GENEIG Procedure

Computes the generalized eigenexpansion of a system Ax = lBx.

Usage

GENEIG, a, b, alpha, beta

Input Parameters

a—Two-dimensional array of size n-by-n containing coefficient matrix A.

b—Two-dimensional array of size n-by-n containing coefficient matrix B.

Output Parameters

alpha—One-dimensional array of size n containing scalars a_i. If b_i ¹ 0, l_i = a_i /b_i for i = 0, ..., n – 1 are the eigenvalues of the system.

beta—One-dimensional array of size n.

Input Keywords

Double—If present and nonzero, double precision is used.

Output Keywords

Vectors—Named variable into which a two-dimensional array of size n-by-n containing eigenvectors of the problem is stored. Each vector is normalized to have Euclidean length equal to one.

Discussion

The function GENEIG uses the QZ algorithm to compute the eigenvalues and eigenvectors of the generalized eigensystem Ax = lBx, where A and B are matrices of order n. The eigenvalues for this problem can be infinite, so a and b are returned instead of l. If b is nonzero, l = a/b.

The first step of the QZ algorithm is to simultaneously reduce A to upper-Hessenberg form and B to upper-triangular form. Then, orthogonal transformations are used to reduce A to quasi-upper-triangular form while keeping B upper triangular. The generalized eigenvalues and eigenvectors for the reduced problem are then computed.

The function GENEIG is based on the QZ algorithm due to Moler and Stewart (1973), as implemented by the EISPACK routines QZHES, QZIT and QZVAL; see Garbow et al. (1977).

Example 1

In this example, the eigenvalue, l, of system Ax = lBx is computed, where:

 

a  =  TRANSPOSE([[1.0, 0.5, 0.0], [-10.0, 2.0, 0.0], $

   [5.0, 1.0, 0.5]])

b  =  TRANSPOSE([[0.5, 0.0, 0.0], [3.0, 3.0, 0.0], $

   [4.0, 0.5, 1.0]])

; Compute eigenvalues

GENEIG, a, b, alpha, beta

; Print eigenvalues

PM, alpha/beta, Title = 'Eigenvalues'

; PV-WAVE prints the following:

; Eigenvalues

; ( 0.833334, 1.99304)

; ( 0.833333, -1.99304)

; ( 0.500000, 0.00000)

Example 2

This example finds the eigenvalues and eigenvectors of the same eigensystem given in the last example.

a  =  TRANSPOSE([[1.0, 0.5, 0.0], [-10.0, 2.0, 0.0], $

   [5.0, 1.0, 0.5]])

b  =  TRANSPOSE([[0.5, 0.0, 0.0], [3.0, 3.0, 0.0], $

   [4.0, 0.5, 1.0]])

; Compute eigenvalues

GENEIG, a, b, alpha, beta, Vectors = vectors

; Print eigenvalues

PM, alpha/beta, Title = 'Eigenvalues'

; PV-WAVE prints the following:

; Eigenvalues

; ( 0.833332, 1.99304)

; ( 0.833332, -1.99304)

; ( 0.500000, -0.00000)

; Print eigenvectors

PM, vectors, Title = 'Eigenvectors'

; PV-WAVE prints the following:

; Eigenvectors

; ( -0.197112, 0.149911)( -0.197112, -0.149911)

; ( -1.53306e-08, 0.00000)

; ( -0.0688163, -0.567750)( -0.0688163, 0.567750)

; ( -4.75248e-07, 0.00000)

; ( 0.782047, 0.00000)( 0.782047, 0.00000)

; ( 1.00000, 0.00000)

Example 3

In this example, the eigenvalue, l, of system Ax = lBx is solved, where:

 

a  =  TRANSPOSE([$

 [COMPLEX(1.0, 0.0), COMPLEX(0.5, 1.0), COMPLEX(0.0, 5.0)], $

 [COMPLEX(-10.0, 0.0), COMPLEX(2.0, 1.0), COMPLEX(0.0, 0.0)], $

 [COMPLEX(5.0, 1.0), COMPLEX(1.0, 0.0), COMPLEX(0.5, 3.0)]])

b  =  TRANSPOSE([$

 [COMPLEX(0.5, 0.0), COMPLEX(0.0, 0.0), COMPLEX(0.0, 0.0)], $

 [COMPLEX(3.0, 3.0), COMPLEX(3.0, 3.0), COMPLEX(0.0, 1.0)], $

 [COMPLEX(4.0, 2.0), COMPLEX(0.5, 1.0), COMPLEX(1.0, 1.0)]])

; Compute eigenvalues

GENEIG, a, b, alpha, beta

; Print eigenvalues

PM, alpha/beta, Title  =  'Eigenvalues'

; PV-WAVE prints the following:

; Eigenvalues

; ( -8.18016, -25.3799)

; ( 2.18006, 0.609113)

; ( 0.120108, -0.389223)

Example 4

This example finds the eigenvalues and eigenvectors of the same eigensystem given in the last example.

a  =  TRANSPOSE([$

 [COMPLEX(1.0, 0.0), COMPLEX(0.5, 1.0), COMPLEX(0.0, 5.0)], $

 [COMPLEX(-10.0, 0.0), COMPLEX(2.0, 1.0), COMPLEX(0.0, 0.0)], $

 [COMPLEX(5.0, 1.0), COMPLEX(1.0, 0.0), COMPLEX(0.5, 3.0)]])

b  =  TRANSPOSE([$

 [COMPLEX(0.5, 0.0), COMPLEX(0.0,0.0), COMPLEX(0.0, 0.0)], $

 [COMPLEX(3.0,3.0), COMPLEX(3.0,3.0), COMPLEX(0.0, 1.0)], $

 [COMPLEX(4.0, 2.0), COMPLEX(0.5, 1.0), COMPLEX(1.0, 1.0)]])

; Compute eigenvalues

GENEIG, a, b, alpha, beta, Vectors = vectors

; Print eigenvalues

PM, alpha/beta, Title = 'Eigenvalues'

; PV-WAVE prints the following:

; Eigenvalues

; ( -8.18018, -25.3799)

; ( 2.18006, 0.609112)

; ( 0.120109, -0.389223)

; Print eigenvecters

PM, vectors, Title = 'Eigenvectors'

; PV-WAVE prints the following:

; Eigenvectors

; ( -0.326709, -0.124509)( -0.300678, -0.244401)

; ( 0.0370698, 0.151778)

; ( 0.176670, 0.00537758)( 0.895923, 0.00000)

; ( 0.957678, 0.00000)

; ( 0.920064, 0.00000)( -0.201900, 0.0801192)

; ( -0.221511, 0.0968290)