A Padé approximate linearization algorithm for solving the quadratic eigenvalue problem with low-rank damping

TL;DR: LuLu et al. as discussed by the authors proposed a Pad´e Approximate Linearization (PAL) algorithm to solve the quadratic eigenvalue problem with low-rank damping.

Abstract: A Pad´e Approximate Linearization Algorithm for Solving the Quadratic Eigenvalue Problem with Low-Rank Damping Ding Lu ∗ Xin Huang † Zhaojun Bai ‡ Yangfeng Su § June 30, 2014 Abstract The low-rank damping term appears commonly in quadratic eigenvalue problems arising from physical simulations. To exploit the low-rank damping property, we propose a Pad´e Approximate Linearization (PAL) algorithm. The advantage of the PAL algorithm is that the dimension of the resulting linear eigenvalue problem is only n + lm, which is generally substantially smaller than the dimension 2n of the linear eigenvalue problem produced by a direct linearization approach, where n is the dimension of the quadratic eigenvalue problem, l and m are the rank of the damping matrix and the order of a Pad´e approximant, respectively. Numerical examples show that by exploiting the low-rank damping property, the PAL algorithm runs 33 – 47% faster than the direct linearization approach for solving modest size quadratic eigenvalue problems. Introduction We consider the quadratic eigenvalue problem (QEP) Q(λ)x ≡ (λ 2 M + λC + K)x = 0, where M , C and K are n × n matrices, referred to as mass, damping and stiffness matrices, respectively, in structural dynamics analysis. The low-rank damping property refers to the case where the damping matrix C is of rank l, l ≪ n and admits the rank-revealing decomposition C = EF T , where E and F are n × l full column rank matrices. The QEP with the low-rank damping arises frequently from analysis of structural dynam- ics [10, 18] and structural-acoustic interaction [3, 6, 32]. In these applications, the damping School of Mathematical Sciences, Fudan University, Shanghai 200433, P. R. China. (dinglu@fudan.edu.cn). Part of this work was done while this author was visiting at University of California, Davis, supported by China Scholarship Council. School of Mathematical Sciences, Fudan University, Shanghai 200433, China. (xinhuang@fudan.edu.cn). Part of this work was done while this author was visiting at University of California, Davis, supported by China Scholarship Council. Department of Computer Science and Department of Mathematics, University of California, Davis, CA 95616, USA. (bai@cs.ucdavis.edu). School of Mathematical Sciences, Fudan University, Shanghai 200433, China. (yfsu@fudan.edu.cn).