SymEigen Class

SymEigen Class

Computes the eigenvalues and eigenvectors of a real symmetric matrix.

Inheritance Hierarchy

SystemObject
Imsl.MathSymEigen

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class SymEigen

<SerializableAttribute>
Public Class SymEigen

[SerializableAttribute]
public ref class SymEigen

[<SerializableAttribute>]
type SymEigen =  class end

The SymEigen type exposes the following members.

Constructors

	Name	Description
	SymEigen(Double)	Constructs the eigenvalues and the eigenvectors for a real symmetric matrix.
	SymEigen(Double, Boolean)	Constructs the eigenvalues and (optionally) the eigenvectors for a real symmetric matrix.

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Methods

	Name	Description
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	GetValues	Returns the eigenvalues.
	GetVectors	Return the eigenvectors of a symmetric matrix of type double.
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	PerformanceIndex	Returns the performance index of a real symmetric eigensystem.
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Remarks

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 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).

Reference

Imsl.Math Namespace

Other Resources

Example