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 h = 0.01 \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 O ( h 2 ) \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 O ( h 4 ) \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 0 . 01 4 = 10 - 8 \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.


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    Title :

    High-Precision Numerical Algorithms and Implementation in Fractional Calculus


    Contributors:
    Xue, Dingyü (author) / Bai, Lu (author)

    Published in:

    Fractional Calculus ; Chapter : 4 ; 101-138


    Publication date :

    2024-05-04


    Size :

    38 pages




    Type of media :

    Article/Chapter (Book)


    Type of material :

    Electronic Resource


    Language :

    English





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