A Comparison of Simpson’s Rule Generalization through Lagrange and Hermite Interpolating Polynomials

Simpson's Rule is a widely used numerical integration technique, but it cannot be applied to unequally spaced data. This paper presents a new generalization of Simpson's Rule using both Lagrange and Hermite interpolating polynomials to address this limitation. I provide a geometric interpretation of the method, showing its relationship to the area calculation of a trapezoid and a triangle, where the accuracy is significantly influenced by the chosen interpolating polynomial for midpoint determination. A comprehensive comparative analysis across various functions reveals that the Hermite-based approach consistently exhibits higher accuracy and stability than the Lagrange method, particularly with an increasing number of subintervals. This improved performance stems from the Hermite polynomial's ability to better approximate the function's behavior between data points. The findings highlight the effectiveness of the proposed Hermite-based generalization of Simpson's Rule in improving the accuracy of numerical integration for unequally spaced data, which is commonly encountered in practical applications.