An adaptive hybrid quadrature scheme: Combining Simpson’s rule and Gaussian quadrature for enhanced numerical integration

This study develops a novel adaptive hybrid quadrature that combines Simpson's 1/3 rule with Gauss-Legendre quadrature to overcome the classical difficulties in performing numerical integration. Classical methods may encounter challenges in achieving a good balance between computational cost and precision, especially when it comes to functions characterized by strongly varying behaviors across their domains. We address these issues via an intelligent adaptation mechanism that reallocates computing resources dynamically on localized function features. We rigorously analyse its convergence properties analytically and prove optimal error estimates in the sense of fourth order accuracy with a strong performance improvement. The hybrid error estimation methodology is based on the mathematical inconsistency of polynomial interpolation and orthogonal polynomial approximation which provides an effective device for local error evaluation. Extensive numerical results indicate that the proposed scheme is consistently better than several existing schemes with significant reduction in function evaluations and acceptable accuracy for different test functions. The proposed framework reduces computational costs by up to 62% when compared to traditional adaptive methods. It maintains similar precision. We carefully examine implementation details, complexity analysis, and practical deployment factors. This work is particularly relevant for scientific computing applications that require high-precision integration in computational physics, engineering simulations, and financial mathematics.