org.apache.commons.math3.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)
Version:
\$Id: SingularValueDecomposition.java 1244107 2012-02-14 16:17:55Z erans \$
MathWorld, Wikipedia
• ### Constructor Summary

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

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 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