Module imagingbook.common

Package imagingbook.common.math.eigen

Class GeneralizedSymmetricEigenDecomposition

java.lang.Object

imagingbook.common.math.eigen.GeneralizedSymmetricEigenDecomposition

All Implemented Interfaces:

RealEigenDecomposition

public class GeneralizedSymmetricEigenDecomposition
extends Object
implements RealEigenDecomposition

Solves the generalized symmetric eigenproblem of the form A x = λ B x, where matrices A, B are symmetric and B is positive definite (see Sec. 11.0.5. of [1]). See Appendix Sec. B.5.2 of [2] for more details. The methods defined by this class are analogous to the conventional eigendecomposition (see EigenDecomposition).

[1] Press, Teukolsky, Vetterling, Flannery: "Numerical Recipes". Cambridge University Press, 3rd ed. (2007).
[2] W. Burger, M.J. Burge, Digital Image Processing – An Algorithmic Introduction, 3rd ed, Springer (2022).

Version:

2022/06/11

See Also:

• GeneralizedEigenDecomposition
• EigenDecompositionJama

Constructor Summary

Constructors

Constructor	Description
GeneralizedSymmetricEigenDecomposition(RealMatrix A, RealMatrix B)	Constructor, using default parameters.
GeneralizedSymmetricEigenDecomposition(RealMatrix A, RealMatrix B, double rsth, double apth)	Constructor, using specific parameters.

Method Summary

Modifier and Type	Method	Description
RealMatrix	getD()	Gets the block diagonal matrix D of the decomposition.
RealVector	getEigenvector(int k)	Returns the k-th eigenvector, i.e., the k-th column vector of the matrix returned by RealEigenDecomposition.getV().
double[]	getImagEigenvalues()
double	getRealEigenvalue(int k)	Returns the real part of the k-th eigenvalue
double[]	getRealEigenvalues()	Returns a vector holding the real parts of the eigenvalues
RealMatrix	getV()	Return the matrix of eigenvectors, which are its column vectors.
boolean	hasComplexEigenvalues()	Returns whether the calculated eigenvalues are complex or real.

Methods inherited from class java.lang.Object

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

Constructor Details

GeneralizedSymmetricEigenDecomposition

public GeneralizedSymmetricEigenDecomposition(RealMatrix A,
 RealMatrix B,
 double rsth,
 double apth)

Constructor, using specific parameters. Solves the generalized symmetric eigenproblem of the form A x = λ B x, where matrices A, B are symmetric and B is positive definite. An exception is thrown if A is not symmetric and the Cholesky decomposition throws an exception if B is either not symmetric, not positive definite or singular.

Parameters:

A - real symmetric matrix

B - real symmetric and positive definite matrix

rsth - relative symmetry threshold

apth - absolute positivity threshold

GeneralizedSymmetricEigenDecomposition

public GeneralizedSymmetricEigenDecomposition(RealMatrix A,
 RealMatrix B)

Constructor, using default parameters. Solves the generalized symmetric eigenproblem of the form A x = λ B x, where matrices A, B * are symmetric and B is positive definite

Parameters:

A - real symmetric matrix

B - real symmetric and positive definite matrix

Method Details

getRealEigenvalues

public double[] getRealEigenvalues()

Description copied from interface: RealEigenDecomposition

Returns a vector holding the real parts of the eigenvalues

Specified by:

getRealEigenvalues in interface RealEigenDecomposition

Returns:

real(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

getImagEigenvalues

public double[] getImagEigenvalues()

hasComplexEigenvalues

public boolean hasComplexEigenvalues()

Description copied from interface: RealEigenDecomposition

Returns whether the calculated eigenvalues are complex or real.

Specified by:

hasComplexEigenvalues in interface RealEigenDecomposition

Returns:

true if any of the eigenvalues is complex, false otherwise

getD

public RealMatrix getD()

Description copied from interface: RealEigenDecomposition

Gets the block diagonal matrix D of the decomposition. Real eigenvalues are on the diagonal while complex values are on 2x2 blocks {{real_pos imaginary}, {neg_imaginary, real}}.

Specified by:

getD in interface RealEigenDecomposition

Returns:

matrix D

getEigenvector

public RealVector getEigenvector(int k)

Description copied from interface: RealEigenDecomposition

Returns the k-th eigenvector, i.e., the k-th column vector of the matrix returned by RealEigenDecomposition.getV().

Specified by:

getEigenvector in interface RealEigenDecomposition

Parameters:

k - index of the eigenvector (0-based)

Returns:

the k-th eigenvector (instance of RealVector)

getV

public RealMatrix getV()

Description copied from interface: RealEigenDecomposition

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

Specified by:

getV in interface RealEigenDecomposition

Returns:

the matrix of eigenvectors