Optimal quadrature formulas of Euler–Maclaurin type

Abstract

In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space L2(m)(0,1) is considered. Here the quadrature sum consists of values of the integrand at nodes and values of derivatives of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number N and for any [m2]≥s≥1 using Sobolev method which is based on discrete analogue of the differential operator d2m/dx2m. In particular, for s=[m/2] optimality of the classical Euler–Maclaurin quadrature formula is obtained. Starting from [m/2] > s new optimal quadrature formulas are obtained. Finally, numerical results which confirm theoretical are presented.