Enhanced Numerical Techniques for Selective Integration Using Error Correction Methods

Classical numerical integration methods, such as Simpson’s rule and Gaussian quadrature, perform well for smooth functions but lose accuracy near discontinuities. This paper introduces a Selective Error-Correcting Adaptive Quadrature algorithm (SECAQ), a method designed to address this issue. SECAQ identifies intervals with large local error or sharp changes, estimates jump sizes using one-sided evaluations, and applies compact correction terms activated only near the discontinuity. These localized corrections restore smoothness within each subinterval and can enable the base quadrature rule to achieve its full theoretical accuracy. Numerical tests involving jumps, cusps, and oscillatory functions demonstrate that SECAQ restores fourth-order convergence for Simpson’s rule and achieves errors close to machine precision. Compared to the standard adaptive Simpson’s method, SECAQ reduces the number of function evaluations by up to 60% while maintaining low computational overhead. The method is particularly useful when discontinuity locations or jump sizes are known or can be reliably estimated