org.apache.commons.math.linear
Class EigenDecompositionImpl

java.lang.Object
  org.apache.commons.math.linear.EigenDecompositionImpl

All Implemented Interfaces:

EigenDecomposition

public class EigenDecompositionImpl
extends Object
implements EigenDecomposition

Calculates the eigen decomposition of a real symmetric matrix.

The eigen decomposition of matrix A is a set of two matrices: V and D such that A = V D V^T. A, V and D are all m × m matrices.

As of 2.0, this class supports only symmetric matrices, and hence computes only real realEigenvalues. This implies the D matrix returned by getD() is always diagonal and the imaginary values returned getImagEigenvalue(int) and getImagEigenvalues() are always null.

When called with a RealMatrix argument, this implementation only uses the upper part of the matrix, the part below the diagonal is not accessed at all.

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

Since:

2.0

Version:

$Id: EigenDecompositionImpl.java 1139906 2011-06-26 18:42:32Z luc $

Constructor Summary

EigenDecompositionImpl(double[] main, double[] secondary, double splitTolerance)
Calculates the eigen decomposition of the symmetric tridiagonal matrix.

EigenDecompositionImpl(RealMatrix matrix, double splitTolerance)
Calculates the eigen decomposition of the given symmetric matrix.

Method Summary

RealMatrix	getD() Returns the block diagonal matrix D of the decomposition.
double	getDeterminant() Return the determinant of the matrix
RealVector	getEigenvector(int i) Returns a copy of the ith eigenvector of the original matrix.
double	getImagEigenvalue(int i) Returns the imaginary part of the ith eigenvalue of the original matrix.
double[]	getImagEigenvalues() Returns a copy of the imaginary parts of the eigenvalues of the original matrix.
double	getRealEigenvalue(int i) Returns the real part of the ith eigenvalue of the original matrix.
double[]	getRealEigenvalues() Returns a copy of the real parts of the eigenvalues of the original matrix.
DecompositionSolver	getSolver() Get a solver for finding the A × X = B solution in exact linear sense.
RealMatrix	getV() Returns the matrix V of the decomposition.
RealMatrix	getVT() Returns 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

Constructor Detail

EigenDecompositionImpl

public EigenDecompositionImpl(RealMatrix matrix,
                              double splitTolerance)

Calculates the eigen decomposition of the given symmetric matrix.

Parameters:

matrix - Matrix to decompose. It must be symmetric.

splitTolerance - Dummy parameter (present for backward compatibility only).

Throws:

NonSymmetricMatrixException - if the matrix is not symmetric.

MaxCountExceededException - if the algorithm fails to converge.

EigenDecompositionImpl

public EigenDecompositionImpl(double[] main,
                              double[] secondary,
                              double splitTolerance)

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 triadiagonal form

secondary - Secondary of the tridiagonal form

splitTolerance - Dummy parameter (present for backward compatibility only).

Throws:

MaxCountExceededException - if the algorithm fails to converge.

Method Detail

getV

public RealMatrix getV()

Returns 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).

Specified by:

getV in interface EigenDecomposition

Returns:

the V matrix

getD

public RealMatrix getD()

Returns 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 +imaginary}, {-imaginary, real} }.

Specified by:

getD in interface EigenDecomposition

Returns:

the D matrix

See Also:

EigenDecomposition.getRealEigenvalues(), EigenDecomposition.getImagEigenvalues()

getVT

public RealMatrix getVT()

Returns 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).

Specified by:

getVT in interface EigenDecomposition

Returns:

the transpose of the V matrix

getRealEigenvalues

public double[] getRealEigenvalues()

Returns a copy of the real parts of the eigenvalues of the original matrix.

Specified by:

getRealEigenvalues in interface EigenDecomposition

Returns:

a copy of the real parts of the eigenvalues of the original matrix

See Also:

EigenDecomposition.getD(), EigenDecomposition.getRealEigenvalue(int), EigenDecomposition.getImagEigenvalues()

getRealEigenvalue

public double getRealEigenvalue(int i)

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

Specified by:

getRealEigenvalue in interface EigenDecomposition

Parameters:

i - index of the eigenvalue (counting from 0)

Returns:

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

See Also:

EigenDecomposition.getD(), EigenDecomposition.getRealEigenvalues(), EigenDecomposition.getImagEigenvalue(int)

getImagEigenvalues

public double[] getImagEigenvalues()

Returns a copy of the imaginary parts of the eigenvalues of the original matrix.

Specified by:

getImagEigenvalues in interface EigenDecomposition

Returns:

a copy of the imaginary parts of the eigenvalues of the original matrix

See Also:

EigenDecomposition.getD(), EigenDecomposition.getImagEigenvalue(int), EigenDecomposition.getRealEigenvalues()

getImagEigenvalue

public double getImagEigenvalue(int i)

Returns the imaginary part of the i^th eigenvalue of the original matrix.

Specified by:

getImagEigenvalue in interface EigenDecomposition

Parameters:

i - index of the eigenvalue (counting from 0)

Returns:

imaginary part of the i^th eigenvalue of the original matrix

See Also:

EigenDecomposition.getD(), EigenDecomposition.getImagEigenvalues(), EigenDecomposition.getRealEigenvalue(int)

getEigenvector

public RealVector getEigenvector(int i)

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

Specified by:

getEigenvector in interface EigenDecomposition

Parameters:

i - index of the eigenvector (counting from 0)

Returns:

copy of the i^th eigenvector of the original matrix

See Also:

EigenDecomposition.getD()

getDeterminant

public double getDeterminant()

Return the determinant of the matrix

Specified by:

getDeterminant in interface EigenDecomposition

Returns:

determinant of the matrix

getSolver

public DecompositionSolver getSolver()

Get a solver for finding the A × X = B solution in exact linear sense.

Specified by:

getSolver in interface EigenDecomposition

Returns:

a solver