发表机构
Instituto de Matemática Multidisciplinar; Universitat Politècnica de València; Instituto de Telecomunicaciones y Aplicaciones Multimedia; Instituto de Instrumentación para Imagen Molecular（多学科数学研究所; 瓦伦西亚理工大学; 电信与多媒体应用研究所; 分子成像仪器研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出MATLAB工具，扩展双乘积节省的矩阵多项式求值方案至特定阶数及以上多项式，经实验验证其精度不低于Paterson–Stockmeyer方法并实现理论缩减。
AI 中文摘要
计算矩阵函数的数值近似常依赖于高效计算高次矩阵多项式。尽管计算复杂度长期由Paterson–Stockmeyer（PS）方法主导，但近期理论进展已证明可消除两次矩阵乘积（2M）的求值方案可行。现有文献仅记录了该2M缩减在孤立情形下的稳定实例，如矩阵指数和矩阵对数的特定阶数泰勒近似，但针对任意多项式的通用方法尚未建立。为解决这一局限，本研究提出一种软件驱动的程序，将这些计算节省扩展至阶数m∈{18,21,24,26,27,28}及所有m≥30的多项式，要求主要为首项系数非零。由于底层求值系数需通过求解非线性方程组（SNEs）确定，选择数值稳定的解集至关重要。我们引入一种自动验证例程，旨在筛选并验证适用于浮点执行的稳健系数集。主要贡献是一个MATLAB实现，利用可变精度算法处理底层SNEs、验证稳定性并估计精度界。涉及多种矩阵函数的数值实验表明，所开发的实现保持了经典PS方法的数值精度，部分情形下还提升了精度，同时系统实现了2M的理论缩减。
英文摘要
Computing numerical approximations of matrix functions frequently relies on the efficient evaluation of high-degree matrix polynomials. Although computational bounds are historically governed by the Paterson--Stockmeyer (PS) method, recent theoretical developments have demonstrated the viability of evaluation schemes that eliminate two matrix products ($2M$). Existing literature documents stable instances of this $2M$ reduction only for isolated cases, such as specific degrees of Taylor approximations for the matrix exponential and the matrix logarithm. However, a generalized approach for arbitrary polynomials remains unestablished. To address this limitation, this work presents a software-driven procedure that extends these computational savings to polynomials of degrees $m \in \{18, 21, 24, 26, 27, 28\}$ and all $m \ge 30$, requiring primarily a non-zero leading coefficient. Since the underlying evaluation coefficients must be determined by solving systems of nonlinear equations (SNEs), selecting a numerically stable solution set is critical. We introduce an automated verification routine designed to filter and validate robust coefficient sets for floating-point execution. The primary contribution is a MATLAB implementation leveraging variable precision arithmetic to handle the underlying SNEs, verify stability, and project precision bounds. Numerical experiments involving various matrix functions verify that the developed implementation preserves or, in some instances, enhances the numerical accuracy of the classic PS method, while systematically achieving the theoretical reduction of $2M$.
Comments27 pages, 3 figures, This version improves the presentation and reproducibility of the work, updates several numerical results and adds some implementation details
