Quadrature rules using an arbitrary fixed order of derivatives

Abstract

In this paper we consider quadrature formulas which use the derivative of only an arbitrary fixed order $\left(m\right)$ of function $f$ at the nodes. One of the advantages of the new approach is that we can increase the precision degree of the $n$-point quadrature formulas from $2n-1$ to $2n+m-1$. Furthermore we give an asymptotic estimation for the rate of convergence of this formula. Some examples will be given to support the results.