Package com.imsl.math

Class SymEigen

java.lang.Object
com.imsl.math.SymEigen
public class SymEigen extends Object
Computes the eigenvalues and eigenvectors of a real symmetric matrix. Orthogonal similarity transformations are used to reduce the matrix to an equivalent symmetric tridiagonal matrix. These transformations are accumulated. An implicit rational QR algorithm is used to compute the eigenvalues of this tridiagonal matrix. The eigenvectors are computed using the eigenvalues as perfect shifts, Parlett (1980, pages 169, 172). The reduction routine is based on the EISPACK routine TRED2. See Smith et al. (1976) for the EISPACK routines. Further details, some timing data, and credits are given in Hanson et al. (1990).

Let M = the number of eigenvalues, \(\lambda\) = the array of eigenvalues, and \(x_j\) is the associated eigenvector with jth eigenvalue.

Also, let \(\varepsilon\) be the machine precision. The performance index, \(\tau\), is defined to be

$$\tau = \mathop{\max}\limits_{1 \le j \le M} \frac{\left\| Ax_j-\lambda _j x_j \right\|_1 }{10N\varepsilon \left\| A \right\|_1 \left\| x_j \right\|_1}$$

While the exact value of \(\tau\) is highly machine dependent, the performance of SymEigen is considered excellent if \(\tau\lt 1\), good if \(1 \le 100\), and poor if \(\tau> 100\). The performance index was first developed by the EISPACK project at Argonne National Laboratory; see Smith et al. (1976, pages 124-125).

See Also:

  • Constructor Summary

    Constructors
    Constructor
    Description
    SymEigen(double[][] a)
    Constructs the eigenvalues and the eigenvectors for a real symmetric matrix.
    SymEigen(double[][] a, boolean computeVectors)
    Constructs the eigenvalues and (optionally) the eigenvectors for a real symmetric matrix.

  • Method Summary

    Modifier and Type
    Method
    Description
    double[]
    getValues()
    Returns the eigenvalues.
    double[][]
    getVectors()
    Return the eigenvectors of a symmetric matrix of type double.
    double
    performanceIndex(double[][] a)
    Returns the performance index of a real symmetric eigensystem.

    Methods inherited from class java.lang.Object

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

  • Constructor Details

    • SymEigen

      public SymEigen(double[][] a)
      Constructs the eigenvalues and the eigenvectors for a real symmetric matrix.
      Parameters:
      a - is the symmetric matrix whose eigensystem is to be constructed.

    • SymEigen

      public SymEigen(double[][] a, boolean computeVectors)
      Constructs the eigenvalues and (optionally) the eigenvectors for a real symmetric matrix.
      Parameters:
      a - a double symmetric matrix whose eigensystem is to be constructed
      computeVectors - a boolean, true if the eigenvectors are to be computed
      Throws:
      IllegalArgumentException - is thrown when the lengths of the rows of the input matrix are not uniform.

  • Method Details

    • getValues

      public double[] getValues()
      Returns the eigenvalues.
      Returns:
      a double array containing the eigenvalues sorted by absolute value in descending order. If the algorithm fails to converge on an eigenvalue, that eigenvalue is set to NaN.

    • getVectors

      public double[][] getVectors()
      Return the eigenvectors of a symmetric matrix of type double.
      Returns:
      a double array containing the eigenvectors. The j-th column of the eigenvector matrix corresponds to the j-th eigenvalue. The eigenvectors are normalized to have Euclidean length one. If the eigenvectors were not computed by the constructor, then null is returned.

    • performanceIndex

      public double performanceIndex(double[][] a)
      Returns the performance index of a real symmetric eigensystem.
      Parameters:
      a - a double symmetric 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.
      Throws:
      IllegalArgumentException - is thrown when the lengths of the rows of the input matrix are not uniform.