Inequalities of Ostrowski–Grüss type and applications

Abstract

Some new inequalities of Ostrowski-Grüss type are derived. They are applied to the error analysis for some Gaussian and Gaussian-like quadrature formulas. 1. Introduction. In this paper we derive some new inequalities of Ostrowski-Grüss type. They are applied to the error analysis for some Gaussian and Gaussian-like quadrature formulas. We consider Gauss-Legendre, Chebyshev, Radau and Lobatto quadratures. A similar error analysis for rules of Newton-Cotes type can be found in the literature. In particular, the mid-point, trapezoid and Simpson rules have been investigated recently ([1], [2], [4], [8], [11]) with a view to obtaining bounds on the quadrature rule in terms of a variety of norms involving, at most, the first derivative. The present work brings results for the above mentioned Gaussian and Gaussian-like quadrature rules giving explicit error bounds. We use Peano type kernels and an approach via inequalities. The error bounds are expressed in terms of first derivatives. The general approach used in the past required the assumption of bounded derivatives of degree higher than one. Our error bounds are in general (but not always) better than the Peano bounds. We also consider composite quadrature rules. Our analysis allows the determination of a partition that would assure a prescribed error tolerance. In Section 2 we establish some inequalities of Ostrowski-Grüss type. Such inequalities are considered in [2], [4], [8] and [11]. In Sections 3-6 we apply these inequalities to obtain explicit error bounds for the above mentioned quadrature rules. In Section 7 we give a general procedure for forming

Error inequalities for a quadrature formula and applications

“…In Section 3, we establish some error bounds for the optimal formula. Similar estimations can be found in [6,7,9], where some different quadrature formulas are considered. These estimations ensure that we can apply the optimal quadrature formula to different classes of functions.…”

“…In particular, the midpoint, trapezoid, and Simpson rules have been investigated more recently [1][2][3][4][5][6] with the view of obtaining bounds on the quadrature rule in terms of a variety of norms involving, at most, the first derivative. Gauss-like quadrature rules are considered in [7,8] from an inequalities point of view. These results enlarge the applicability of the mentioned quadrature rules.…”

Pure and Applied Mathematics

“…Formulae using such interpolation with evenly spaced nodes are referred to as Newton-Cotes formulae. The Gaussian quadrature formulae, which are optimal and converge rapidly by selecting the node points carefully that need not be equally spaced, are investigated in [94].…”

Error inequalities for an optimal quadrature formula

“…We can also seek a quadrature formula such that it is exact for polynomials of maximal degree (Gauss formulas). Gauss-like quadrature formulas are considered in [12].…”