High-Precision Numerical Algorithms and Implementation in Fractional Calculus

Dingyü Xue () and Lu Bai ()
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Dingyü Xue: Northeastern University
Lu Bai: Shenyang University, School of Information Engineering

Chapter 4 in Fractional Calculus, 2024, pp 101-138 from Springer

Abstract: 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 sizeStep size h. If h is large, the computational error is also large. For example, imprecisely, if $$h = 0.01$$ h = 0.01 , the computational error is almost 0.01. If there is an algorithm with $$O(h^2)$$ O ( h 2 ) , called a second-order algorithm, it is possible to obtain a computational error of 0.0001, while a fourth-order algorithm $$O(h^4)$$ O ( h 4 ) may bring the error down to $$0.01^4 = 10^{-8}$$ 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.

Date: 2024
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DOI: 10.1007/978-981-99-2070-9_4

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