eigHermComplex¶

Computes the eigenexpansion of a complex Hermitian matrix A.

Synopsis¶

eigHermComplex (a)

Required Arguments¶

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

Return Value¶

The n 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.

range, float elow, float ehigh (Input)

Return eigenvalues and optionally eigenvectors that lie in the interval with lower limit elow and upper limit ehigh.

Default: (elow, ehigh) = (−∞, +∞).

returnNumber (Output)

The number of output eigenvalues and eigenvectors in the range elow, ehigh.

Description¶

The function eigHermComplex computes the eigenvalues of a complex Hermitian matrix by a two-phase process. The matrix is reduced to tridiagonal form by elementary orthogonal similarity transformations. Then, the eigenvalues are computed using a rational QR or bisection algorithm. Eigenvectors are calculated as required.

Examples¶

Example 1¶

from numpy import *
from pyimsl.math.eigHermComplex import eigHermComplex
from pyimsl.math.writeMatrix import writeMatrix

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

# Compute eigenvalues
eval = eigHermComplex(a)

# Print eigenvalues
writeMatrix("Eigenvalues", eval)

Output¶

 
             Eigenvalues
          1            2            3
      15.38       -10.63        -0.75

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.eigHermComplex import eigHermComplex
from pyimsl.math.writeMatrix import writeMatrix
from pyimsl.math.writeMatrixComplex import writeMatrixComplex

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

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

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

Output¶

 
             Eigenvalues
          1            2            3
      15.38       -10.63        -0.75
 
                     Eigenvectors
                           1                          2
1  (     0.0631,    -0.4075)  (    -0.0598,    -0.3117)
2  (     0.7703,     0.0000)  (    -0.5939,     0.1841)
3  (     0.4668,     0.1366)  (     0.7160,     0.0000)
 
                           3
1  (     0.8539,     0.0000)
2  (    -0.0313,    -0.1380)
3  (     0.0808,    -0.4942)

Warning Errors¶

`IMSL_LOST_ORTHOGONALITY`	The iteration for at least one eigenvector failed to converge. Some of the eigenvectors may be inaccurate.
`IMSL_NEVAL_MXEVAL_MISMATCH`	The determined number of eigenvalues in the interval (#, #) is #. However, the input value for the maximum number of eigenvalues in this interval is #.

Fatal Errors¶

`IMSL_SLOW_CONVERGENCE_GEN`	The iteration for the eigenvalues did not converge.
`IMSL_HERMITIAN_DIAG_REAL`	The matrix element A (#, #) = #. The diagonal of a Hermitian matrix must be real.