论文标题
低内存Krylov子空间方法用于最佳有理矩阵函数近似
Low-memory Krylov subspace methods for optimal rational matrix function approximation
论文作者
论文摘要
我们描述了一种基于兰开斯的算法，用于近似于矢量的有理矩阵函数的乘积。该算法称为最佳理性矩阵函数近似（Lancos-OR）的Lanczos方法，它从给定的Krylov子空间中返回最佳的近似，具体取决于有理功能的分母，并且可以使用较大的Krylov子空间的信息来计算。我们还提供了一个低内存的实现，该实现仅需要存储与有理函数的分母程度成正比的许多向量。最后，我们表明可以使用Lanczos-OR来得出用于计算其他矩阵函数的算法，包括矩阵符号函数和基于正交的合理函数近似值。在许多情况下，它改善了先前方法的近似质量，包括标准的兰开斯方法，几乎没有其他计算开销。
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.
