Midpoint-Type Rules from an Inequalities Point f View [chapter]

Pietro Cerone, Sever Dragomir
2000 Handbook of Analytic Computational Methods in Applied Mathematics  
The article investigates interior point rules which contain the midpoint as a special case, and obtains explicit bounds through the use of a Peano kernel approach and the modern theory of inequalities. Thus the simplest open Newton-Cotes rules are examined. 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
more » ... d 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. It is demonstrated that the bounds of the approximations are equivalent to those obtained from a Peano kernel that produces Trapezoidal type rules. 1991 Mathematics Subject Classification. Primary 41A55; Secondary 26D15.
doi:10.1201/9781420036053.ch4 fatcat:smsoxua2h5bmxn527ivn23kf34