论文标题
矩阵函数的低排名更新II：理性Krylov方法
Low-rank updates of matrix functions II: Rational Krylov methods
论文作者
论文摘要
这项工作开发了新的理性Krylov方法，用于更新大规模矩阵函数F（a）当A受到低级别修改时。它在此上下文上扩展了我们以前的工作，以对多项式Krylov方法进行，为此我们提供了简化的收敛分析。对于理性案例，我们的收敛分析是基于确切的结果，该结果与Bernstein and Van Loan连接到了有理矩阵函数的排名一项更新。我们演示了导出的误差边界在指导krylov方法中的指数函数和Markov函数中指导极点的有用性。矩阵符号功能的低排名更新需要额外关注；为此，我们开发和分析了方法的组合以及一个平方的技巧。指出了Sylvester矩阵方程的此类更新与现有的Rational Krylov子空间方法之间的奇怪联系。
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.
