com.numericalmethod.suanshu.matrix.doubles.factorization.svd
Class GloubKahanSVD

java.lang.Object
  com.numericalmethod.suanshu.matrix.doubles.factorization.svd.GloubKahanSVD

All Implemented Interfaces:

SVDDecomposition

public class GloubKahanSVD
extends java.lang.Object
implements SVDDecomposition

This class implements the SVD decomposition of a tall matrix using the Gloub-Kahan SVD algorithm.

There are two steps.

1. In the first step, the matrix is reduced to a bidiagonal matrix using a sequence of Householder transformations.
2. In the second step, we iteratively reduce the super-diagonal of the bidiagonal matrix to 0s, using a sequence of Givens transformations.

If only the singular values are needed, we can save computation by not computing U and V.

See Also:

"Algorithm 8.6.2. Matrix Computations, 3rd edition. Golub G. H., van Loan C. F."

Field Summary

double

epsilon a precision parameter: when a number |x| ≤ ε, it is considered 0

Constructor Summary

GloubKahanSVD(Matrix A, boolean doUV, boolean normalize, double epsilon)
Perform the Gloub-Kahan SVD decomposition.

Method Summary

DiagonalMatrix	D() Get a copy of D as in U' %*% A %*% V = D U %*% D %*% V' = A
double[]	singularValues() Get an array of the normalized, hence positive, singular values.
Matrix	U() Get a copy of U as in U' %*% A %*% V = D U %*% D %*% V' = A
Matrix	Ut() Get a copy of U.t() as in U.t() %*% A %*% V = D U %*% D %*% V' = A
Matrix	V() Get a copy of V as in U' %*% A %*% V = D U %*% D %*% V' = A

Methods inherited from class java.lang.Object

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

Field Detail

epsilon

public final double epsilon

a precision parameter: when a number |x| ≤ ε, it is considered 0

Constructor Detail

GloubKahanSVD

public GloubKahanSVD(Matrix A,
                     boolean doUV,
                     boolean normalize,
                     double epsilon)

Perform the Gloub-Kahan SVD decomposition.

Parameters:

A - a tall matrix

doUV - true if only the singular values are wanted; U and V are not computed

normalize - true if to sort the singular values in descending order and make them positive

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0

Throws:

java.lang.IllegalArgumentException - if A is not tall

Method Detail

singularValues

public double[] singularValues()

Description copied from interface: SVDDecomposition

Get an array of the normalized, hence positive, singular values. It may differ from D if this class is constructed with normalization off.

Specified by:

singularValues in interface SVDDecomposition

Returns:

the singular values

D

public DiagonalMatrix D()

Description copied from interface: SVDDecomposition

Get a copy of D as in

 U' %*% A %*% V = D
 U %*% D %*% V' = A
 

Specified by:

D in interface SVDDecomposition

Returns:

a copy of D

U

public Matrix U()

Description copied from interface: SVDDecomposition

Get a copy of U as in

 U' %*% A %*% V = D
 U %*% D %*% V' = A
 

Specified by:

U in interface SVDDecomposition

Returns:

a copy of U

Ut

public Matrix Ut()

Description copied from interface: SVDDecomposition

Get a copy of U.t() as in

 U.t() %*% A %*% V = D
 U %*% D %*% V' = A
 

Specified by:

Ut in interface SVDDecomposition

Returns:

a copy of U.t()

V

public Matrix V()

Description copied from interface: SVDDecomposition

Get a copy of V as in

 U' %*% A %*% V = D
 U %*% D %*% V' = A
 

Specified by:

V in interface SVDDecomposition

Returns:

a copy of V