com.imsl.math
Class Eigen

java.lang.Object
  com.imsl.math.Eigen

public class Eigen
extends Object

Collection of Eigen System functions.

Eigen computes the eigenvalues and eigenvectors of a real matrix. The matrix is first balanced. Orthogonal similarity transformations are used to reduce the balanced matrix to a real upper Hessenberg matrix. The implicit double-shifted QR algorithm is used to compute the eigenvalues and eigenvectors of this Hessenberg matrix. The eigenvectors are normalized such that each has Euclidean length of value one. The largest component is real and positive.

The balancing routine is based on the EISPACK routine BALANC. The reduction routine is based on the EISPACK routines ORTHES and ORTRAN. The QR algorithm routine is based on the EISPACK routine HQR2. See Smith et al. (1976) for the EISPACK routines. Further details, some timing data, and credits are given in Hanson et al. (1990).

While the exact value of the performance index, , is highly machine dependent, the performance of Eigen is considered excellent if , good if , and poor if .

The performance index was first developed by the EISPACK project at Argonne National Laboratory; see Smith et al. (1976, pages 124-125).

See Also:

Example

Nested Class Summary

static class

Eigen.DidNotConvergeException The iteration did not converge

Constructor Summary

Eigen(double[][] a)
Constructs the eigenvalues and the eigenvectors of a real square matrix.

Eigen(double[][] a, boolean computeVectors)
Constructs the eigenvalues and (optionally) the eigenvectors of a real square matrix.

Method Summary

Complex[]	getValues() Returns the eigenvalues of a matrix of type double.
Complex[][]	getVectors() Returns the eigenvectors.
double	performanceIndex(double[][] a) Returns the performance index of a real eigensystem.

Methods inherited from class java.lang.Object

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

Constructor Detail

Eigen

public Eigen(double[][] a)
      throws Eigen.DidNotConvergeException

Constructs the eigenvalues and the eigenvectors of a real square matrix.

Parameters:

a - is the double square matrix whose eigensystem is to be constructed

Throws:

Eigen.DidNotConvergeException - is thrown when the algorithm fails to converge on the eigenvalues of the matrix.

Eigen

public Eigen(double[][] a,
             boolean computeVectors)
      throws Eigen.DidNotConvergeException

Constructs the eigenvalues and (optionally) the eigenvectors of a real square matrix.

Parameters:

a - is the double square matrix whose eigensystem is to be constructed

computeVectors - is true if the eigenvectors are to be computed

Throws:

Eigen.DidNotConvergeException - is thrown when the algorithm fails to converge on the eigenvalues of the matrix.

Method Detail

getValues

public Complex[] getValues()

Returns the eigenvalues of a matrix of type double.

Returns:

a Complex array containing the eigenvalues of this matrix in descending order

getVectors

public Complex[][] getVectors()

Returns the eigenvectors.

Returns:

A Complex matrix containing the eigenvectors. The eigenvector corresponding to the j-th eigenvalue is stored in the j-th column. Each vector is normalized to have Euclidean length one.

performanceIndex

public double performanceIndex(double[][] a)

Returns the performance index of a real eigensystem.

Parameters:

a - a double matrix

Returns:

A double scalar value indicating how well the algorithms which have computed the eigenvalue and eigenvector pairs have performed. A performance index less than 1 is considered excellent, 1 to 100 is good, while greater than 100 is considered poor.