Optimization for Software Implementation of Fractional Calculus Numerical Methods in an Embedded System

Abstract

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.

Complexity Analysis of Three-Dimensional Fractional-Order Chaotic System Based on Entropy Theory

“…In recent years, fractional-order differential operators are introduced into nonlinear dynamical system, and the study of chaos in fractional-order nonlinear dynamical systems becomes a hot topic. At present, there are many definitions of the fractional derivatives, including Grünwald-Letnikov (G-L) definition [38,39], Riemann-Liouville (R-L) definition [40,41] and Caputo definition [42,43].…”

Computational Complexity-based Fractional-Order Neural Network Models for the Diagnostic Treatments and Predictive Transdifferentiability of Heterogeneous Cancer Cell Propensity

“…One example of work handled within that view is by (Raubitzek et al 2022) providing exemplary applications to segment MRI brain scans, for stroke, to be applied as input for a machine learning algorithm. Another work addresses practical software optimization methods to implement fractional-order backward difference, sum, and differintegral operator, which are dependent on the Grünwald-Letnikov definition regarding the evaluation of fractional-order differential equations in embedded systems owing to their more convenient form in contrast with Caputo and Riemann-Liouville definitions (Matusiak 2020).…”