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

Since:
2.0
MathWorld, Wikipedia
• ### 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` `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
`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)
`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)
`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)
`getV()`
• #### getCovariance

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