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.
Highlights
The numerical integration of a function f : [a, b] → R is often performed by applying the quadrature rule: b m
If f is 3-convex on [a, b], g : [−1, 1] → R given by g(t) = f a + b + b − a t is 3-convex on [−1, 1]
We shall prove that E[g] 0 for any 3-convex function g : [−1, 1] → R
Summary
The numerical integration of a function f : [a, b] → R is often performed by applying the quadrature rule:. In most cases it is not enough to approximate the integral by a simple quadrature rule, since Q[f ; a, b] − I[f ; a, b] > ε. For the composite Simpson’s rule over f ∈ C4[a, b] the following estimation is widely known Is valid for all n ∈ N and for all functions f ∈ C4[a, b] with f (4) of a constant sign This means that if S2n[f ; a, b] − Sn[f ; a, b] < ε, S2n[f ; a, b] − I[f ; a, b] < ε, so we arrive at the approximation of the integral by S2n[f ; a, b]. It is worth mentioning that none of these stopping criteria are valid for all integrable functions. Notice that an extensive survey () of stopping criteria was given by Gonnet [5] (this interesting study is available on the arXiv repository, arXiv:1003.4629)
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