We describe a Lanczos-based algorithm for approximating the product of a rational matrix function with a vector. This algorithm, which we call the Lanczos method for optimal rational matrix function approximation (Lanczos-OR), returns the optimal approximation from a given Krylov subspace in a norm depending on the rational function's denominator, and can be computed using the information from a slightly larger Krylov subspace. We also provide a low-memory implementation which only requires storing a number of vectors proportional to the denominator degree of the rational function. Finally, we show that Lanczos-OR can be used to derive algorithms for computing other matrix functions, including the matrix sign function and quadrature based rational function approximations. In many cases, it improves on the approximation quality of prior approaches, including the standard Lanczos method, with little additional computational overhead.

翻译：我们描述了一种基于Lanczos的算法，用于近似有理矩阵函数与向量的乘积。该算法称为Lanczos最优有理矩阵函数逼近方法（Lanczos-OR），能够从给定Krylov子空间中返回依赖于有理函数分母的范数下的最优逼近，并且可利用稍大的Krylov子空间的信息进行计算。我们还提供了一种低内存实现，仅需存储与有理函数分母次数成正比的向量数量。最后，我们展示了Lanczos-OR可用于推导计算其他矩阵函数的算法，包括矩阵符号函数和基于求积的有理函数逼近。在许多情况下，它在几乎没有额外计算开销的情况下提高了先前方法（包括标准Lanczos方法）的逼近质量。