Solving a structured quadratic eigenvalue problem by a structure-preserving doubling algorithm

Abstract

In studying the vibration of fast trains, we encounter a palindromic quadratic eigenvalue problem (QEP) $(\lambda^2 A^T + \lambda Q + A)z = 0$, where $A, Q \in \mathbb{C}^{n \times n}$ and $Q^T = Q$. Moreover, the matrix $Q$ is block tridiagonal and block Toeplitz, and the matrix $A$ has only one nonzero block in the upper-right corner. So most of the eigenvalues of the QEP are zero or infinity. In a linearization approach, one typically starts with deflating these known eigenvalues, for the sake of efficiency. However, this initial deflation process involves the inverses of two potentially ill-conditioned matrices. As a result, large error might be introduced into the data for the reduced problem. In this paper we propose using the solvent approach directly on the original QEP, without any deflation process. We apply a structure-preserving doubling algorithm to compute the stabilizing solution of the matrix equation $X+A^TX^{-1}A=Q$, whose existence is guaranteed by a result on the Wiener--Hopf factorization of rational matrix functions associated with semi-infinite block Toeplitz matrices and a generalization of Bendixson's theorem to bounded linear operators on Hilbert spaces. The doubling algorithm is shown to be well defined and quadratically convergent. The complexity of the doubling algorithm is drastically reduced by using the Sherman--Morrison--Woodbury formula and the special structures of the problem. Once the stabilizing solution is obtained, all nonzero finite eigenvalues of the QEP can be found efficiently and with the automatic reciprocal relationship, while the known eigenvalues at zero or infinity remain intact.