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影响因子：2.4

DOI码：10.1016/j.cam.2021.113740

发表刊物：Journal of Computational and Applied Mathematics

关键字：Large-scale eigenvalue problem; Arnoldi method; Refined Arnoldi method; Improved refined Arnoldi method; Ritz vector; Refined Ritz vector

摘要：The Arnoldi method is a commonly used technique for finding a few eigenpairs of large, sparse and nonsymmetric matrices. Recently, a new variant of Arnoldi method (NVRA) was proposed. In NVRA, the modified Ritz vector is used to take the place of the Ritz vector by solving a minimization problem. Moreover, it was shown that if the refined Arnoldi method converges, then the NVRA method also converges. The contribution of this work is as follows. First, we point out that the convergence theory of the NVRA method is incomplete. More precisely, the cosine of the angle between the refined Ritz vector and the Ritz vector may not be uniformly lower-bounded, and it can be arbitrarily close to zero in theory. Consequently, the modified Ritz vector may fail to converge even when the search subspace is good enough. A remedy to the convergence of the NVRA method is given. Second, we show that the linear system for solving the modified Ritz vector in the NVRA method will become more and more ill-conditioned as the refined Ritz vector converges. If the Ritz vector also tends to converge as the refined Ritz vector does so, the ill-conditioning of the linear system will have little influence on the convergence of the modified Ritz vector, and the modified Ritz vector can improve the Ritz vector substantially. Otherwise, the ill-conditioning may have significant influence on the convergence of the modified Ritz vector. Third, to fix the NVRA method, we propose an improved refined Arnoldi method that uses improved refined Ritz vector to take the place of the modified Ritz vector. Theoretical results indicate that the improved refined Ritz method is often better than the refined Ritz method. Numerical experiments illustrate the numerical behavior of the improved refined Ritz method, and demonstrate the effectiveness of our theoretical analysis.

备注：中科院 2 区

第一作者：Bo Feng; Gang Wu

论文类型：SCI

通讯作者：Gang Wu

论文编号：113740

学科门类：理学

一级学科：数学

文献类型：SCI

卷号：400

是否译文：否

发表时间：2021-03-25

收录刊物：SCI

发布期刊链接：https://doi.org/10.1016/j.cam.2021.113740