jsat.linear

Class SingularValueDecomposition

java.lang.Object

jsat.linear.SingularValueDecomposition

All Implemented Interfaces:

java.io.Serializable, java.lang.Cloneable

public class SingularValueDecomposition
extends java.lang.Object
implements java.lang.Cloneable, java.io.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

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 VT.
SingularValueDecomposition(Matrix A, int maxIterations) Creates a new SVD of the matrix A such that A = U Σ VT.
SingularValueDecomposition(Matrix U, Matrix V, double[] s) Sets the values for a SVD explicitly.

Method Summary

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 UT 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, java.util.concurrent.ExecutorService threadpool) Solves the linear system of equations for A x = B by using the equation x = A-1 B = V S-1 UT 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 UT 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, 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 UT 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 UT 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,
                    java.util.concurrent.ExecutorService threadpool)

Solves the linear system of equations for A x = B by using the equation
x = A-1 B = V S-1 UT 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 java.lang.Object