# An algorithm for quadratic eigenproblems with low rank damping

Taslaman, Leo (2014) An algorithm for quadratic eigenproblems with low rank damping. [MIMS Preprint]

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 develop a new algorithm based on a method proposed by Wang and Zhao [SIAM J. Matrix Anal. Appl. 12-4 (1991), pp. 654--660], which can compute all eigenvalues of definite generalized eigenvalue problems with semidefinite coefficient matrices in a backward stable and symmetry preserving manner. We use this new algorithm to compute the solution to the undamped problem, and then use this solution in order 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 lower order terms to the overall flop count. Numerical experiments show that the proposed algorithm is both fast and accurate. Finally we discuss the application to large scale quadratics and the possibility of generalizations to other problems.