real > &solver)
	Constructor.
virtual uint	converged () const
	Returns the number of converged eigen pairs.
virtual const concepts::Array < concepts::Vector< Real > * > &	getEF ()
virtual const concepts::Array < Real > &	getEV ()
	Returns an array with the eigen values.
virtual uint	iterations () const
	Returns the number of iterations.
virtual	~Constrained ()
Protected Member Functions
virtual std::ostream &	info (std::ostream &os) const
	Returns information in an output stream.
Private Member Functions
void	compute_ ()
	The real compute routine.
Private Attributes
concepts::Operator< Real > &	A_
	Stiffness matrix.
const concepts::SparseMatrix < Real > &	C_
	Matrix of constraints. $C^\top \vec x = g = 0$ is enforced.
bool	computed_
	If false, the code will have to compute, otherwise not.
concepts::Array< Real >	eigenvectors_
	Storage space for eigenvectors.
concepts::Array < concepts::Vector< Real > * >	ev_
	References into storage for eigenvectors.
concepts::TrivExtendRestrict < Real > *	ext
uint	it_
	Statistics.
uint	k_conv_
uint	kmax_
	Maximal number of eigenvalues which can be computed.
concepts::Array< Real >	lambda_
	Storage space for eigenvalues.
concepts::Operator< Real > &	M_
	Mass matrix.
concepts::Permutation< Real > *	Pr
std::auto_ptr < concepts::Compose< Real > >	PrQgt
std::auto_ptr < concepts::Permutation< Real > >	Prt
sparseqr::GivensRotations< Real > *	Qg
std::auto_ptr < concepts::Compose< Real > >	QgPrt
std::auto_ptr < concepts::Compose< Real > >	QgPrtext
sparseqr::GivensRotations< Real > *	Qgt
concepts::TrivExtendRestrict < Real > *	restr
std::auto_ptr < concepts::Compose< Real > >	restrPrQgt
std::auto_ptr < concepts::Compose< Real > >	restrPrQgtA
std::auto_ptr < concepts::Compose< Real > >	restrPrQgtAQgPrtext
std::auto_ptr < concepts::Compose< Real > >	restrPrQgtM
std::auto_ptr < concepts::Compose< Real > >	restrPrQgtMQgPrtext
SolverFabric< Real > &	solver_
	Fabric for the eigenvalue solver.

Detailed Description

Solves a generalized eigenvalue problem subject to linear, homogeneous constraints.

This class solve a generalized eigenvalue problem of the form

$A \vec x = \lambda M \vec x$

subject to the constraints $C^\top \vec x = g = 0$ where $C \in R^{n \times m}$ and $A, M \in R^{n \times n}$ . This is important for solving eigenvalue problems with essential boundary conditions like $\vec u \times \vec n = 0$ .

For the case $ g = 0 $ , this is described in [1 section 12.6, 2]. For the much more complicated case of $g \neq 0$ , see [3, 4]. The first case should be sufficient for most applications.

See also:: [1] Gene H. Golub and Charles F. van Loan, Matrix Computations, The Johns Hopkins University Press, 1983.
[2] Gene H. Golub, Some Modified Matrix Eigenvalue Problems, SIAM Review, 15 (1973), No. 2, 318--334.
[3] Walter Gander, Gene H. Golub and Urs von Matt, A Constrained Eigenvalue Problem, in Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms, Gene H. Golub and P. Van Dooren (eds.), NATO ASI Series, Vol. F70, 1991, 677--686.
[4] Walter Gander, Gene H. Golub and Urs von Matt, A Constrained Eigenvalue Problem, Linear Algebra and Its Applications, 114/115 (1989), 815--839.

Test:: test::ConstrainedEVP

Author:: Philipp Frauenfelder, 2002

Definition at line 51 of file constrained.hh.

Constructor & Destructor Documentation

eigensolver::Constrained::Constrained	(	concepts::Operator< Real > &	A,
		concepts::Operator< Real > &	M,
		const concepts::SparseMatrix< Real > &	C,
		SolverFabric< Real > &	solver
	)		`[inline]`