Numerical Methods For Large-scale Symmetric Generalized Eigenvalue Problems And Their Applications

Large-scale symmetric generalized eigenvalue problems have a wide range of applications in engineering,computational physics,and biological sciences.For some practical problems,such as modal analysis,the matrices are often large,sparse,block-structured,symmetric,positive definite,and generally only need to solve a few extreme eigenpairs.Traditional transform algorithms cannot make full use of these characteristics,and are difficult to solve due to the limitation of large amount of computation and high storage capacity.The Chebyshev-Davidson algorithm is an effective method for solving the smallest or largest few eigenpairs of large-scale symmetric generalized eigenvalue problems in projection algorithms.First,we introduce the transformation process of large-scale symmetric generalized eigenvalue problems in modal analysis,and the principle of Chebyshev-Davidson algorithm.Secondly,we improve the Chebyshev-Davidson algorithm in two aspects to reduce storage requirements and speed up convergence.On the one hand,the selection method of filter parameters is optimized.In each iteration,we use the k-step Lanczos algorithm to estimate the upper bound of the eigenvalues of the intermediate matrix to adjust the Chebyshev filter parameters,so that the algorithm achieves faster convergence.On the other hand,considering that the Ritz value converges but the Ritz vector may converge slowly or even fail to converge,we incorporate refinement strategies into algorithms and analyze the convergence of the algorithm.In each iteration,replacing the Ritz vector with the refinement vector can obtain a better approximation of the eigenvectors.The refinement process only needs to increase the amount of negligible calculations to achieve the correction effect by borrowing the existing calculation results in the iterative process.Finally,we use several experiments to show the shortcomings of the original ChebyshevDavidson algorithm,the improvement effect of the improved Chebyshev-Davidson algorithm in terms of iteration steps and running time,the effect comparisons of selecting filter parameters by different methods,the influence of the maximum projected subspace dimension on the convergence of the algorithm,and the application in modal analysis.Several examples show that the improved Chebyshev-Davidson algorithm works better than the original algorithm.