IDL Analyst Reference Guide: Eigensystem Analysis

IMSL_GENEIG

The IMSL_GENEIG procedure computes the generalized eigenexpansion of a system Ax = lBx.

Syntax

IMSL_GENEIG, a, b, alpha, beta [, /DOUBLE] [, VECTORS=variable]

Arguments

a

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

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.

b

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

beta

One-dimensional array of size n.

Keywords

DOUBLE

If present and nonzero, double precision is used.

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 IMSL_GENEIG function 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 QZ algorithm first simultaneously reduces A to upper-Hessenberg form and B to upper-triangular form, then it uses orthogonal transformations 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 IMSL_GENEIG function 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).

Examples

Example 1

This example computes the eigenvalue, l, of system Ax = lBx, 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  
IMSL_GENEIG, a, b, alpha, beta  
; Print eigenvalues  
PM, alpha/beta, Title = 'Eigenvalues'  
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  
IMSL_GENEIG, a, b, alpha, beta, Vectors = vectors  
; Print eigenvalues  
PM, alpha/beta, Title = 'Eigenvalues'  
Eigenvalues  
   ( 0.833332, 1.99304)  
   ( 0.833332, -1.99304)  
   ( 0.500000, -0.00000)  
; Print eigenvectors  
PM, vectors, Title = 'Eigenvectors'  
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

This example solves the eigenvalue, l, of system Ax = lBx, 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  
IMSL_GENEIG, a, b, alpha, beta  
; Print eigenvalues  
PM, alpha/beta, Title  =  'Eigenvalues'  
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  
IMSL_GENEIG, a, b, alpha, beta, Vectors = vectors  
; Print eigenvalues  
PM, alpha/beta, Title = 'Eigenvalues'  
Eigenvalues  
   ( -8.18018, -25.3799)  
   ( 2.18006, 0.609112)  
   ( 0.120109, -0.389223)  
; Print eigenvecters  
PM, vectors, Title = 'Eigenvectors'  
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)  

Version History

6.4	Introduced

  IDL Online Help (March 06, 2007)