A new subspace iteration algorithm for solving generalized eigenvalue problems in vibration analysis

Abstract

Large scale generalized eigenvalue problems (GEP) arise in many applications, such as the vibration analysis, quantum mechanics, electronic structure calculation. A class of subspace iteration method, the generalized Chebyshev–Davidson (gCD) algorithm, was recently proposed to solve GEP. In the gCD algorithm, the Chebyshev polynomial filter technique is incorporated in the subspace iteration. One of the advantages of the gCD algorithm is that it only concerns matrix vector product, making it suitable for solving large-scale problems. In this paper, based on gCD algorithm, a new subspace iteration algorithm is constructed. In the proposed algorithm, we combine the Chebyshev filter and inexact Rayleigh quotient iteration techniques to enlarge the subspace in the iteration, and the obtained algorithm is named as Chebyshev-RQI subspace (CRS) method. Numerical results for both two dimensional and three dimensional vibration analysis problems show that CRS algorithm is more effective than gCD algorithm measured by iteration numbers and computing time. Specifically, when these two methods are used to compute the smallest 20 eigenpairs of the tested vibration models, CRS converges at least 3.9 times faster than gCD measured by number of iteration and up to 2.5 times faster than gCD measured by solution time. Furthermore, CRS algorithm is more robust than gCD because in some cases, gCD cannot converge while CRS always converges for all test cases.