High-Precision Numerical Algorithms and Implementation in Fractional Calculus

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 \(h = 0.01\) , 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.