A method for evaluating matrix polynomials have recently been developed that require one fewer matrix product ($1M$) than the Paterson--Stockmeyer (PS) method. Since the computational cost for large-scale matrices is asymptotically determined by the number of matrix products, this reduction directly affects the total execution time. However, the coefficients in these optimized formulas emerge as solutions to systems of nonlinear polynomial equations, resulting in multiple potential solution sets. An inappropriate selection of these coefficients can lead to numerical instability in floating-point arithmetic. This paper presents a systematic framework and a MATLAB implementation, MatrixPolEval1, used to obtain and validate stable coefficient sets for matrix polynomials of degrees $m \in \{8, 10, 12\}$ and above. The framework introduces structural variants to maintain stability even when the original configuration fails to yield a robust solution. The provided tool identifies stable coefficient sets using variable precision arithmetic (VPA) and provides a reliability indicator for expected accuracy. Numerical experiments on polynomials arising in applications, including the matrix exponential and geometric series, show that the framework achieves the $1M$ saving while maintaining numerical accuracy comparable to the PS method.

翻译：近期发展出一种矩阵多项式求值方法，其所需矩阵乘积数比Paterson-Stockmeyer（PS）法少一个（1M）。由于大规模矩阵的计算代价渐近地由矩阵乘积次数决定，这一减少直接影响总执行时间。然而，这些优化公式中的系数以非线性多项式方程组解的形式出现，从而产生多组潜在解。对这些系数的不当选取可能导致浮点运算中的数值不稳定性。本文提出一个系统性框架及MATLAB实现MatrixPolEval1，用于获取并验证次数m∈{8,10,12}及更高阶矩阵多项式的稳定系数集合。该框架引入结构变体，即便原始配置未能产生稳健解时也能维持稳定性。所提供工具使用可变精度算术（VPA）识别稳定系数集合，并提供预期精度的可靠性指标。在包括矩阵指数和几何级数等应用问题中出现的多项式上进行的数值实验表明，该框架在实现1M节省的同时，保持了与PS方法相当的数值精度。