Some New Time and Cost Efficient Quadrature Formulas to Compute Integrals Using Derivatives with Error Analysis

In this research, some new and efficient quadrature rules are proposed involving the combination of function and its first derivative evaluations at equally spaced data points with the main focus on their computational efficiency in terms of cost and time usage. The methods are theoretically derived, and theorems on the order of accuracy, degree of precision and error terms are proved. The proposed methods are semi-open-type rules with derivatives. The order of accuracy and degree of precision of the proposed methods are higher than the classical rules for which a systematic and symmetrical ascendancy has been proved. Various numerical tests are performed to compare the performance of the proposed methods with the existing methods in terms of accuracy, precision, leading local and global truncation errors, numerical convergence rates and computational cost with average CPU usage. In addition to the classical semi-open rules, the proposed methods have also been compared with some Gauss–Legendre methods for performance evaluation on various integrals involving some oscillatory, periodic and integrals with derivative singularities. The analysis of the results proves that the devised techniques are more efficient than the classical semi-open Newton–Cotes rules from theoretical and numerical perspectives because of promisingly reduced functional cost and lesser execution times. The proposed methods compete well with the spectral Gauss–Legendre rules, and in some cases outperform. Symmetric error distributions have been observed in regular cases of integrands, whereas asymmetrical behavior is evidenced in oscillatory and highly nonlinear cases.