Asian Journal of Mathematics and Computer Research

In this paper, we propose a novel generalized quadrature rule, denoted by SM₁₅ (f), constructed by combining the Anti-Lobatto 4-point rule and the Anti-Gauss 3-point rule through a generalized quadrature framework. A detailed theoretical investigation of the proposed rule is carried out, including convergence analysis and the derivation of appropriate truncation error estimates. The analytical results reveal that the proposed quadrature rule possesses a higher degree of precision and significantly improved accuracy compared with its constituent quadrature rules. To assess the practical performance of the method, several numerical experiments are performed on a variety of test integrals. The obtained results demonstrate that the proposed rule yields highly accurate approximations with considerably reduced truncation errors, thereby confirming its reliability, stability, and computational efficiency. Comparative error analysis and numerical illustrations further establish the superiority of the proposed quadrature rule over the existing component rules. Consequently, the generalized quadrature rule SM₁₅ (f) emerges as an efficient and powerful technique for high-precision numerical integration problems arising in applied mathematics and scientific computing.