Some Efficient Derivative-Based Semi-Open Quadrature Rules with Error Analysis

This study suggests four innovative and effective quadrature techniques that combine functional evaluations and their first-order derivatives for data points that are identically spaced, with an emphasis on computational efficiency regarding time and cost utilisation. All the techniques are theoretically derived, and the theorems concerning accuracy, precision, and error terms are also established. The suggested approaches are derivative-based semi-open-type rules. Compared to the conventional rules, the proposed methods are more accurate and possess higher precision degree. Several numerical experiments are conducted to compare the accuracy, truncation errors, rates of convergence, cost evaluation and average execution times of the new approaches compared with the conventional methods. Because of their promisingly lower computational costs, the results of the analysis demonstrate that the developed methods are more efficient than the original methods from both theoretical and numerical aspects.