Error bounds for quadrature methods involving lower order derivatives

Abstract

Quadrature methods for approximating the definite integral of a function fðtÞ over an interval ½a, b are in common use. Examples of such methods are the Newton-Cotes formulas (midpoint, trapezoidal and Simpson methods etc.) and the Gauss-Legendre quadrature rules, to name two types of quadrature. Error bounds for these approximations involve higher order derivatives. For the Simpson method, in particular, the error bound involves a fourth-order derivative. Discounting the fact that calculating a fourth-order derivative requires a lot of differentiation, the main concern is that an error bound for the Simpson method, for example, is only relevant for a function that is four times differentiable, a rather stringent condition. This paper caters for functions for which derivatives exist only of order lower than normally required. A number of quadrature methods are considered and error bounds derived involving only lower order derivatives that can be used depending on the smoothness of the function.

Error bounds for Gaussian quadrature rules using linear kernels

“…First of all, we point out that new results for the error bounds of Gauss-Legendre quadrature can only be derived from Theorem 2, because the results of Theorem 1 have been previously found in the literature when the corresponding bounds are just real numbers, see, for example [5,6]. The Legendre polynomials…”

New error bounds for Gauss-Legendre quadrature rules

“…are just in terms of the two functions α(x), β(x) not in terms of f and its derivatives while other works ( [1][2][3][4][5][6][7], [17,23]) contain a variety of norms ( like || f || 1 ,|| f || 2 and || f || ∞ ), which are rather difficult to calculate.…”

“…A main problem in an n-point interpolatory quadrature is that the remaining term depends on at least an n-order derivative of the integrand function which is of no use if the integrand is not smooth enough and requires a lot of differentiation and calculation for large n. Hence, many authors prefer to use other techniques including lower order derivatives [1,3,4,6,17,23]. For example, if only f ∈ C 1 [a, b], then the error term would strongly tend to zero for a large class of quadrature rules [5,7]. One of the important cases of interpolatory quadratures is the Gauss-Legendre formula which can be constructed via Hermite interpolation [16].…”

A representation of the Peano kernel for some quadrature rules and applications

“…In recent years, a number of authors have considered error analyses of some known and new quadrature rules (Fedotov & Dragomir 1999;Cerone 2001;Cheng 2001;Pečarić & Varošanec 2001;Dragomir 2003;Engelbrecht et al 2003;Matić 2003). Such rules can be obtained by integrating interpolation polynomials.…”