Revisited Optimal Error Bounds for Interpolatory Integration Rules

Abstract

We present a unified way to obtain optimal error bounds for general interpolatory integration rules. The method is based on the Peano form of the error term when we use Taylor’s expansion. These bounds depend on the regularity of the integrand. The method of integration by parts “backwards” to obtain bounds is also discussed. The analysis includes quadrature rules with nodes outside the interval of integration. Best error bounds for composite integration rules are also obtained. Some consequences of symmetry are discussed.

Standard and Corrected Numerical Differentiation Formulae

“…The first approach will be called the standard direct approach, while the second way is the integration by parts "backwards". This second approach presented in [8,9,10], usually presented for numerical integration [11], can be used in general when we consider the method of undetermined coefficients [12]. Let us note some bounds were already presented for specific formulae elsewhere, for example in [13].…”

A new approach to numerical integration based on Coifman wavelets

“…Because most integrals cannot be determined via analytical methods, the numerical integration methods have taken a growing interest of many researchers for approximating the value of a definite integral. To see some quadrature rules based on polynomials, one can refer to [1][2][3]. In recent years, wavelets have gained a lot of popularity and have become a standard tool for many disciplines.…”