SingularValueDecomposition (Java Statistical Analysis Tool 0.0.8 API)

java.lang.Object
- jsat.linear.SingularValueDecomposition

All Implemented Interfaces:

Serializable, Cloneable
```
public class SingularValueDecomposition
extends Object
implements Cloneable, Serializable
```
The Singular Value Decomposition (SVD) of a matrix A_m,n = U_m,n Σ_n,n V^T_n,n, where S is the diagonal matrix of the singular values sorted in descending order and are all non negative.
The SVD of a matrix has many practical uses, but is expensive to compute.

Implementation adapted from the Public Domain work of JAMA: A Java Matrix Package
NOTE: The current implementation has been revised and is now passing all test cases. However, it is still being tested. Use with awareness that it used to be bugged. Note left at revision 597

Author:

Edward Raff

See Also:

Serialized Form

Constructor Summary

Constructors
Constructor and Description
`SingularValueDecomposition(Matrix A)` Creates a new SVD of the matrix `A` such that A = U Σ V^T.
`SingularValueDecomposition(Matrix A, int maxIterations)` Creates a new SVD of the matrix `A` such that A = U Σ V^T.
`SingularValueDecomposition(Matrix U, Matrix V, double[] s)` Sets the values for a SVD explicitly.

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`double`	`absDet()` Computes the absolute value of the determinant for the full matrix.
`SingularValueDecomposition`	`clone()`
`double`	`getCondition()` Returns the condition number of the matrix.
`double[]`	`getInverseSingularValues()` Returns an array containing the inverse singular values.
`double[]`	`getInverseSingularValues(double tol)` Returns an array containing the inverse singular values.
`double`	`getNorm2()` Returns the 2 norm of the matrix, which is the maximal singular value.
`double`	`getPseudoDet()` Computes the pseudo determinant of the matrix, which corresponds to absolute value of the determinant of the full rank square sub matrix that contains all non zero singular values.
`double`	`getPseudoDet(double tol)` Computes the pseudo determinant of the matrix, which corresponds to absolute value of the determinant of the full rank square sub matrix that contains all non singular values > `tol`.
`Matrix`	`getPseudoInverse()` Returns the Moore–Penrose pseudo inverse of the matrix.
`Matrix`	`getPseudoInverse(double tol)` Returns the Moore–Penrose pseudo inverse of the matrix.
`int`	`getRank()` Returns the numerical rank of the matrix.
`int`	`getRank(double tol)` Returns the numerical rank of the matrix.
`Matrix`	`getS()` Returns the diagonal matrix S such that the SVD product results in the original matrix.
`double[]`	`getSingularValues()` Returns a copy of the sorted array of the singular values, include the near zero ones.
`Matrix`	`getU()` Returns the backing matrix U of the SVD.
`Matrix`	`getV()` Returns the backing matrix V of the SVD.
`boolean`	`isFullRank()` Indicates whether or not the input matrix was of full rank, full rank matrices are more numerically stable.
`Matrix`	`solve(Matrix B)` Solves the linear system of equations for A x = B by using the equation `x = A^-1 B = V S^-1 U^T B` When A is not full rank, this results in a more numerically stable approximation that minimizes the least squares error.
`Matrix`	`solve(Matrix b, ExecutorService threadpool)` Solves the linear system of equations for A x = B by using the equation `x = A^-1 B = V S^-1 U^T B` When A is not full rank, this results in a more numerically stable approximation that minimizes the least squares error.
`Vec`	`solve(Vec b)` Solves the linear system of equations for A x = b by using the equation `x = A^-1 b = V S^-1 U^T b` When A is not full rank, this results in a more numerically stable approximation that minimizes the least squares error.

Methods inherited from class java.lang.Object
equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Constructor Detail
  - SingularValueDecomposition
```
public SingularValueDecomposition(Matrix A)
```
    Creates a new SVD of the matrix A such that A = U Σ V^T. The matrix A will be modified and used as temp space when computing the SVD.
    
    Parameters:
    
    A - the matrix to create the SVD of
  - SingularValueDecomposition
```
public SingularValueDecomposition(Matrix A,
                                  int maxIterations)
```
    Creates a new SVD of the matrix A such that A = U Σ V^T. The matrix A will be modified and used as temp space when computing the SVD.
    
    Parameters:
    
    A - the matrix to create the SVD of
    
    maxIterations - the maximum number of iterations to perform per singular value till convergence.
  - SingularValueDecomposition
```
public SingularValueDecomposition(Matrix U,
                                  Matrix V,
                                  double[] s)
```
    Sets the values for a SVD explicitly. This is not a copy constructor, and will hold the given values.
    
    Parameters:
    
    U - the U matrix of an SVD
    
    V - the V matrix of an SVD
    
    s - the singular sorted by magnitude of an SVD
- Method Detail
  - getU
```
public Matrix getU()
```
    Returns the backing matrix U of the SVD. Do not alter this matrix.
    
    Returns:
    
    the matrix U of the SVD
  - getV
```
public Matrix getV()
```
    Returns the backing matrix V of the SVD. Do not later this matrix.
    
    Returns:
    
    the matrix V of the SVD
  - getSingularValues
```
public double[] getSingularValues()
```
    Returns a copy of the sorted array of the singular values, include the near zero ones.
    
    Returns:
    
    a copy of the sorted array of the singular values, including the near zero ones.
  - getS
```
public Matrix getS()
```
    Returns the diagonal matrix S such that the SVD product results in the original matrix. The diagonal contains the singular values.
    
    Returns:
    
    a dense diagonal matrix containing the singular values
  - getNorm2
```
public double getNorm2()
```
    Returns the 2 norm of the matrix, which is the maximal singular value.
    
    Returns:
    
    the 2 norm of the matrix
  - getCondition
```
public double getCondition()
```
    Returns the condition number of the matrix. The condition number is a positive measure of the numerical instability of the matrix. The larger the value, the less stable the matrix. For singular matrices, the result is Double.POSITIVE_INFINITY.
    
    Returns:
    
    the condition number of the matrix
  - getRank
```
public int getRank()
```
    Returns the numerical rank of the matrix. Near zero values will be ignored.
    
    Returns:
    
    the rank of the matrix
  - isFullRank
```
public boolean isFullRank()
```
    Indicates whether or not the input matrix was of full rank, full rank matrices are more numerically stable.
    
    Returns:
    
    true if the matrix was of full tank
  - getRank
```
public int getRank(double tol)
```
    Returns the numerical rank of the matrix. Values <= than tol will be ignored.
    
    Parameters:
    
    tol - the cut of for singular values
    
    Returns:
    
    the rank of the matrix
  - getInverseSingularValues
```
public double[] getInverseSingularValues()
```
    Returns an array containing the inverse singular values. Near zero values are converted to zero.
    
    Returns:
    
    an array containing the inverse singular values
  - getInverseSingularValues
```
public double[] getInverseSingularValues(double tol)
```
    Returns an array containing the inverse singular values. Values that are <= tol are converted to zero.
    
    Parameters:
    
    tol - the cut of for singular values
    
    Returns:
    
    an array containing the inverse singular values
  - getPseudoInverse
```
public Matrix getPseudoInverse()
```
    Returns the Moore–Penrose pseudo inverse of the matrix. The pseudo inverse for a matrix is unique. If a matrix is non singular, the pseudo inverse is the inverse.
    
    Returns:
    
    the pseudo inverse of the matrix
  - getPseudoInverse
```
public Matrix getPseudoInverse(double tol)
```
    Returns the Moore–Penrose pseudo inverse of the matrix. The pseudo inverse for a matrix is unique. If a matrix is non singular, the pseudo inverse is the inverse.
    
    Parameters:
    
    tol - the tolerance for singular values to ignore
    
    Returns:
    
    the pseudo inverse of the matrix
  - getPseudoDet
```
public double getPseudoDet()
```
    Computes the pseudo determinant of the matrix, which corresponds to absolute value of the determinant of the full rank square sub matrix that contains all non zero singular values.
    
    Returns:
    
    the pseudo determinant.
  - getPseudoDet
```
public double getPseudoDet(double tol)
```
    Computes the pseudo determinant of the matrix, which corresponds to absolute value of the determinant of the full rank square sub matrix that contains all non singular values > tol.
    
    Parameters:
    
    tol - the cut of for singular values
    
    Returns:
    
    the pseudo determinant
  - absDet
```
public double absDet()
```
    Computes the absolute value of the determinant for the full matrix.
    
    Returns:
    
    abs(determinant)
  - solve
```
public Vec solve(Vec b)
```
    Solves the linear system of equations for A x = b by using the equation
    x = A^-1 b = V S^-1 U^T b
    When A is not full rank, this results in a more numerically stable approximation that minimizes the least squares error.
    
    Parameters:
    
    b - the vector to solve for
    
    Returns:
    
    the vector that gives the least squares solution to A x = b
  - solve
```
public Matrix solve(Matrix B)
```
    Solves the linear system of equations for A x = B by using the equation
    x = A^-1 B = V S^-1 U^T B
    When A is not full rank, this results in a more numerically stable approximation that minimizes the least squares error.
    
    Parameters:
    
    B - the matrix to solve for
    
    Returns:
    
    the matrix that gives the least squares solution to A x = B
  - solve
```
public Matrix solve(Matrix b,
                    ExecutorService threadpool)
```
    Solves the linear system of equations for A x = B by using the equation
    x = A^-1 B = V S^-1 U^T B
    When A is not full rank, this results in a more numerically stable approximation that minimizes the least squares error.
    
    Parameters:
    
    b - the matrix to solve for
    
    threadpool -
    
    Returns:
    
    the matrix that gives the least squares solution to A x = B
  - clone
```
public SingularValueDecomposition clone()
```
    Overrides:
    
    clone in class Object

Class SingularValueDecomposition

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

SingularValueDecomposition

SingularValueDecomposition

SingularValueDecomposition

Method Detail

getU

getV

getSingularValues

getS

getNorm2

getCondition

getRank

isFullRank

getRank

getInverseSingularValues

getInverseSingularValues

getPseudoInverse

getPseudoInverse

getPseudoDet

getPseudoDet

absDet

solve

solve

solve

clone