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

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.



Quadratures and adaptive methods of numerical integration
The numerical integration of a function f : [a, b] → R is often performed by applying the quadrature rule: where the coefficients λ 1 , . . . , λ m ∈ [0, 1] and the weights w 1 , . . . , w m ∈ R (usually positive) are fixed. For example, the following assignment is the familiar simple Simpson's rule: In this paper we also deal with the simple Chebyshev's rule For the sake of the record, let us also list simple two-point Gauss' rule Every quadrature rule induces the so-called composite rule, which is created by subdividing the interval [a, b] into n subintervals with equally spaced endpoints: ], then we sum over k ∈ {0, . . . , n − 1} to arrive at the rule: In this way we get Simpson's and Chebyshev's composite rules, denoted respectively S n and C n . Let ε > 0 be a fixed tolerance. In most cases it is not enough to approximate the integral by a simple quadrature rule, since In order to achieve the desired precision we apply the composite rule Q n with n large enough to have Such a method of approximation is called adaptive. How to find such n? If the function f is regular enough, the error terms of the simple quadratures are known and the error bounds could be easily found. For instance, for the composite Simpson's rule over f ∈ C 4 [a, b] the following estimation is widely known (cf. e.g. [4,7] ) If the term on the right hand side is less than ε, then a suitable n can be easily determined. Then, of course, Nevertheless, this approach could be complicated because of difficulties in the estimation of the maximum of |f (4) |, or even impossible, if f is not regular enough. Another approach to adaptive integration is based on successive incrementation of n. Then the natural question arises how large should be n to achieve the desired precision, i.e., when the incrementation could be finished. Various stopping criteria are known in the literature. Let us briefly mention two of them for Simpson's method. The first one was given by Lyness [9]; it used the so-called local errors Another stopping criterion was proposed by Clenshaw and Curtis [3] and was investigated by Rowland and Varol [12], who have proved that the stopping inequality is valid for all n ∈ N and for all functions This is the starting point for our considerations. It is worth mentioning that none of these stopping criteria are valid for all integrable functions. Indeed, there are functions satisfying these criteria with an error of approximation of the integral greater than the tolerance ε. Notice that an extensive survey (not only) of stopping criteria was given by Gonnet [5] (this interesting study is also available on the arXiv repository, arXiv:1003.4629).

Convex functions of higher order
Hopf [6] proposed to consider functions with non-negative divided differences and subsequently Popoviciu [10,11] investigated properties of functions in this class. He termed the function f to be n-convex, if where the divided differences are defined recursively by [ He stated many basic properties of such functions. He proved among others, However the function x → x n + (where x + = max{x, 0}) is n-convex on R but it is not n-times differentiable at 0. He also showed [10, p. 18] that the composition of an n-convex function with an affine function remains n-convex on the suitable interval, whenever n is an odd number. Another result of Popoviciu tells us that if f is (n Then functions with f (4) of a constant sign (see the previous subsection) are necessarily either 3-convex, or 3-concave. One could ask whether the stopping inequality (3) holds for all 3-convex and 3-concave functions. The positive answer could be derived from the paper [1] by Bessenyei and Páles, who developed the smoothing technique for n-convex functions. In Sect. 2 as a tool we apply the spline approximation of n-convex functions. Below we recall a result due to Bojanić and Roulier [2].
where p is a polynomial of degree at most n, a i > 0, c i ∈ [a, b] (i = 1, . . . , m).

Inequalities between simple quadratures
Bessenyei and Páles proved that the inequality (of Hermite-Hadamard type) [14, Proposition 14]. Applying the affine substitution is 3-convex on [−1, 1] (see the remark in the previous subsection). Then (5) together with (4) lead immediately to the inequality which holds true for any 3-convex function f : [a, b] → R.

The aim of the paper
In Sect. 2 we improve the part of the inequality (8), which is This allows us to introduce in Sect. 3 a certain adaptive method and give its stopping criterion. In Sect. 4 we compare it to Rowland and Varol's adaptive method with stopping criterion (3). It is to be noted that our approach requires considerably fewer subdivisions than that of Rowland and Varol.

The inequality of Hermite-Hadamard type
The inequality we establish below improves inequality (9 Proof. The inequality on the left is nothing but (5). We prove the inequality on the right. In order to simplify the notation, let us skip the endpoints of the where f c (x) = (x − c) 3 + . First we claim that ϕ is an even function. Define also Because of the form of the operators C, I, S, the operator E is symmetric, that is, E[g](−x) = E[g](x) (we emphasize here that the operator E is applied to g as a function of x). Hence and ϕ is an even function on [−1, 1]. Consequently, it is enough to prove (11) for all c ∈ [0, 1]. For this, let us write ϕ(c) explicitly: Finally, if c ∈ [0, 1], then Multiplying both sides by 12 we get ψ(c) := 12ϕ(c) = 4 Obviously ψ(c) 0 as long as Then for 0 < c < Remark 3. Let us note that in the above proof the function 1 6 ϕ is in fact the Peano kernel of E. So, Bojanié and Roulier's result in the proof of Theorem 1 could be replaced by the Peano Kernel Theorem for 3-convex functions g ∈ C 4 [−1, 1] Then, by Bessenyei and Páles' smoothing technique (mentioned in the Introduction, cf. [1]), it is enough to extend the result to arbitrary 3convex functions g : [−1, 1] → R. This alternative approach is used in the recent paper [8] by Komisarski and the present author. Proof. Let f : [a, b] → R be a 3-convex function. We proceed as in Sect. 1.3. The function g : [−1, 1] → R given by (7) is 3-convex on [−1, 1]. We apply Theorem 2 to g. The substitution (6) gives the desired conclusion. Let Below we give the inequality which plays the key role in the next section. It is an immediate consequence of Corollary 5 and the above definition of Q[f ; a, b].

An adaptive method and its stopping criterion
Let the quadrature rule Q be defined by (12). In the way we described in the Introduction (cf. (1) If f is 3-concave, we then apply the above inequality to the 3-convex function −f and use the linearity of the operators Q n , I, S n : In both cases we obtain inequality (13) and the proof is complete.
Our next result is derived trivially from Theorem 7. The adaptive method we propose goes by the following steps. 1. Take a function f : [a, b] → R which is either 3-convex, or 3-convace. 2. Fix a tolerance ε > 0 and take n = 1.