On a certain adaptive method of approximate integration and its stopping criterion

Abstract

We introduce a new quadrature rule based on Chebyshev's and Simpson's rules. The corresponding composite rule induces the adaptive method of approximate integration. We propose a stopping criterion for this method and we prove that if it is satisfied for a function which is either 3-convex or 3-concave, then the integral is approximated with the prescribed tolerance. Nevertheless, we give an example of a function which does satisfy our criterion, but the approximation error exceeds the assumed tolerance. The numerical experiments (performed by a computer program created by the author) show that integration of 3-convex functions with our method requires considerably fewer steps than the adaptive Simpson's method with a classical stopping criterion. As a tool in our investigations we present a certain inequality of Hermite-Hadamard type.

On optimal inequalities between three-point quadratures

“…We get a similar conclusion on any interval [a, b]. Then the results of the present paper improve on the results of [9].…”

“…Finally we consider the method of Rowland and Varol with the stopping criterion given by ( 6). In fact it was done in [9], so we recall the results here.…”

“…For a more extensive discussion on adaptive integration see the second-named author's recent paper [9].…”

“…We apply the adaptive method which we described above. Our aim is to compare their results with the results of the experiments with quadratures based on the Chebyshev quadrature and Simpson's rule (in the notation of our paper this is the case of z = √ 2 2 and t = 1 2 ) performed in [9]. If for z = √ 2 2 we apply the optimal t z given by Theorem 9, the number of needed subdivisions of the interval of integration is less then the analogous number obtained for t = 1 2 .…”

“…In our experiments we use the same functions and intervals of integration as in [9]. We also use (in fact) the same computer program which we created in the Python programming language for the purposes of [9] (of course some slight modifications needed for the present paper were done).…”