Module imagingbook.common

Package imagingbook.common.math.eigen

Class Eigensolver2x2

java.lang.Object

imagingbook.common.math.eigen.Eigensolver2x2

All Implemented Interfaces:

RealEigenDecomposition

public class Eigensolver2x2
extends Object
implements RealEigenDecomposition

Implements an efficient, closed form algorithm for calculating the real eigenvalues (λ) and eigenvectors (x) of a 2x2 matrix of the form

   | a b |
   | c d |
 

There are typically (but not always) two pairs of real-valued solutions ⟨λ₀, x₀⟩, ⟨λ₁, x₁⟩ such that A·x_k = λ_k·x_k. The resulting eigensystems are ordered such that λ₀ ≥ λ₁. Eigenvectors are not normalized, i.e., no unit vectors (any scalar multiple of an Eigenvector is an Eigenvector too). Non-real eigenvalues are not handled. Clients should call method hasComplexEigenvalues() to check if the eigenvalue calculation was successful.

This implementation is inspired by Ch. 5 ("Consider the Lowly 2x2 Matrix") of [1]. Note that Blinn uses the notation x·A = λ·x for the matrix-vector product (as common in computer graphics), while this implementation uses A·x = λ·x. Thus x is treated as a column vector and matrix A is transposed (elements b/c are exchanged). See Appendix Sec. B.5 of [2] for more details.

This implementation is considerably faster (ca. factor 5) than the general solution (e.g., EigenDecompositionJama) for 2x2 matrices.

[1] Blinn, Jim: "Jim Blinn's Corner: Notation, Notation, Notation", Morgan Kaufmann (2002).
[2] W. Burger, M.J. Burge, Digital Image Processing – An Algorithmic Introduction, 3rd ed, Springer (2022).

Version:

2022/02/18

See Also:

• EigenDecompositionJama
• EigenDecompositionApache

Constructor Summary

Constructors

Constructor	Description
Eigensolver2x2(double[][] A)	Constructor, takes a 2x2 matrix.
Eigensolver2x2(double a, double b, double c, double d)	Constructor, takes the individual elements of a 2x2 matrix A:

Method Summary

Modifier and Type	Method	Description
RealVector	getEigenvector(int k)	Returns the k-th eigenvector, i.e., the k-th column vector of the matrix returned by RealEigenDecomposition.getV().
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.
String	toString()

Methods inherited from class java.lang.Object

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

Methods inherited from interface imagingbook.common.math.eigen.RealEigenDecomposition

getD

Constructor Details

Eigensolver2x2

public Eigensolver2x2(double[][] A)

Constructor, takes a 2x2 matrix.

Parameters:

A - a 2x2 matrix

Eigensolver2x2

public Eigensolver2x2(double a,
 double b,
 double c,
 double d)

Constructor, takes the individual elements of a 2x2 matrix A:

   | a b |
   | c d | 

Parameters:

a - matrix element A[0,0]

b - matrix element A[0,1]

c - matrix element A[1,0]

d - matrix element A[1,1]

Method Details

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

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

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

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)

toString

public String toString()

Overrides:

toString in class Object