Mathematics and Computer Science: Research Updates Vol. 11

Among the various classical quadrature rules, some of the most widely used are the Trapezoidal Rule, Simpson’s Rule, Newton–Cotes Rules, Clenshaw–Curtis Quadrature Rule, and Gauss–Legendre Quadrature Rule. Each of these methods approximates the given integral by replacing the integrand with a simpler function, such as a polynomial, whose integral can be evaluated easily. This paper presents a generalised quadrature rule SM₁₄ (f) of degree of precision seven by combining two well-known quadrature formulas, namely the Clenshaw–Curtis five-point quadrature rule and the Gauss–Legendre three-point quadrature rule, each possessing precision five. The proposed rule is constructed using the generalised quadrature technique to achieve higher accuracy with improved computational efficiency. The theoretical superiority of the developed rule over its constituent formulas is established through rigorous error analysis. Furthermore, the performance and dominance of the proposed quadrature rule are numerically verified by considering different types of test integrals. The proposed rule also demonstrates exactness for certain test integrals where the base rules fail or show larger errors. Hence, the developed generalised quadrature rule provides a more efficient, stable, and accurate technique for numerical integration, making it a valuable contribution to computational mathematics.