Derivation of a New Numerical Method for Approximating Single Integrals Based on Boole and Simpson's Rules

Abstract

This research aims to derive a new numerical method and its correction terms (error
values) for approximating single integrals using Boole’s and Simpson’s formulas. This
method addresses cases where the integrand is continuous, yet its derivative exhibits a
singularity (or irregularity) at one of the endpoints of the integration interval. To
accelerate convergence, the Romberg acceleration technique was employed.
A numerical comparison was conducted between the proposed method and established
numerical integration techniques. This comparison utilized a set of numerical examples
to evaluate accuracy, the rate of convergence, the number of sub-intervals required to
achieve a specific level of precision, and computational efficiency. The results
demonstrate the proposed method's ability to achieve high levels of accuracy with fewer
sub-intervals in certain cases, highlighting its numerical efficiency in approximating
single integrals. The proposed method has been named the Boole–Simpson method and
is designated by the symbol (BS).