eigSymgen¶

Computes the generalized eigenexpansion of a system \(Ax = \lambda Bx\). The matrices A and B are real and symmetric, and B is positive definite.

Synopsis¶

eigSymgen (a, b)

Required Arguments¶

float a[[]] (Input): Array of size n ×n containing the symmetric coefficient matrix A.
float b[[]] (Input): Array of size n ×n containing the positive definite symmetric coefficient matrix B.

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 eigenvectors of the problem.

Description¶

The function eigSymgen computes the eigenvalues of a symmetric, positive definite eigenvalue problem by a three-phase process (Martin and Wilkinson 1971). The matrix B is reduced to factored form using the Cholesky decomposition. These factors are used to form a congruence transformation that yields a symmetric real matrix whose eigenexpansion is obtained. The problem is then transformed back to the original coordinates. Eigenvectors are calculated and transformed as required.

Examples¶

Example 1¶

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

a = [[1.1, 1.2, 1.4],
     [1.2, 1.3, 1.5],
     [1.4, 1.5, 1.6]]
b = [[2.0, 1.0, 0.0],
     [1.0, 2.0, 1.0],
     [0.0, 1.0, 2.0]]

# Solve for eigenvalues
eval = eigSymgen(a, b)

# Print eigenvalues
writeMatrix("Eigenvalues", eval)

Output¶

 
             Eigenvalues
          1            2            3
      1.386       -0.058       -0.003

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.eigSymgen import eigSymgen
from pyimsl.math.writeMatrix import writeMatrix

a = [[1.1, 1.2, 1.4],
     [1.2, 1.3, 1.5],
     [1.4, 1.5, 1.6]]
b = [[2.0, 1.0, 0.0],
     [1.0, 2.0, 1.0],
     [0.0, 1.0, 2.0]]
evec = []

# Solve for eigenvalues and eigenvectors
eval = eigSymgen(a, b, vectors=evec)

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

Output¶

 
             Eigenvalues
          1            2            3
      1.386       -0.058       -0.003
 
              Eigenvectors
             1            2            3
1       0.6431      -0.1147      -0.6817
2      -0.0224      -0.6872       0.7266
3       0.7655       0.7174      -0.0858

Warning Errors¶

IMSL_SLOW_CONVERGENCE_SYM The iteration for an eigenvalue failed to converge in 100 iterations before deflating.

Fatal Errors¶

`IMSL_SUBMATRIX_NOT_POS_DEFINITE`	The leading # by # submatrix of the input matrix is not positive definite.
`IMSL_MATRIX_B_NOT_POS_DEFINITE`	Matrix `B` is not positive definite.