Some Remarks on the Trapezoid Rule In Numerical Integration

In this paper, by the use of some classical results from the Theory of Inequalities, we point out quasi-trapezoid quadrature formulae for which the error of approximation is smaller than in the classical case. Examples are given to demonstrate that the bounds obtained within this paper may be tighter than the classical ones. Some applications for special means are also given

Summation-By-Parts Operators and High-Order Quadrature

Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by parts. This property can be useful in constructing energy-stable discretizations of partial differential vequations. SBP operators are defined by a weight matrix and a difference operator, with the latter designed to approximate $d/dx$ to a specified order of accuracy. The accuracy of the weight matrix as a quadrature rule is not explicitly part of the SBP definition. We show that SBP weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes. The accuracy of SBP quadrature extends to curvilinear domains provided the Jacobian is approximated with the same SBP operator used for the quadrature. This quadrature has significant implications for SBP-based discretizations; for example, the discrete norm accurately approximates the $L^{2}$ norm for functions, and multi-dimensional SBP discretizations accurately mimic the divergence theorem.Comment: 18 pages, 3 figure

Quadrature formulae of Euler-Maclaurin type based on generalized Euler polynomials of level m

This article deals with some properties -which are, to the best of our knowledge, new- of the generalized Euler polynomials of level m. These properties include a new recurrence relation satisfied by these polynomials and quadrature formulae of Euler-Maclaurin type based on them. Numerical examples are also given.The work of the first author (YQ) has been supported by Decanato de Investigación y Desarrollo, Universidad Simón Bolívar, grant DID-USB (S1-IC-CB-003-16). Also, the first author thanks the hospitality of Coordinación de Matemáticas of Universidad del Atlántico during her visit for the Fall Semester 2016. The work of the second author (AU) has been supported by Universidad del Atlántico, Colombia, grant Impacto Caribe (IC-002627-2015)

Some integral inequalities for functions with (n−1)st derivatives of bounded variation

AbstractIn this paper, we generalize Cerone’s results, and a unified treatment of error estimates for a general inequality satisfying f(n−1) being of bounded variation is presented. We derive the estimates for the remainder terms of the mid-point, trapezoid, and Simpson formulas. All constants of the errors are sharp. Applications in numerical integration are also given

On generalizations of Ostrowski inequality via Euler harmonic identities

Copyright © 2002 L. J. Dedić et al. This work is licensed under a Creative Commons License.Some generalizations of Ostrowski inequality are given, by using some Euler identities involving harmonic sequences of polynomials.L. J. Dedić, M. Matić, J. Pečarić, and A. Vukeli