Numerical Methods for the Tridiagonal Hyperbolic Quadratic Eigenvalue Problem

We consider numerical methods for the computation of the eigenvalues of the tridiagonal hyperbolic quadratic eigenvalue problem. The eigenvalues are computed as zeros of the characteristic polynomial using the bisection, Laguerre’s method, the Ehrlich–Aberth method, and the Durand–Kerner method. Initial approximations are provided by a divide-and-conquer approach using rank two modifications. The above methods need a stable and efficient computation of the characteristic polynomial and its derivatives. We discuss how to obtain these values using the three-term recurrences, the QR factorization, and the LU factorization.