Fast Numerical Simpson and Boole Integration by Using the Derivatives at the Boundaries
This article shows the extension of the closed Newton-Cotes numerical integration of Simpson’s and Boole’s rule by using the odd derivatives of the function at the boundaries of the integration interval. The derivatives can be used to efficiently increase the convergence order of numerical integration and a fast decrease of the error. Furthermore, due to its simplicity, it is very easy to write into program code, which is also shown. The error estimation is given and proven. Also, the method is confirmed with two different examples for numerical integration, of π and of the integral of the Gaussian distribution. Here, the method is compared to some common numerical integration methods, showing comparably faster convergence.
Michael Engler
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