The Paterson--Stockmeyer method is an evaluation scheme for matrix polynomials with scalar coefficients that arise in many state-of-the-art algorithms based on polynomial or rational approximation, for example, those for computing transcendental matrix functions. We derive a mixed-precision version of the Paterson--Stockmeyer method that is particularly useful for evaluating matrix polynomials with scalar coefficients of decaying magnitude. The new method is mainly of interest in the arbitrary precision arithmetic, and it is particularly attractive for high-precision computations. The key idea is to perform computations on data of small magnitude in low precision, and rounding error analysis is provided for the use of lower-than-working precisions. We focus on the evaluation of the Taylor approximants of the matrix exponential and show the applicability of our method to the existing scaling and squaring algorithms. We also demonstrate through experiments the general applicability of our method to other problems, such as computing the polynomials from the Pad\'e approximant of the matrix exponential and the Taylor approximant of the matrix cosine. Numerical experiments show our mixed-precision Paterson--Stockmeyer algorithms can be more efficient than its fixed-precision counterpart while delivering the same level of accuracy.

翻译：Paterson–Stockmeyer方法是针对标量系数矩阵多项式的求值方案，广泛存在于基于多项式或有理逼近的先进算法中，例如计算超越矩阵函数的算法。本文推导出混合精度的Paterson–Stockmeyer方法，特别适用于求解具有衰减量级标量系数的矩阵多项式。新方法主要适用于任意精度算术计算领域，尤其在高精度计算中具有显著优势。其核心思想是对小量级数据采用低精度计算，并针对低于工作精度的计算提供了舍入误差分析。我们重点研究矩阵指数泰勒逼近式的求值问题，展示了该方法在现有缩放与平方算法中的适用性。通过实验进一步证明了该方法对其他问题的普适性，例如计算矩阵指数的Padé逼近多项式及矩阵余弦的泰勒逼近式。数值实验表明，混合精度Paterson–Stockmeyer算法在保持相同精度水平的同时，可较固定精度版本获得更高的计算效率。