org.jmatrices.dbl.decomposition
Class SingularValueDecomposition

java.lang.Object
  org.jmatrices.dbl.decomposition.SingularValueDecomposition

public class SingularValueDecomposition

extends java.lang.Object

SingularValueDecomposition

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

http://kwon3d.com/theory/jkinem/svd.html http://web.mit.edu/be.400/www/SVD/Singular_Value_Decomposition.htm http://www.matheverywhere.com/mei/courseware/mgm/svdB/ http://www.netlib.org/lapack/lug/node53.html http://www.mppmu.mpg.de/~schieck/svd.pdf http://www.cs.ut.ee/~toomas_l/linalg/lin2/node14.html

A x X = B
X = A^-1 x B
A = U x S x V^T (Singular value decomposition of A)
A-1 = V x S^-1 x U^T (Generalized inverse or pseudoinverse of A)
X = V x S^-1 x U^T x B

The code is basically JAMA code with modifications made to fit in the scheme of things.

Author: purangp

Date: 12.03.2004 Time: 23:09:45

Constructor Summary

SingularValueDecomposition(Matrix matrix)
Construct the singular value decomposition

Method Summary

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

Constructor Detail

SingularValueDecomposition

public SingularValueDecomposition(Matrix matrix)

Construct the singular value decomposition

Parameters:

matrix - Rectangular 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 nonnegligible singular values.