The matrix exponential is a fundamental operator in scientific computing and system simulation, with applications ranging from control theory and quantum mechanics to modern generative machine learning. While Padé approximants combined with scaling and squaring have long served as the standard, recent Taylor-based methods, which utilize polynomial evaluation schemes that surpass the classical Paterson--Stockmeyer technique, offer superior accuracy and reduced computational complexity. This paper presents an optimized Taylor-based algorithm for the matrix exponential, specifically designed for the high-throughput requirements of generative AI flows. We provide a rigorous error analysis and develop a dynamic selection strategy for the Taylor order and scaling factor to minimize computational effort under a prescribed error tolerance. Extensive numerical experiments demonstrate that our approach provides significant acceleration and maintains high numerical stability compared to existing state-of-the-art implementations. These results establish the proposed method as a highly efficient tool for large-scale generative modeling.

翻译：矩阵指数是科学计算与系统仿真中的基本算子，其应用范围涵盖控制理论、量子力学乃至现代生成式机器学习。尽管长期以来，结合缩放平方技术的帕德逼近被视为标准方法，但近期基于泰勒级数的方法，通过采用超越经典Paterson–Stockmeyer技术的多项式求值方案，提供了更高的精度和更低的计算复杂度。本文提出了一种针对矩阵指数的优化泰勒基算法，专门为满足生成式AI流程的高吞吐量需求而设计。我们提供了严格的误差分析，并开发了一种针对泰勒阶数和缩放因子的动态选择策略，以在给定误差容限下最小化计算量。大量的数值实验表明，与现有最先进的实现相比，我们的方法能显著加速计算并保持高度的数值稳定性。这些结果确立了所提方法作为大规模生成式建模的高效工具。