org.apache.commons.math4.linear

## Class 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 × VT. 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 VT 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:

Since:
2.0 (changed to concrete class in 3.0)
MathWorld, Wikipedia
• ### Constructor Summary

Constructors
Constructor and Description
SingularValueDecomposition(RealMatrix matrix)
Calculates the compact Singular Value Decomposition of the given matrix.
• ### Method Summary

All Methods
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 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
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)
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)
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)
getV()
• #### getCovariance

public RealMatrix getCovariance(double minSingularValue)
Returns the n × n covariance matrix.

The covariance matrix is V × J × VT 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 euclidean 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