eigSym¶

Computes the eigenexpansion of a real symmetric matrix A.

Synopsis¶

eigSym (a)

Required Arguments¶

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

Return Value¶

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

Optional Arguments¶

vectors (Output)

An array of size n × n containing the 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 eigSym computes the eigenvalues of a symmetric real 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 (Parlett 1980, pp. 169 - 173).

Examples¶

Example 1¶

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

a = [[7., -8., -8.],
     [-8., -16., -18.],
     [-8., -18., 13.]]

# Compute eigenvalues
eval = eigSym(a)

# Print eigenvalues
writeMatrix("Eigenvalues", eval)

Output¶

 
             Eigenvalues
          1            2            3
     -27.90        22.68         9.22

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.eigSym import eigSym
from pyimsl.math.writeMatrix import writeMatrix

a = [[7., -8., -8.],
     [-8., -16., -18.],
     [-8., -18., 13.]]
vectors = []
returnNumber = []

# Compute eigenvalues and eigenvectors
eval = eigSym(a, vectors=vectors, returnNumber=returnNumber)

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

Output¶

 
             Eigenvalues
          1            2            3
     -27.90        22.68         9.22
 
              Eigenvectors
             1            2            3
1       0.2945      -0.2722       0.9161
2       0.8521      -0.3591      -0.3806
3       0.4326       0.8927       0.1262

Warning Errors¶

`IMSL_SLOW_CONVERGENCE_SYM`	The iteration for the eigenvalue failed to converge in 100 iterations before deflating.
`IMSL_SLOW_CONVERGENCE_2`	Inverse iteration did not converge. Eigenvector is not correct for the specified eigenvalue.
`IMSL_LOST_ORTHOGONALITY_2`	The eigenvectors have lost orthogonality.
`IMSL_NO_EIGENVALUES_RETURNED`	The number of eigenvalues in the specified interval exceeds `mxeval`. The argument `returnNumber` contains the number of eigenvalues in the interval. No eigenvalues will be returned.