Optimization for Software Implementation of Fractional Calculus Numerical Methods in an Embedded System
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 fracti...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-05-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/22/5/566 |
| _version_ | 1797567612964569088 |
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| author | Mariusz Matusiak |
| author_facet | Mariusz Matusiak |
| author_sort | Mariusz Matusiak |
| collection | DOAJ |
| description | 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<sup>®</sup> Cortex<sup>®</sup>-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<sup>®</sup> core. |
| first_indexed | 2024-03-10T19:44:22Z |
| format | Article |
| id | doaj.art-08f86db62a1a434f84cf567e4379f37f |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T19:44:22Z |
| publishDate | 2020-05-01 |
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| series | Entropy |
| spelling | doaj.art-08f86db62a1a434f84cf567e4379f37f2023-11-20T00:53:48ZengMDPI AGEntropy1099-43002020-05-0122556610.3390/e22050566Optimization for Software Implementation of Fractional Calculus Numerical Methods in an Embedded SystemMariusz Matusiak0Institute of Applied Computer Science, Lodz University of Technology, 90-924 Lodz, PolandIn 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<sup>®</sup> Cortex<sup>®</sup>-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<sup>®</sup> core.https://www.mdpi.com/1099-4300/22/5/566fractional-order differential equationsGrünwald–Letnikov differintegralfractional-order backward difference/sumArm<sup>®</sup> Cortex<sup>®</sup>-MSTM32<sup>TM</sup> microcontroller implementation |
| spellingShingle | Mariusz Matusiak Optimization for Software Implementation of Fractional Calculus Numerical Methods in an Embedded System Entropy fractional-order differential equations Grünwald–Letnikov differintegral fractional-order backward difference/sum Arm<sup>®</sup> Cortex<sup>®</sup>-M STM32<sup>TM</sup> microcontroller implementation |
| title | Optimization for Software Implementation of Fractional Calculus Numerical Methods in an Embedded System |
| title_full | Optimization for Software Implementation of Fractional Calculus Numerical Methods in an Embedded System |
| title_fullStr | Optimization for Software Implementation of Fractional Calculus Numerical Methods in an Embedded System |
| title_full_unstemmed | Optimization for Software Implementation of Fractional Calculus Numerical Methods in an Embedded System |
| title_short | Optimization for Software Implementation of Fractional Calculus Numerical Methods in an Embedded System |
| title_sort | optimization for software implementation of fractional calculus numerical methods in an embedded system |
| topic | fractional-order differential equations Grünwald–Letnikov differintegral fractional-order backward difference/sum Arm<sup>®</sup> Cortex<sup>®</sup>-M STM32<sup>TM</sup> microcontroller implementation |
| url | https://www.mdpi.com/1099-4300/22/5/566 |
| work_keys_str_mv | AT mariuszmatusiak optimizationforsoftwareimplementationoffractionalcalculusnumericalmethodsinanembeddedsystem |