In this article, some practical software optimization methods for implementations of fractional order backward difference, sum, and differintegral operator based on Grünwald–Letnikov definition are presented. These numerical algorithms are of great interest in the context of the evaluation of fractional-order differential equations in embedded systems, due to their more convenient form compared to Caputo and Riemann–Liouville definitions or Laplace transforms, based on the discrete convolution operation. A well-known difficulty relates to the non-locality of the operator, implying continually increasing numbers of processed samples, which may reach the limits of available memory or lead to exceeding the desired computation time. In the study presented here, several promising software optimization techniques were analyzed and tested in the evaluation of the variable fractional-order backward difference and derivative on two different Arm® Cortex®-M architectures. Reductions in computation times of up to 75% and 87% were achieved compared to the initial implementation, depending on the type of Arm® core.

中文翻译：

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嵌入式系统中分数阶微积分数值方法的软件实现优化

在本文中，提出了一些基于 Grünwald-Letnikov 定义的用于分数阶后向差分、求和和微分积分算子实现的实用软件优化方法。与基于离散卷积运算的 Caputo 和 Riemann-Liouville 定义或拉普拉斯变换相比，这些数值算法在评估嵌入式系统中的分数阶微分方程的上下文中非常有趣。一个众所周知的困难与算子的非局部性有关，这意味着处理样本的数量不断增加，这可能会达到可用内存的限制或导致超出所需的计算时间。在这里介绍的研究中，在评估两种不同 Arm® Cortex®-M 架构上的可变分数阶后向差分和导数时，分析和测试了几种有前途的软件优化技术。与初始实施相比，计算时间最多减少 75% 和 87%，具体取决于 Arm® 内核的类型。