org.apache.commons.math.linear

Interface SingularValueDecomposition

All Known Implementing Classes:

SingularValueDecompositionImpl

public interface SingularValueDecomposition

An interface to classes that implement an algorithm to calculate the Singular Value Decomposition of a real 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 interface 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

See Also:

MathWorld, Wikipedia

Method Summary

Modifier and Type	Method and Description
double	getConditionNumber() Return the condition number of the matrix.
RealMatrix	getCovariance(double minSingularValue) Returns the n × n covariance matrix.
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.

Method Detail

getU

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

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

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

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

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

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

RealMatrix getCovariance(double minSingularValue)
                  throws java.lang.IllegalArgumentException

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:

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

getNorm

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

double getConditionNumber()

Return the condition number of the matrix.

Returns:

condition number of the matrix

getRank

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

DecompositionSolver getSolver()

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

Returns:

a solver