eigGenComplex¶

Computes the eigenexpansion of a complex matrix A.

Synopsis¶

eigGenComplex (a)

Required Arguments¶

complex a[[]] (Input): Array of size n ×n containing the matrix.

Return Value¶

The n complex eigenvalues of the matrix. If no value can be computed, then None is returned.

Optional Arguments¶

vectors (Output): An array of size n ×n containing eigenvectors of the matrix.

Description¶

The function eigGenComplex computes the eigenvalues of a complex matrix by a two-phase process. The matrix is reduced to upper Hessenberg form by elementary Gauss transformations. Then, the eigenvalues are computed using an explicitly shifted LR algorithm. Eigenvectors are calculated during the iterations for the eigenvalues (Martin and Wilkinson 1971).

Examples¶

Example 1¶

from numpy import *
from pyimsl.math.eigGenComplex import eigGenComplex
from pyimsl.math.writeMatrixComplex import writeMatrixComplex

a = [[5 + 9j, 5 + 5j, -6 - 6j, -7 - 7j],
     [3 + 3j, 6 + 10j, -5 - 5j, -6 - 6j],
     [2 + 2j, 3 + 3j, -1 + 3j, -5 - 5j],
     [1 + 1j, 2 + 2j, -3 - 3j, 0 + 4j]]

# Compute eigenvalues
eval = eigGenComplex(a)

# Print eigenvalues
writeMatrixComplex("Eigenvalues", eval)

Output¶

 
                     Eigenvalues
                        1                          2
(          4,          8)  (          3,          7)
 
                        3                          4
(          2,          6)  (          1,          5)

Example 2¶

This example is a variation of the first example. Here, the eigenvectors are computed as well as the eigenvalues.

from numpy import *
from pyimsl.math.eigGenComplex import eigGenComplex
from pyimsl.math.writeMatrixComplex import writeMatrixComplex

a = [[5 + 9j, 5 + 5j, -6 - 6j, -7 - 7j],
     [3 + 3j, 6 + 10j, -5 - 5j, -6 - 6j],
     [2 + 2j, 3 + 3j, -1 + 3j, -5 - 5j],
     [1 + 1j, 2 + 2j, -3 - 3j, 0 + 4j]]
evec = []

# Compute eigenvalues and eigenvectors
eval = eigGenComplex(a, vectors=evec)

# Print eigenvalues and eigenvectors
writeMatrixComplex("Eigenvalues", eval)
writeMatrixComplex("Eigenvectors", evec)

Output¶

 
                     Eigenvalues
                        1                          2
(          4,          8)  (          3,          7)
 
                        3                          4
(          2,          6)  (          1,          5)
 
                     Eigenvectors
                           1                          2
1  (     0.5774,     0.0000)  (     0.5774,     0.0000)
2  (     0.5774,    -0.0000)  (     0.5774,    -0.0000)
3  (     0.5774,    -0.0000)  (    -0.0000,     0.0000)
4  (    -0.0000,     0.0000)  (     0.5774,    -0.0000)
 
                           3                          4
1  (     0.3780,     0.0000)  (     0.7559,     0.0000)
2  (     0.7559,     0.0000)  (     0.3780,    -0.0000)
3  (     0.3780,     0.0000)  (     0.3780,    -0.0000)
4  (     0.3780,     0.0000)  (     0.3780,    -0.0000)

Fatal Errors¶

IMSL_SLOW_CONVERGENCE_GEN

The iteration for an eigenvalue did not converge after

# iterations.