Module imagingbook.common

Package imagingbook.common.math.eigen

Class EigenDecompositionJama

java.lang.Object

imagingbook.common.math.eigen.EigenDecompositionJama

All Implemented Interfaces:

RealEigenDecomposition

public class EigenDecompositionJama
extends Object
implements RealEigenDecomposition

Eigenvalues and eigenvectors of a real matrix. This code has been ported from https://math.nist.gov/javanumerics/jama/ (Version 1.0.3), with the public API (and some internals) adapted to Apache Commons Math. Most comments below were taken from the original sources.

This is intended as a temporary substitute for Apache Commons Math's implementation (EigenDecomposition, whose symmetry tolerance appears to be too low and thus returns complex eigenvalues for close-to-symmetric matrices although solutions with real eigenvalues exist.

If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal and the eigenvector matrix V is orthogonal. I.e. A = V.times(D.times(V.transpose())) and V.times(V.transpose()) equals the identity matrix.

If A is not symmetric, then the eigenvalue matrix D is block diagonal with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The columns of V represent the eigenvectors in the sense that A*V = V*D, i.e. A.times(V) equals V.times(D). The matrix V may be badly conditioned, or even singular, so the validity of the equation A = V*D*inverse(V) depends upon V.cond().

See Appendix Sec. B.5 of [1] for more details.

[1] W. Burger, M.J. Burge, Digital Image Processing – An Algorithmic Introduction, 3rd ed, Springer (2022).

Version:

2022/06/19

See Also:

• EigenDecompositionApache
• Eigensolver2x2

Field Summary

Fields

Modifier and Type	Field	Description
static final double	DefaultSymmetryTolerance
static final double	DefaultZeroTolerance
static final int	SchurMaxIterations

Constructor Summary

Constructors

Constructor	Description
EigenDecompositionJama(RealMatrix M)	Constructor using default tolerance setting.
EigenDecompositionJama(RealMatrix M, double symmetryTol, double zeroTol)	Constructor.

Method Summary

Modifier and Type	Method	Description
RealMatrix	getD()	Gets the block diagonal matrix D of the decomposition.
double	getDeterminant()	Calculates and returns the determinant of the decomposed matrix.
RealVector	getEigenvector(int k)	Returns a copy of the specified eigenvector, i.e., the associated column vector of the matrix returned by getV().
double	getImagEigenvalue(int k)
double[]	getImagEigenvalues()	Return the imaginary parts of the eigenvalues
double	getRealEigenvalue(int k)	Returns the real part of the k-th eigenvalue
double[]	getRealEigenvalues()	Return the real parts of the eigenvalues
RealMatrix	getV()	Return the matrix of eigenvectors, which are its column vectors.
RealMatrix	getVT()	Return the transpose of the eigenvector matrix.
boolean	hasComplexEigenvalues()	Returns whether the calculated eigenvalues are complex or real.
boolean	isSymmetric()	Returns true if the decomposed matrix is considered symmetric.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Details

DefaultSymmetryTolerance

public static final double DefaultSymmetryTolerance

See Also:

• Constant Field Values

DefaultZeroTolerance

public static final double DefaultZeroTolerance

See Also:

• Constant Field Values

SchurMaxIterations

public static final int SchurMaxIterations

See Also:

• Constant Field Values

Constructor Details

EigenDecompositionJama

public EigenDecompositionJama(RealMatrix M,
 double symmetryTol,
 double zeroTol)
                       throws MaxIterationsExceededException

Constructor. Checks for symmetry, then constructs the eigenvalue decomposition.

Parameters:

M - matrix to be decomposed

symmetryTol - absolute threshold for determining matrix symmetry

zeroTol - absolute threshold for determining zero matrix entries

Throws:

MaxIterationsExceededException - if the maximum number of iterations is exceeded (see SchurMaxIterations)

EigenDecompositionJama

public EigenDecompositionJama(RealMatrix M)

Constructor using default tolerance setting. See also DefaultSymmetryTolerance, DefaultZeroTolerance.

Parameters:

M - matrix to be decomposed

Method Details

isSymmetric

public boolean isSymmetric()

Returns true if the decomposed matrix is considered symmetric. See also EigenDecompositionJama(RealMatrix, double, double), DefaultSymmetryTolerance.

Returns:

true if symmetric, false otherwise

getV

public RealMatrix getV()

Return the matrix of eigenvectors, which are its column vectors.

Specified by:

getV in interface RealEigenDecomposition

Returns:

the matrix of eigenvectors

getVT

public RealMatrix getVT()

Return the transpose of the eigenvector matrix. The eigenvectors are the rows of the returned matrix.

Returns:

the transposed matrix of eigenvectors

getEigenvector

public RealVector getEigenvector(int k)

Returns a copy of the specified eigenvector, i.e., the associated column vector of the matrix returned by getV().

Specified by:

getEigenvector in interface RealEigenDecomposition

Parameters:

k - index of the eigenvector (counting from 0).

Returns:

a copy of the k-th eigenvector

getRealEigenvalues

public double[] getRealEigenvalues()

Return the real parts of the eigenvalues

Specified by:

getRealEigenvalues in interface RealEigenDecomposition

Returns:

real(diag(D))

getImagEigenvalues

public double[] getImagEigenvalues()

Return the imaginary parts of the eigenvalues

Returns:

imag(diag(D))

getRealEigenvalue

public double getRealEigenvalue(int k)

Description copied from interface: RealEigenDecomposition

Returns the real part of the k-th eigenvalue

Specified by:

getRealEigenvalue in interface RealEigenDecomposition

Parameters:

k - index of the eigenvalue @param k index of the eigenvector (0-based)

Returns:

real part of the k-th eigenvalue

getImagEigenvalue

public double getImagEigenvalue(int k)

hasComplexEigenvalues

public boolean hasComplexEigenvalues()

Returns whether the calculated eigenvalues are complex or real. The method performs a zero check on each element of the getImagEigenvalues() array and returns true if any element is not equal to zero.

Specified by:

hasComplexEigenvalues in interface RealEigenDecomposition

Returns:

true if any of the eigenvalues is complex, false otherwise

getD

public RealMatrix getD()

Gets the block diagonal matrix D of the decomposition. D is a block diagonal matrix. Real eigenvalues are on the diagonal while complex values are on 2x2 blocks {{real pos imaginary}, {neg imaginary, real}}. WB: Wonder if indexes are safe!

Specified by:

getD in interface RealEigenDecomposition

Returns:

D

getDeterminant

public double getDeterminant()

Calculates and returns the determinant of the decomposed matrix.

Returns:

the determinant of the matrix.