Error bounds for the approximation of matrix functions with rational Krylov methods

IF 1.8 3区 数学 Q1 MATHEMATICS Numerical Linear Algebra with Applications Pub Date : 2024-07-02 DOI:10.1002/nla.2571
Igor Simunec
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Abstract

We obtain an expression for the error in the approximation of and with rational Krylov methods, where is a symmetric matrix, is a vector and the function admits an integral representation. The error expression is obtained by linking the matrix function error with the error in the approximate solution of shifted linear systems using the same rational Krylov subspace, and it can be exploited to derive both a priori and a posteriori error bounds. The error bounds are a generalization of the ones given in Chen et al. for the Lanczos method for matrix functions. A technique that we employ in the rational Krylov context can also be applied to refine the bounds for the Lanczos case.
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用有理克雷洛夫方法逼近矩阵函数的误差边界
我们获得了用有理克雷洛夫方法逼近 和 的误差表达式,其中 是对称矩阵, 是矢量,函数允许积分表示。误差表达式是通过将矩阵函数误差与使用同一有理克雷洛夫子空间的移位线性系统近似解的误差联系起来而得到的,利用它可以得出先验和后验误差边界。这些误差边界是 Chen 等人针对矩阵函数的 Lanczos 方法给出的误差边界的一般化。我们在有理克雷洛夫背景下采用的一种技术也可用于完善 Lanczos 情况下的误差边界。
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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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