org.opensourcephysics.numerics
Class EigenvalueDecomposition

java.lang.Object
  org.opensourcephysics.numerics.EigenvalueDecomposition

All Implemented Interfaces:

java.io.Serializable

public class EigenvalueDecomposition
extends java.lang.Object
implements java.io.Serializable

Eigenvalues and eigenvectors of a real matrix.

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 Also:

Serialized Form

Constructor Summary

EigenvalueDecomposition(double[][] A)
Constructs the EigenvalueDecomposition.

Method Summary

double[][]	getEigenvalueMatrix() Return the block diagonal eigenvalue matrix
double[][]	getEigenvectors() Return the eigenvector matrix
double[]	getImagEigenvalues() Return the imaginary parts of the eigenvalues
double[]	getRealEigenvalues() Return the real parts of the eigenvalues
static double	hypot(double a, double b) sqrt(a^2 + b^2) without under/overflow.
void	recompute(double[][] A) Check for symmetry, then recompute the eigenvalue decomposition

Methods inherited from class java.lang.Object

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

Constructor Detail

EigenvalueDecomposition

public EigenvalueDecomposition(double[][] A)

Constructs the EigenvalueDecomposition. Allocates arrays, checks for symmetry and then constructs the eigenvalue decomposition

Parameters:

A - square matrix

Method Detail

recompute

public void recompute(double[][] A)

Check for symmetry, then recompute the eigenvalue decomposition

Parameters:

A - square matrix

getEigenvectors

public double[][] getEigenvectors()

Return the eigenvector matrix

Returns:

V

getRealEigenvalues

public double[] getRealEigenvalues()

Return the real parts of the eigenvalues

Returns:

real(diag(D))

getImagEigenvalues

public double[] getImagEigenvalues()

Return the imaginary parts of the eigenvalues

Returns:

imag(diag(D))

getEigenvalueMatrix

public double[][] getEigenvalueMatrix()

Return the block diagonal eigenvalue matrix

Returns:

D

hypot

public static double hypot(double a,
                           double b)

sqrt(a^2 + b^2) without under/overflow.