An algorithm for quadratic eigenproblems with low rank damping

We consider quadratic eigenproblems $\left(M\lambda^2+D\lambda+K\right)x=0$, where all coefficient matrices are real and positive semidefinite, $(M,K)$ is regular and $D$ is of low rank. Matrix polynomials of this form appear in the analysis of vibrating structures with discrete dampers. We develop an algorithm for such problems, which first solves the undamped problem $\left(M\lambda^2+K\right)x=0$ and then accommodates for the low rank term $D\lambda$. For the first part, we modify an algorithm proposed by Wang and Zhao [SIAM J. Matrix Anal. Appl. 12-4 (1991), pp. 654--660]. The modified algorithm is then used to solve the undamped problem such that all eigenvalues are computed in a backward stable manner. We then use the solution to the undamped problem to compute all eigenvalues of the original problem, and the associated eigenvectors if requested. To this end, we use an Ehrlich-Aberth iteration that works exclusively with vectors and tall skinny matrices and contributes only with lower order terms to the flop count. Numerical experiments show that the proposed algorithm is both fast and accurate. Finally we discuss the application to the large scale case and the possibility of generalizations