Advanced Family of Newton-Cotes Formulas

Abstract

In this paper, we introduced an advanced family of numerical composite integration formulas of closed Newton–Cotes-type that uses the function values on uniformly spaced intervals only without any derivative values. To increase the accuracy, we divide the given interval into a number of equal subintervals and integrating on each interval by using integration rules with abscissas outside integration interval. Since there are more unknowns when using including function values outside integration interval in addition to function values of on interval, the order of accuracy of these numerical integration formulas is higher than the standard closed Newton-Cotes formulae. These new formulae are obtained using the method of undetermined coefficients which are based on the concept of the precision of the quadrature formula. The error terms are found using the concept of precision. Also, we compared the errors in an advanced family of numerical composite integration formulas with the errors in composite closed Newton–Cotes-type. Finally, we have presented some examples and then mentioned the related MATLAB codes.