AN ALGORITHM FOR QUADRATIC EIGENPROBLEMS WITH LOW RANK DAMPING

作者：Taslaman Leo^*

来源：SIAM Journal on Matrix Analysis and Applications, 2015, 36(1): 251-272.

DOI：10.1137/140969099

• quadratic eigenvalue problem
• eigenvalue algorithm
• matrix polynomial
• discrete damper
• vibrating system
• low rank damping
• definite generalized eigenvalue problem
• semidefinite matrix

We consider quadratic eigenproblems (M lambda(2) + D lambda + K)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 (M lambda(2) + K)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 (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 we 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.

• 出版日期2015