Package org.hipparchus.linear

Class EigenDecompositionSymmetric

java.lang.Object

org.hipparchus.linear.EigenDecompositionSymmetric

public class EigenDecompositionSymmetric
extends Object

Calculates the eigen decomposition of a symmetric real matrix.

The eigen decomposition of matrix A is a set of two matrices: \(V\) and \(D\) such that \(A V = V D\) where $\(A\), \(V\) and \(D\) are all \(m \times m\) matrices.

This class is similar in spirit to the EigenvalueDecomposition class from the JAMA library, with the following changes:

• a getVt method has been added,
• a getEigenvalue method to pick up a single eigenvalue has been added,
• a getEigenvector method to pick up a single eigenvector has been added,
• a getDeterminant method has been added.
• a getSolver method has been added.

As \(A\) is symmetric, then \(A = V D V^T\) where the eigenvalue matrix \(D\) is diagonal and the eigenvector matrix \(V\) is orthogonal, i.e. A = V.multiply(D.multiply(V.transpose())) and V.multiply(V.transpose()) equals the identity matrix.

The columns of \(V\) represent the eigenvectors in the sense that \(A V = V D\), i.e. A.multiply(V) equals V.multiply(D). The matrix \(V\) may be badly conditioned, or even singular, so the validity of the equation \(A = V D V^{-1}\) depends upon the condition of \(V\).

This implementation is based on the paper by A. Drubrulle, R.S. Martin and J.H. Wilkinson "The Implicit QL Algorithm" in Wilksinson and Reinsch (1971) Handbook for automatic computation, vol. 2, Linear algebra, Springer-Verlag, New-York.

See Also:

• MathWorld
• Wikipedia

Field Summary

Fields

Modifier and Type	Field	Description
static final double	DEFAULT_EPSILON	Default epsilon value to use for internal epsilon

Constructor Summary

Constructors

Constructor	Description
EigenDecompositionSymmetric(double[] main, double[] secondary)	Calculates the eigen decomposition of the symmetric tridiagonal matrix.
EigenDecompositionSymmetric(double[] main, double[] secondary, double epsilon, boolean decreasing)	Calculates the eigen decomposition of the symmetric tridiagonal matrix.
EigenDecompositionSymmetric(RealMatrix matrix)	Calculates the eigen decomposition of the given symmetric real matrix.
EigenDecompositionSymmetric(RealMatrix matrix, double epsilon, boolean decreasing)	Calculates the eigen decomposition of the given real matrix.

Method Summary

Modifier and Type	Method	Description
DiagonalMatrix	getD()	Gets the diagonal matrix D of the decomposition.
double	getDeterminant()	Computes the determinant of the matrix.
double	getEigenvalue(int i)	Returns the ith eigenvalue of the original matrix.
double[]	getEigenvalues()	Gets a copy of the eigenvalues of the original matrix.
RealVector	getEigenvector(int i)	Gets a copy of the ith eigenvector of the original matrix.
double	getEpsilon()	Get's the value for epsilon which is used for internal tests (e.g.
DecompositionSolver	getSolver()	Gets a solver for finding the \(A \times X = B\) solution in exact linear sense.
RealMatrix	getSquareRoot()	Computes the square-root of the matrix.
RealMatrix	getV()	Gets the matrix V of the decomposition.
RealMatrix	getVT()	Gets 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

Field Details

DEFAULT_EPSILON

public static final double DEFAULT_EPSILON

Default epsilon value to use for internal epsilon

See Also:

• Constant Field Values

Constructor Details

EigenDecompositionSymmetric

public EigenDecompositionSymmetric(RealMatrix matrix)

Calculates the eigen decomposition of the given symmetric real matrix.

This constructor uses the default epsilon and decreasing order for eigenvalues.

Parameters:

matrix - Matrix to decompose.

Throws:

MathIllegalStateException - if the algorithm fails to converge.

MathRuntimeException - if the decomposition of a general matrix results in a matrix with zero norm

EigenDecompositionSymmetric

public EigenDecompositionSymmetric(RealMatrix matrix,
 double epsilon,
 boolean decreasing)
                            throws MathRuntimeException

Calculates the eigen decomposition of the given real matrix.

Supports decomposition of a general matrix since 3.1.

Parameters:

matrix - Matrix to decompose.

epsilon - Epsilon used for internal tests (e.g. is singular, eigenvalue ratio, etc.)

decreasing - if true, eigenvalues will be sorted in decreasing order

Throws:

MathIllegalStateException - if the algorithm fails to converge.

MathRuntimeException - if the decomposition of a general matrix results in a matrix with zero norm

Since:

3.0

EigenDecompositionSymmetric

public EigenDecompositionSymmetric(double[] main,
 double[] secondary)

Calculates the eigen decomposition of the symmetric tridiagonal matrix.

The Householder matrix is assumed to be the identity matrix.

This constructor uses the default epsilon and decreasing order for eigenvalues.

Parameters:

main - Main diagonal of the symmetric tridiagonal form.

secondary - Secondary of the tridiagonal form.

Throws:

MathIllegalStateException - if the algorithm fails to converge.

EigenDecompositionSymmetric

public EigenDecompositionSymmetric(double[] main,
 double[] secondary,
 double epsilon,
 boolean decreasing)

Calculates the eigen decomposition of the symmetric tridiagonal matrix. The Householder matrix is assumed to be the identity matrix.

Parameters:

main - Main diagonal of the symmetric tridiagonal form.

secondary - Secondary of the tridiagonal form.

epsilon - Epsilon used for internal tests (e.g. is singular, eigenvalue ratio, etc.)

decreasing - if true, eigenvalues will be sorted in decreasing order

Throws:

MathIllegalStateException - if the algorithm fails to converge.

Since:

3.0

Method Details

getV

public RealMatrix getV()

Gets the matrix V of the decomposition. V is an orthogonal matrix, i.e. its transpose is also its inverse. The columns of V are the eigenvectors of the original matrix. No assumption is made about the orientation of the system axes formed by the columns of V (e.g. in a 3-dimension space, V can form a left- or right-handed system).

Returns:

the V matrix.

getD

public DiagonalMatrix getD()

Gets the diagonal matrix D of the decomposition. D is a diagonal matrix.

Returns:

the D matrix.

See Also:

• getEigenvalues()

getEpsilon

public double getEpsilon()

Get's the value for epsilon which is used for internal tests (e.g. is singular, eigenvalue ratio, etc.)

Returns:

the epsilon value.

getVT

public RealMatrix getVT()

Gets the transpose of the matrix V of the decomposition. V is an orthogonal matrix, i.e. its transpose is also its inverse. The columns of V are the eigenvectors of the original matrix. No assumption is made about the orientation of the system axes formed by the columns of V (e.g. in a 3-dimension space, V can form a left- or right-handed system).

Returns:

the transpose of the V matrix.

getEigenvalues

public double[] getEigenvalues()

Gets a copy of the eigenvalues of the original matrix.

Returns:

a copy of the eigenvalues of the original matrix.

See Also:

• getD()
• getEigenvalue(int)

getEigenvalue

public double getEigenvalue(int i)

Returns the i^th eigenvalue of the original matrix.

Parameters:

i - index of the eigenvalue (counting from 0)

Returns:

real part of the i^th eigenvalue of the original matrix.

See Also:

• getD()
• getEigenvalues()

getEigenvector

public RealVector getEigenvector(int i)

Gets a copy of the i^th eigenvector of the original matrix.

Note that if the the i^th is complex this method will throw an exception.

Parameters:

i - Index of the eigenvector (counting from 0).

Returns:

a copy of the i^th eigenvector of the original matrix.

See Also:

• getD()

getDeterminant

public double getDeterminant()

Computes the determinant of the matrix.

Returns:

the determinant of the matrix.

getSquareRoot

public RealMatrix getSquareRoot()

Computes the square-root of the matrix. This implementation assumes that the matrix is positive definite.

Returns:

the square-root of the matrix.

Throws:

MathRuntimeException - if the matrix is not symmetric or not positive definite.

getSolver

public DecompositionSolver getSolver()

Gets a solver for finding the \(A \times X = B\) solution in exact linear sense.

Returns:

a solver