SIAM Journal on Numerical Analysis, Volume 59, Issue 3, Page 1325-1347, January 2021.
This work develops novel rational Krylov methods for updating a large-scale matrix function $f(A)$ when $A$ is subject to low-rank modifications. It extends our previous work in this context on polynomial Krylov methods, for which we present a simplified convergence analysis. For the rational case, our convergence analysis is based on an exactness result that is connected to work by Bernstein and Van Loan on rank-one updates of rational matrix functions. We demonstrate the usefulness of the derived error bounds for guiding the choice of poles in the rational Krylov method for the exponential function and Markov functions. Low-rank updates of the matrix sign function require additional attention; we develop and analyze a combination of our methods with a squaring trick for this purpose. A curious connection between such updates and existing rational Krylov subspace methods for Sylvester matrix equations is pointed out.

中文翻译：

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矩阵函数的低阶更新II：有理Krylov方法

SIAM数值分析学报，第59卷，第3期，第1325-1347页，2021年1月。
这项工作开发了新颖的有理Krylov方法，用于在$ A $受到低秩修改的情况下更新大规模矩阵函数$ f（A）$。它扩展了我们在此背景下关于多项式Krylov方法的工作，为此，我们提出了一个简化的收敛分析。对于有理情况，我们的收敛性分析基于一个准确性结果，该结果与Bernstein和Van Loan在有理矩阵函数的秩更新上的工作有关。我们证明了导出的误差界限对于指导指数函数和马尔可夫函数的有理Krylov方法中极点选择的有用性。矩阵符号函数的低位更新需要额外注意；为此，我们开发并分析了我们的方法与平方技巧的组合。