SingularValueDecomposition (Apache Commons Math 3.6.1 API)

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)

See Also:
MathWorld, Wikipedia

Constructor Summary

Constructors
Constructor and Description

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

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
    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

Class SingularValueDecomposition

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

SingularValueDecomposition

Method Detail

getU

getUT

getS

getSingularValues

getV

getVT

getCovariance

getNorm

getConditionNumber

getInverseConditionNumber

getRank

getSolver