Package 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

Constructor Summary

Constructors

Constructor	Description
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

Modifier and Type	Method	Description
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

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

Constructor Details

EigenDecompositionImpl

public EigenDecompositionImpl(RealMatrix matrix,
 double splitTolerance)
                       throws InvalidMatrixException

Calculates the eigen decomposition of the given symmetric matrix.

Parameters:

matrix - The symmetric matrix to decompose.

splitTolerance - dummy parameter, present for backward compatibility only.

Throws:

InvalidMatrixException - (wrapping a ConvergenceException if algorithm fails to converge

EigenDecompositionImpl

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

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:

InvalidMatrixException - (wrapping a ConvergenceException if algorithm fails to converge

Method Details

getV

public RealMatrix getV()
                throws InvalidMatrixException

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

Throws:

InvalidMatrixException

getD

public RealMatrix getD()
                throws InvalidMatrixException

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

Throws:

InvalidMatrixException

See Also:

• EigenDecomposition.getRealEigenvalues()
• EigenDecomposition.getImagEigenvalues()

getVT

public RealMatrix getVT()
                 throws InvalidMatrixException

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

Throws:

InvalidMatrixException

getRealEigenvalues

public double[] getRealEigenvalues()
                            throws InvalidMatrixException

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

Throws:

InvalidMatrixException

See Also:

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

getRealEigenvalue

public double getRealEigenvalue(int i)
                         throws InvalidMatrixException,
ArrayIndexOutOfBoundsException

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

Throws:

InvalidMatrixException

ArrayIndexOutOfBoundsException

See Also:

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

getImagEigenvalues

public double[] getImagEigenvalues()
                            throws InvalidMatrixException

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

Throws:

InvalidMatrixException

See Also:

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

getImagEigenvalue

public double getImagEigenvalue(int i)
                         throws InvalidMatrixException,
ArrayIndexOutOfBoundsException

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

Throws:

InvalidMatrixException

ArrayIndexOutOfBoundsException

See Also:

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

getEigenvector

public RealVector getEigenvector(int i)
                          throws InvalidMatrixException,
ArrayIndexOutOfBoundsException

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

Throws:

InvalidMatrixException

ArrayIndexOutOfBoundsException

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