`polyeig`

Eigenvalues of a polynomial eigenvalue problem.

Syntax

e = polyeig(A0, A1, ..., Ap)

Returns the values e for which (A0 + e*A1 + ... + e^p*Ap)*x = 0 has a nonzero solution x.

Description

With p = 1 the problem is (A0 + e*A1)x = 0, a generalized eigenproblem; with A1 = eye(n) it gives the negatives of the eigenvalues of A0. With p = 2 it is the quadratic eigenvalue problem that models vibrating systems with mass, damping and stiffness: polyeig(K, C, M).

The problem is solved by linearization into a larger generalized eigenvalue problem of size n*p.

Input arguments

A0, ..., Ap: Square matrices of the same order n, the coefficients of the matrix polynomial, lowest power first.

Output arguments

e: Column vector with n*p eigenvalues (some may be Inf if Ap is singular).

Example

A linear case equals minus eig

A0 = [1 2; 3 4];
e = polyeig(A0, eye(2))
-eig(A0)

Output

e = 2×1

    0.3723
   -5.3723
ans = 2×1

    0.3723
   -5.3723

Notes

A second output, [X, e] = polyeig(...), also returns the eigenvectors as the columns of X.

polyeig