The accuracy of the algorithm described earlier is at the O(h) level, also known as the first-order algorithm. The computational error is closely related to the step sizeStep size h. If h is large, the computational error is also large. For example, imprecisely, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h = 0.01$$\end{document}, the computational error is almost 0.01. If there is an algorithm with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(h^2)$$\end{document}, called a second-order algorithm, it is possible to obtain a computational error of 0.0001, while a fourth-order algorithm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(h^4)$$\end{document} may bring the error down to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.01^4 = 10^{-8}$$\end{document}. It follows that if we want to obtain a numerical solution with high accuracy, we need to increase the order of the algorithm.
High-Precision Numerical Algorithms and Implementation in Fractional Calculus
Fractional Calculus ; Chapter : 4 ; 101-138
2024-05-04
38 pages
Article/Chapter (Book)
Electronic Resource
English
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