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Erschienen in:
2024 | OriginalPaper | Buchkapitel
4. High-Precision Numerical Algorithms and Implementation in Fractional Calculus
verfasst von : Dingyü Xue, Lu Bai
Erschienen in: Fractional Calculus
Verlag: Springer Nature Singapore
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Abstract
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 size h. If h is large, the computational error is also large. For example, imprecisely, if \(h = 0.01\), the computational error is almost 0.01. If there is an algorithm with \(O(h^2)\), called a second-order algorithm, it is possible to obtain a computational error of 0.0001, while a fourth-order algorithm \(O(h^4)\) may bring the error down to \(0.01^4 = 10^{-8}\). It follows that if we want to obtain a numerical solution with high accuracy, we need to increase the order of the algorithm.