Trapezoidal-Type Rules from an Inequalities Point of View

Abstract

Abstract. The article investigates trapezoid type rules and obtains explicit bounds through the use of a Peano kernel approach and the modern theory of inequalities. Both Riemann-Stieltjes and Riemann integrals are evaluated with a variety of assumptions about the integrand enabling the characterisation of the bound in terms of a variety of norms. Perturbed quadrature rules are obtained through the use of Grüss, Chebychev and Lupaş inequalities, producing a variety of tighter bounds. The implementation is demonstrated through the investigation of a variety of composite rules based on inequalities developed. The analysis allows the determination of the partition required that would assure that the accuracy the result would be within a prescribed error tolerance.

Generalizations of a weighted trapezoidal inequality for monotonic functions and applications

“…(b -a) 3 and the remainder R T if, h) satisfies the estimation [2] i=0 with /, := x i+ \ -x t for i = 0, 1, . .…”

“…For some recent results which generalize, improve and extend this classic inequality (1.1), see the papers [2][3][4][5][6][7][8].…”

“…Recently, Cerone and Dragomir [3] proved the following two trapezoidal-type inequalities for monotonic functions: …”

“…https://doi.org/10.1017/S1446181100003217 [3] A weighted trapezoidal inequality for monotonic functions n-l p .…”

A corrected quadrature formula and applications

“…For examples: Cerone and Dragomir [1,2] considered the corrected midpoint and trapezoid quadrature rules; and with Agarwal [3,4] they considered Simpson's rule.…”