Solving problems involving numerical integration (Ⅱ): Modified Simpson's methods for equal intervals of odd numbers

Abstract

<abstract> <p>The trapezium and Simpson's methods are widely used for numerical integration. In most circumstances, Simpson's method is more accurate than the trapezium method but only applicable to cases with equal intervals of even numbers. This technical note reports the formulation of two modified Simpson's methods, the trapezium-corrected Simpson's method (TCSM) and cubic-corrected Simpson's method (CCSM), as general-purpose symmetric formulas to solve numerical integrations with equal intervals of odd numbers (<italic>n</italic> ≥ 5) with the same level of accuracy as that of Simpson's method applied to the even number near <italic>n</italic>. Error analysis in terms of the order of error bound and case studies in this note demonstrate and validate the usefulness of the proposed formulas for solving different types of theoretical problems and real-world applications. In terms of accuracy of approximation for cases with equal intervals of odd numbers, CCSM performs better than TCSM by at least one order in error bound whereas TCSM performs better than the trapezium method by at least one order in error bound.</p> </abstract>