Optimal Quadrature Formulas with Derivative in the Space L₂^(m)(0,1)

Abdullo R. Hayotov^1, , Farhod A. Nuraliev¹ and Kholmat M. Shadimetov¹

¹Institute of Mathematics, National University of Uzbekistan, Do‘rmon yo‘li str., Tashkent, Uzbekistan

Cite this paper:
Abdullo R. Hayotov, Farhod A. Nuraliev and Kholmat M. Shadimetov. Optimal Quadrature Formulas with Derivative in the Space L₂^(m)(0,1). American Journal of Numerical Analysis. 2014; 2(4):115-127. doi: 10.12691/ajna-2-4-4

Abstract

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the space . In this paper the quadrature sum consists of values of the integrand and its first derivative at nodes. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number and for any using S.L. Sobolev method which is based on discrete analogue of the differential operator. In particular, for m=2,3 optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from m=4 new optimal quadrature formulas are obtained.

Keywords:
optimal quadrature formulas the error functional the extremal function the Sobolev space the optimal coefficients

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