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 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, particularly when the norm of the input matrix (which in practical algorithms is often scaled towards to origin) is sufficiently small. We also demonstrate through experiments the general applicability of our method to the computation of 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方法是一种用于矩阵多项式求值的方案，这些多项式的标量系数出现在许多基于多项式或有理逼近的最新算法中（例如计算超越矩阵函数的算法）。我们推导了一种混合精度的Paterson–Stockmeyer方法，特别适用于求值具有衰减量级标量系数的矩阵多项式。其核心思想是对小量级数据以低精度执行计算，并针对使用低于工作精度的情形提供了舍入误差分析。我们重点研究了矩阵指数泰勒逼近的求值，并展示了该方法在现有缩放平方法算法中的适用性——特别是当输入矩阵范数（在实际算法中通常被向原点缩放）足够小时。通过实验，我们进一步证明了该方法在计算矩阵指数Padé逼近多项式与矩阵余弦泰勒逼近多项式时的广泛适用性。数值实验表明，我们的混合精度Paterson–Stockmeyer算法在保持相同精度水平的同时，比固定精度算法更高效。