de.lmu.ifi.dbs.elki.math.linearalgebra

Class SingularValueDecomposition

java.lang.Object

de.lmu.ifi.dbs.elki.math.linearalgebra.SingularValueDecomposition

public class SingularValueDecomposition
extends Object

Singular Value Decomposition.

For an m-by-n matrix A with m >= n, the singular value decomposition is an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and an n-by-n orthogonal matrix V so that A = U*S*V'.

The singular values, sigma[k] = S[k][k], are ordered so that sigma[0] >= sigma[1] >= ... >= sigma[n-1].

The singular value decomposition always exists, so the constructor will never fail. The matrix condition number and the effective numerical rank can be computed from this decomposition.

Field Summary

Fields

Modifier and Type	Field and Description
private int	m Row and column dimensions.
private int	n Row and column dimensions.
private double[]	s Array for internal storage of singular values.
private double[][]	U Arrays for internal storage of U and V.
private double[][]	V Arrays for internal storage of U and V.

Constructor Summary

Constructors

Constructor and Description
SingularValueDecomposition(double[][] Arg) Constructor.
SingularValueDecomposition(Matrix Arg) Construct the singular value decomposition

Method Summary

Methods

Modifier and Type	Method and Description
double	cond() Two norm condition number
Matrix	getS() Return the diagonal matrix of singular values
double[]	getSingularValues() Return the one-dimensional array of singular values
Matrix	getU() Return the left singular vectors
Matrix	getV() Return the right singular vectors
double	norm2() Two norm
int	rank() Effective numerical matrix rank

Methods inherited from class java.lang.Object

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

Field Detail

U

private double[][] U

Arrays for internal storage of U and V.

V

private double[][] V

Arrays for internal storage of U and V.

s

private double[] s

Array for internal storage of singular values.

m

private int m

Row and column dimensions.

n

private int n

Row and column dimensions.

Constructor Detail

SingularValueDecomposition

public SingularValueDecomposition(Matrix Arg)

Construct the singular value decomposition

Parameters:

Arg - Rectangular matrix

SingularValueDecomposition

public SingularValueDecomposition(double[][] Arg)

Constructor.

Parameters:

Arg - Rectangular input matrix

Method Detail

getU

public Matrix getU()

Return the left singular vectors

Returns:

U

getV

public Matrix getV()

Return the right singular vectors

Returns:

V

getSingularValues

public double[] getSingularValues()

Return the one-dimensional array of singular values

Returns:

diagonal of S.

getS

public Matrix getS()

Return the diagonal matrix of singular values

Returns:

S

norm2

public double norm2()

Two norm

Returns:

max(S)

cond

public double cond()

Two norm condition number

Returns:

max(S)/min(S)

rank

public int rank()

Effective numerical matrix rank

Returns:

Number of non-negligible singular values.