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

Class SingularValueDecomposition

java.lang.Object

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

public class SingularValueDecomposition
extends java.lang.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 ozthogonal 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.

Since:

0.1

Author:

Arthur Zimek

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.

Method Summary

Modifier and Type	Method and Description
double	cond() Two norm condition number
private int	convergence(int k, int pp, boolean wantu, boolean wantv)
private void	deflate(double[] e, int p, int k, boolean wantv)
private void	generateU(int nu, int nct)
private void	generateV(int nu, double[] e, int nrt)
double[][]	getS() Return the diagonal matrix of singular values
double[]	getSingularValues() Return the one-dimensional array of singular values
double[][]	getU() Return the left singular vectors
double[][]	getV() Return the right singular vectors
double	norm2() Two norm
private void	qrStep(double[] e, int p, int k, boolean wantu, boolean wantv)
int	rank() Effective numerical matrix rank
private void	split(double[] e, int p, int k, boolean wantu)

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(double[][] Arg)

Constructor.

Parameters:

Arg - Rectangular input matrix

Method Detail

generateU

private void generateU(int nu,
                       int nct)

generateV

private void generateV(int nu,
                       double[] e,
                       int nrt)

deflate

private void deflate(double[] e,
                     int p,
                     int k,
                     boolean wantv)

split

private void split(double[] e,
                   int p,
                   int k,
                   boolean wantu)

qrStep

private void qrStep(double[] e,
                    int p,
                    int k,
                    boolean wantu,
                    boolean wantv)

convergence

private int convergence(int k,
                        int pp,
                        boolean wantu,
                        boolean wantv)

getU

public double[][] getU()

Return the left singular vectors

Returns:

U

getV

public double[][] 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 double[][] 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.