Lanczos Algorithm for Eigenvalues / Eigenvectors of a Symmetric Matrix

Classes
struct	LanczosOptions
	An instance of this class, taken by LanczosEigenSolver(), is a collection of options for the Lanczos procedure to solve symmetric eigenvalue problems. More...
struct	LanczosInfo
	LanczosEigenSolver() returns an instance of this class which gives the information of the Lanczos procedure. More...

Typedefs
typedef Vector(*	NEXT_VECTOR) (const Vector &)

Functions
LanczosInfo	LanczosEigenSolver (Vector &eigVal, Matrix &eigVec, NEXT_VECTOR NextKrylovVector, mkIndex n, mkIndex neigWanted, const char eigPart[], LanczosOptions &opts)
	Lanczos eigensolver, the most general form. More...
LanczosInfo	LanczosEigenSolver (Vector &eigVal, Matrix &eigVec, NEXT_VECTOR NextKrylovVector, mkIndex n, mkIndex neigWanted, const char eigPart[])
	The same as LanczosEigenSolver(), but with the default LanczosOptions.
LanczosInfo	LanczosEigenSolver (Vector &eigVal, Matrix &eigVec, const SymmetricMatrix &A, mkIndex neigWanted, const char eigPart[], LanczosOptions &opts)
	The same as LanczosEigenSolver(), but with the NextKrylovVector() defined as A*v.
LanczosInfo	LanczosEigenSolver (Vector &eigVal, Matrix &eigVec, const SymmetricMatrix &A, mkIndex neigWanted, const char eigPart[])
	The same as LanczosEigenSolver(), but with the NextKrylovVector() defined as A*v and with default LanczosOptions.
LanczosInfo	LanczosEigenSolver (Vector &eigVal, Matrix &eigVec, const SparseMatrix &A, mkIndex neigWanted, const char eigPart[], LanczosOptions &opts)
	The same as LanczosEigenSolver(), but with the NextKrylovVector() defined as A*v.
LanczosInfo	LanczosEigenSolver (Vector &eigVal, Matrix &eigVec, const SparseMatrix &A, mkIndex neigWanted, const char eigPart[])
	The same as LanczosEigenSolver(), but with the NextKrylovVector() defined as A*v and with default LanczosOptions.

Detailed Description

Typedef Documentation

NEXT_VECTOR

typedef Vector(* NEXT_VECTOR) (const Vector &)

The definition of a function / operator with input a vector and output a vector. The operator should be self-adjoint if it is used for the Lanczos or conjugated gradient methods.

• In a standard Lanczos or conjugate gradient procedure, the input is v and output is A*v.
• In a filtered Lanczos or conjugate gradient procedure, the input is v and output is p(A)*v, with p a polynomial.

In both cases, A is a symmetric matrix. It can be dense or sparse.

Function Documentation

LanczosEigenSolver()

LanczosInfo LanczosEigenSolver	(	Vector &	eigVal,
		Matrix &	eigVec,
		NEXT_VECTOR	NextKrylovVector,
		mkIndex	n,
		mkIndex	neigWanted,
		const char	eigPart[],
		LanczosOptions &	opts
	)

Lanczos eigensolver, the most general form.

Parameters

NextKrylovVector	is the function which defines a self-adjoint operator, e.g. A*v with A a symmetric matrix.
n	is the length of the input/output vectors of NextKrylovVector().
neigWanted	is the number of eigenvalues to be sought.
eigPart	is a string "LA", "SA", or "BE". "LA" - largest algebraic, for largest eigenvalues; "SA" - smallest algebraic, for smallest eigenvalues; "BE" - both ends, one more from high end if nev is odd.
opts	is a collection of Lanczos options.
eigVal	is the output vector of length neigWanted containing the computed eigenvalues.
eigVec	is the output n-by-neigWanted matrix with columns as eigenvectors, in the order as elements in eigVal, if opts.wantEigVec==true.

Returns

An instance of LanczosInfo which gives the information of the Lanczos procedure.