Stokhos Package Browser (Single Doxygen Collection): Stokhos::Lanczos< vectorspace_type, operator_type > Class Template Reference

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Applies Lanczos procedure to a given matrix. More...

#include <Stokhos_Lanczos.hpp>

Public Types
typedef operator_type::ordinal_type	ordinal_type

typedef operator_type::value_type	value_type

typedef Teuchos::SerialDenseVector< ordinal_type, value_type >	vector_type

typedef Teuchos::SerialDenseMatrix< ordinal_type, value_type >	matrix_type

Static Public Member Functions
static void	compute (ordinal_type k, const vectorspace_type &vs, const operator_type &A, const vector_type &u_init, matrix_type &u, Teuchos::Array< value_type > &alpha, Teuchos::Array< value_type > &beta, Teuchos::Array< value_type > &nrm_sqrd)
	Compute Lanczos basis.

static void	computeNormalized (ordinal_type k, const vectorspace_type &vs, const operator_type &A, const vector_type &u_init, matrix_type &u, Teuchos::Array< value_type > &alpha, Teuchos::Array< value_type > &beta, Teuchos::Array< value_type > &nrm_sqrd)
	Compute Lanczos basis.

static void	gramSchmidt (ordinal_type k, const vectorspace_type &vs, matrix_type &u, vector_type &u2)
	Gram-Schmidt orthogonalization routine.

Detailed Description

template<typename vectorspace_type, typename operator_type>
class Stokhos::Lanczos< vectorspace_type, operator_type >

Applies Lanczos procedure to a given matrix.

Given a matrix $A$ , a starting vector $u_0$ , and integer $k>0$ , applies the Lanczos procedure to compute an orthogonal basis for the Krylov subspace

$\mathcal{K}(A, u_0, k) = \{ u_0, A_u_0, A^2 u_0, \dots, A^k u_0 \}.$

The basis vectors are given by

$u_{i+1} = Au_i - \alpha_i u_i - \beta_i u_{i-1}, \;\; i=0,\dots,k$

where $u_{-1} = 0$ and

$alpha_i = \frac{u_i^T W (Au_i)}{u_i^T W u_i}, \;\; beta_i = \frac{u_i^T W u_i}{u_{i-1}^T W u_{i-1}}.$

Here $W$ is a diagonal weighting matrix.

Definition at line 121 of file Stokhos_Lanczos.hpp.

Member Typedef Documentation

◆ ordinal_type

template<typename vectorspace_type, typename operator_type>

typedef operator_type::ordinal_type Stokhos::Lanczos< vectorspace_type, operator_type >::ordinal_type

Definition at line 124 of file Stokhos_Lanczos.hpp.

◆ value_type

template<typename vectorspace_type, typename operator_type>

typedef operator_type::value_type Stokhos::Lanczos< vectorspace_type, operator_type >::value_type

Definition at line 125 of file Stokhos_Lanczos.hpp.

◆ vector_type

template<typename vectorspace_type, typename operator_type>

typedef Teuchos::SerialDenseVector<ordinal_type,value_type> Stokhos::Lanczos< vectorspace_type, operator_type >::vector_type

Definition at line 126 of file Stokhos_Lanczos.hpp.

◆ matrix_type

template<typename vectorspace_type, typename operator_type>

typedef Teuchos::SerialDenseMatrix<ordinal_type,value_type> Stokhos::Lanczos< vectorspace_type, operator_type >::matrix_type

Definition at line 127 of file Stokhos_Lanczos.hpp.

Member Function Documentation

◆ compute()

template<typename vectorspace_type, typename operator_type>

static void Stokhos::Lanczos< vectorspace_type, operator_type >::compute	(	ordinal_type	k,
		const vectorspace_type &	vs,
		const operator_type &	A,
		const vector_type &	u_init,
		matrix_type &	u,
		Teuchos::Array< value_type > &	alpha,
		Teuchos::Array< value_type > &	beta,
		Teuchos::Array< value_type > &	nrm_sqrd )

inlinestatic

Compute Lanczos basis.

Definition at line 130 of file Stokhos_Lanczos.hpp.

◆ computeNormalized()

template<typename vectorspace_type, typename operator_type>

static void Stokhos::Lanczos< vectorspace_type, operator_type >::computeNormalized	(	ordinal_type	k,
		const vectorspace_type &	vs,
		const operator_type &	A,
		const vector_type &	u_init,
		matrix_type &	u,
		Teuchos::Array< value_type > &	alpha,
		Teuchos::Array< value_type > &	beta,
		Teuchos::Array< value_type > &	nrm_sqrd )

inlinestatic

Compute Lanczos basis.

Definition at line 194 of file Stokhos_Lanczos.hpp.

◆ gramSchmidt()

template<typename vectorspace_type, typename operator_type>

static void Stokhos::Lanczos< vectorspace_type, operator_type >::gramSchmidt	(	ordinal_type	k,
		const vectorspace_type &	vs,
		matrix_type &	u,
		vector_type &	u2 )

inlinestatic

Gram-Schmidt orthogonalization routine.

Definition at line 251 of file Stokhos_Lanczos.hpp.

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

Stokhos_Lanczos.hpp