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Stokhos::Lanczos< vectorspace_type, operator_type > Class Template Reference

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

#include <Stokhos_Lanczos.hpp>

List of all members.

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.


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