org.apache.commons.math3.linear
Class SingularValueDecomposition

java.lang.Object
  org.apache.commons.math3.linear.SingularValueDecomposition

public class SingularValueDecomposition
extends Object

Calculates the compact Singular Value Decomposition of a matrix.

The Singular Value Decomposition of matrix A is a set of three matrices: U, Σ and V such that A = U × Σ × V^T. Let A be a m × n matrix, then U is a m × p orthogonal matrix, Σ is a p × p diagonal matrix with positive or null elements, V is a p × n orthogonal matrix (hence V^T is also orthogonal) where p=min(m,n).

This class is similar to the class with similar name from the JAMA library, with the following changes:

• the norm2 method which has been renamed as getNorm,
• the cond method which has been renamed as getConditionNumber,
• the rank method which has been renamed as getRank,
• a getUT method has been added,
• a getVT method has been added,
• a getSolver method has been added,
• a getCovariance method has been added.

Since:

2.0 (changed to concrete class in 3.0)

Version:

$Id: SingularValueDecomposition.java 3720 2012-03-16 16:34:17Z CardosoP $

See Also:

MathWorld, Wikipedia

Constructor Summary

SingularValueDecomposition(RealMatrix matrix)
Calculates the compact Singular Value Decomposition of the given matrix.

Method Summary

double	getConditionNumber() Return the condition number of the matrix.
RealMatrix	getCovariance(double minSingularValue) Returns the n × n covariance matrix.
double	getInverseConditionNumber() Computes the inverse of the condition number.
double	getNorm() Returns the L2 norm of the matrix.
int	getRank() Return the effective numerical matrix rank.
RealMatrix	getS() Returns the diagonal matrix Σ of the decomposition.
double[]	getSingularValues() Returns the diagonal elements of the matrix Σ of the decomposition.
DecompositionSolver	getSolver() Get a solver for finding the A × X = B solution in least square sense.
RealMatrix	getU() Returns the matrix U of the decomposition.
RealMatrix	getUT() Returns the transpose of the matrix U of the decomposition.
RealMatrix	getV() Returns the matrix V of the decomposition.
RealMatrix	getVT() Returns the transpose of the matrix V of the decomposition.

Methods inherited from class java.lang.Object

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

Constructor Detail

SingularValueDecomposition

public SingularValueDecomposition(RealMatrix matrix)

Calculates the compact Singular Value Decomposition of the given matrix.

Parameters:

matrix - Matrix to decompose.

Method Detail

getU

public RealMatrix getU()

Returns the matrix U of the decomposition.

U is an orthogonal matrix, i.e. its transpose is also its inverse.

Returns:

the U matrix

See Also:

getUT()

getUT

public RealMatrix getUT()

Returns the transpose of the matrix U of the decomposition.

U is an orthogonal matrix, i.e. its transpose is also its inverse.

Returns:

the U matrix (or null if decomposed matrix is singular)

See Also:

getU()

getS

public RealMatrix getS()

Returns the diagonal matrix Σ of the decomposition.

Σ is a diagonal matrix. The singular values are provided in non-increasing order, for compatibility with Jama.

Returns:

the Σ matrix

getSingularValues

public double[] getSingularValues()

Returns the diagonal elements of the matrix Σ of the decomposition.

The singular values are provided in non-increasing order, for compatibility with Jama.

Returns:

the diagonal elements of the Σ matrix

getV

public RealMatrix getV()

Returns the matrix V of the decomposition.

V is an orthogonal matrix, i.e. its transpose is also its inverse.

Returns:

the V matrix (or null if decomposed matrix is singular)

See Also:

getVT()

getVT

public RealMatrix getVT()

Returns the transpose of the matrix V of the decomposition.

V is an orthogonal matrix, i.e. its transpose is also its inverse.

Returns:

the V matrix (or null if decomposed matrix is singular)

See Also:

getV()

getCovariance

public RealMatrix getCovariance(double minSingularValue)

Returns the n × n covariance matrix.

The covariance matrix is V × J × V^T where J is the diagonal matrix of the inverse of the squares of the singular values.

Parameters:

minSingularValue - value below which singular values are ignored (a 0 or negative value implies all singular value will be used)

Returns:

covariance matrix

Throws:

IllegalArgumentException - if minSingularValue is larger than the largest singular value, meaning all singular values are ignored

getNorm

public double getNorm()

Returns the L₂ norm of the matrix.

The L₂ norm is max(|A × u|₂ / |u|₂), where |.|₂ denotes the vectorial 2-norm (i.e. the traditional euclidian norm).

Returns:

norm

getConditionNumber

public double getConditionNumber()

Return the condition number of the matrix.

Returns:

condition number of the matrix

getInverseConditionNumber

public double getInverseConditionNumber()

Computes the inverse of the condition number. In cases of rank deficiency, the condition number will become undefined.

Returns:

the inverse of the condition number.

getRank

public int getRank()

Return the effective numerical matrix rank.

The effective numerical rank is the number of non-negligible singular values. The threshold used to identify non-negligible terms is max(m,n) × ulp(s₁) where ulp(s₁) is the least significant bit of the largest singular value.

Returns:

effective numerical matrix rank

getSolver

public DecompositionSolver getSolver()

Get a solver for finding the A × X = B solution in least square sense.

Returns:

a solver