A rational‐Chebyshev projection method for nonlinear eigenvalue problems

IF 1.8 3区 数学 Q1 MATHEMATICS
Z. Tang, Yousef Saad
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引用次数: 0

Abstract

This article describes a projection method based on a combination of rational and polynomial approximations for efficiently solving large nonlinear eigenvalue problems. In a first stage the nonlinear matrix function under consideration is approximated by a matrix polynomial in . The error resulting from this polynomial approximation is in turn approximated by rational functions with the help of the Cauchy integral formula. The two approximations are combined and a linearization is performed. A key ingredient of the proposed approach is a projection method that uses subspaces spanned by vectors of the same dimension as that of the original problem instead of that of the linearized problem. A procedure is also presented to automatically select shifts and to partition the region of interest into a few subregions. This allows to subdivide the problem into smaller subproblems that are solved independently. The accuracy of the proposed method is theoretically analyzed and its performance is illustrated with a few test problems that have been discussed in the literature.
非线性特征值问题的有理-切比雪夫投影法
本文介绍了一种基于有理近似和多项式近似相结合的投影方法,用于高效解决大型非线性特征值问题。在第一阶段,所考虑的非线性矩阵函数用.的矩阵多项式近似。多项式近似产生的误差反过来在考希积分公式的帮助下用有理函数近似。这两个近似值被结合起来,并进行线性化处理。所提方法的一个关键要素是投影法,它使用与原始问题维度相同的向量所跨的子空间,而不是线性化问题的子空间。此外,还提出了一种自动选择移位并将相关区域划分为几个子区域的程序。这样就可以将问题细分为独立求解的更小的子问题。从理论上分析了所提方法的准确性,并用文献中讨论过的几个测试问题说明了该方法的性能。
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来源期刊
CiteScore
3.40
自引率
2.30%
发文量
50
审稿时长
12 months
期刊介绍: Manuscripts submitted to Numerical Linear Algebra with Applications should include large-scale broad-interest applications in which challenging computational results are integral to the approach investigated and analysed. Manuscripts that, in the Editor’s view, do not satisfy these conditions will not be accepted for review. Numerical Linear Algebra with Applications receives submissions in areas that address developing, analysing and applying linear algebra algorithms for solving problems arising in multilinear (tensor) algebra, in statistics, such as Markov Chains, as well as in deterministic and stochastic modelling of large-scale networks, algorithm development, performance analysis or related computational aspects. Topics covered include: Standard and Generalized Conjugate Gradients, Multigrid and Other Iterative Methods; Preconditioning Methods; Direct Solution Methods; Numerical Methods for Eigenproblems; Newton-like Methods for Nonlinear Equations; Parallel and Vectorizable Algorithms in Numerical Linear Algebra; Application of Methods of Numerical Linear Algebra in Science, Engineering and Economics.
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