Numerical Methods for the Tridiagonal Hyperbolic Quadratic Eigenvalue Problem
Bor Plestenjak · SIAM Journal on Matrix Analysis and Applications · 2006
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, and the Ehrlich–Aberth method. Initial approximations are provided by a divide‐and‐conquer approach using rank two modifications, and we show that these initial approximations interlace with the exact eigenvalues. The above methods need a stable and efficient evaluation of the quadratic eigenvalue problem’s characteristic polynomial and its derivatives. We discuss how to obtain these values using three‐term recurrences, the QR factorization, and the LU factorization. Numerical results show that the presented methods are more efficient than solving a linearized generalized eigenvalue problem.