and T.M. Rassias
Ostrowski Type Inequalities and Applications in Numerical Integration
(pdf) (412 pages in total)
|It was noted in the preface of the book Inequalities Involving Functions and Their Integrals and
Derivatives, Kluwer Academic Publishers, 1991, by D.S. Mitrinovic,
J.E. Pecaric and A.M. Fink; since the writing of the classical book by Hardy, Littlewood and Polya (1934), the subject of differential and integral inequalities has grown by about
800%. Ten years on, we can confidently assert that this growth will increase even more significantly. Twenty pages of Chapter XV in the above mentioned book are devoted to integral inequalities involving functions with bounded derivatives, or, Ostrowski type inequalities. This is now itself a special domain of the Theory of Inequalities with many powerful results and a large number of applications in Numerical Integration, Probability Theory and Statistics, Information Theory and Integral Operator Theory.
The main aim of this present book, jointly written by the members of the Victoria University node of RGMIA (Research Group in Mathematical Inequalities and Applications), is to present a selected number of results on Ostrowski type inequalities. Results for univariate and multivariate real functions and their natural applications in the error analysis of numerical quadrature for both simple and multiple integrals as well as for the Riemann-Stieltjes integral are given.
In Chapter 1, authored by S.S. Dragomir and T.M. Rassias, generalisations of the Ostrowski integral inequality for mappings of bounded variation and for absolutely continuous functions via kernels with n-branches including applications for general quadrature formulae, are given.
Chapter 2, authored by A. Sofo, builds on the work in Chapter 1. He investigates generalisations of integral inequalities for n-times differentiable mappings. With the aid of the modern theory of inequalities and by use of a general Peano kernel, explicit bounds for interior point rules are obtained. Firstly, he develops integral equalities which are then used to obtain inequalities for n-times differentiable mappings on the Lebesgue spaces L¥[a,b], Lp[a,b] , 1< p < ¥ and L1[a,b]. Secondly, some particular inequalities are obtained which include explicit bounds for perturbed trapezoid, midpoint, Simpson's, Newton-Cotes, left and right rectangle rules. Finally, inequalities are also applied to various composite quadrature rules and the analysis allows the determination of the partition required for the accuracy of the result to be within a prescribed error tolerance.
In Chapter 3, authored by P. Cerone and S.S. Dragomir, a unified treatment of three point quadrature rules is presented in which the classical rules of mid-point, trapezoidal and Simpson type are recaptured as particular cases. Riemann integrals are approximated for the derivative of the integrand belonging to a variety of norms. The Grüss inequality and a number of variants are also presented which provide a variety of inequalities that are suitable for numerical implementation. Mappings that are of bounded total variation, Lipschitzian and monotonic are also investigated with relation to Riemann-Stieltjes integrals. Explicit a priori bounds are provided allowing the determination of the partition required to achieve a prescribed error tolerance.
It is demonstrated that with the above classes of functions, the average of a mid-point and trapezoidal type rule produces the best bounds.
In Chapter 4, authored by P. Cerone, product branches of Peano kernels are used to obtain results suitable for numerical integration. In particular, identities and inequalities are obtained involving evaluations at an interior and at the end points. It is shown how previous work and rules in numerical integration are recaptured as particular instances of the current development. Explicit a priori bounds are provided allowing the determination of the partition required for achieving a prescribed error tolerance. In the main, Ostrowski-Grüss type inequalities are used to obtain bounds on the rules in terms of a variety of norms.
In Chapter 5, authored by N.S. Barnett, P. Cerone and S.S. Dragomir, new results for Ostrowski type inequalities for double and multiple integrals and their applications for cubature formulae are presented. This work is then continued in Chapter 6, authored by G. Hanna, where an Ostrowski type inequality in two dimensions for double integrals on a rectangle region is developed. The resulting integral inequalities are evaluated for the class of functions with bounded first derivative. They are employed to approximate the double integral by one dimensional integrals and function evaluations using different types of norms. If the one-dimensional integrals are not known, they themselves can be approximated by using a suitable rule, to produce a cubature rule consisting only of sampling points.
In addition, some generalisations of an Ostrowski type inequality in two dimensions for n - time differentiable mappings are given. The result is an integral inequality with bounded n - time derivatives. This is employed to approximate double integrals using one dimensional integrals and function evaluations at the boundary and interior points.
In Chapter 7, authored by John Roumeliotis, weighted quadrature rules are investigated. The results are valid for general weight functions. The robustness of the bounds is explored for specific weight functions and for a variety of integrands. A comparison of the current development is made with traditional quadrature rules and it is demonstrated that the current development has some advantages. In particular, this method allows the nodes and weights of an n point rule to be easily obtained, which may be preferential if the region of integration varies. Other explicit error bounds may be obtained in advance, thus making it possible to determine the weight dependent partition required to achieve a certain error tolerance.
Note: This book is not in final form. The authors invite
researchers who have suggested amendments to contact S.S. Dragomir.