smile.math.matrix

Class Lanczos

java.lang.Object

smile.math.matrix.Lanczos

public class Lanczos
extends java.lang.Object

The Lanczos algorithm is a direct algorithm devised by Cornelius Lanczos that is an adaptation of power methods to find the most useful eigenvalues and eigenvectors of an n^th order linear system with a limited number of operations, m, where m is much smaller than n.

Although computationally efficient in principle, the method as initially formulated was not useful, due to its numerical instability. In this implementation, we use partial reorthogonalization to make the method numerically stable.

Constructor Summary

Constructors

Constructor and Description
Lanczos()

Method Summary

Modifier and Type	Method and Description
static EigenValueDecomposition	eigen(Matrix A, int k) Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.
static EigenValueDecomposition	eigen(Matrix A, int k, double kappa) Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.
static SingularValueDecomposition	svd(Matrix A, int k) Find k largest approximate singular triples of a matrix by the Lanczos algorithm.
static SingularValueDecomposition	svd(Matrix A, int k, double kappa) Find k largest approximate singular triples of a matrix by the Lanczos algorithm.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

Lanczos

public Lanczos()

Method Detail

svd

public static SingularValueDecomposition svd(Matrix A,
                                             int k)

Find k largest approximate singular triples of a matrix by the Lanczos algorithm.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

k - the number of singular triples we wish to compute for the input matrix. This number cannot exceed the size of A.

svd

public static SingularValueDecomposition svd(Matrix A,
                                             int k,
                                             double kappa)

Find k largest approximate singular triples of a matrix by the Lanczos algorithm.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

k - the number of singular triples we wish to compute for the input matrix. This number cannot exceed the size of A.

kappa - relative accuracy of ritz values acceptable as singular values.

eigen

public static EigenValueDecomposition eigen(Matrix A,
                                            int k)

Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

k - the number of eigenvalues we wish to compute for the input matrix. This number cannot exceed the size of A.

eigen

public static EigenValueDecomposition eigen(Matrix A,
                                            int k,
                                            double kappa)

Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.

Parameters:

A - the matrix supporting matrix vector multiplication operation.

k - the number of eigenvalues we wish to compute for the input matrix. This number cannot exceed the size of A.

kappa - relative accuracy of ritz values acceptable as eigenvalues.