Simpson's Rule Revisited

In this article we give some refinements of Simpson's Rule in cases when it is not applicable in it's classical form i.e., when the target function is not four times differentiable on a given interval. Some sharp two-sided inequalities for an extended form of Simpson's Rule are also proven.


Introduction
We begin with some notions from Classical Analysis which will be frequently needed in the sequel.
A function h : I ⊂ R → R is said to be convex on an non-empty interval I if the inequality (1.1) h(px + qy) ≤ ph(x) + qh(y) holds for all x, y ∈ I and all non-negative p, q; p + q = 1. If the inequality (1.1) reverses, then h is said to be concave on I. [HLP] The well-known convexity/concavity criteria says that if h ∈ C (2) (I) and h ′′ (x) ≷ 0, x ∈ I, then the function h is convex/concave on I. [HLP] Let h : I ⊂ R → R be a convex function on an interval I and a, b ∈ I with a < b. Then This double inequality is known in the literature as the Hermite-Hadamard (HH) integral inequality for convex functions. See, for example, [NP] and references therein. There is a number of refinements and possible generalizations of HH inequality. Some recent trends can be found in [WQ] and [SE].
If h is a concave function then both inequalities in (1.2) hold in the reversed direction.
Closely connected to the HH inequality is the well-known Simpson's Rule which is of great importance in numerical integration. It says that Lemma 1.3. [U] For an integrable function g, we have File: revis0.tex, printed: 2020-11-30, 2.05 Now, by taking x 1 = a, x 2 = (a+b)/2, x 3 = b, it follows that h = (b−a)/2 and therefore we obtain another form of Simpson's Rule: Note that the equation (1.4) explicitly supposes that g ∈ C (4) (I).
An interesting problem arises if g / ∈ C (4) (I), i.e., if g is not a four times continuously differentiable function on I. How to approximate the expression in this case?
In order to give an answer to this question, we shall consider the class of functions g ∈ C (n) (E) which are continuously differentiable up to n-th order on an interval E := [a, b] ⊂ I. Since g (n) (·) is a continuous function on a closed interval, there exist numbers m n = m n (a, b; g) := min t∈E g (n) (t) and M n = M n (a, b; g) := max t∈E g (n) (t). These numbers will play an important role in further approximations.
Our task in this article is to demonstrate a method which improves Simpson's Rule in some characteristic situations.
For example, let g(·) be a twice differentiable function on E. Some preliminary bounds for T g (a, b) in this case can be obtained by utilizing Hermite-Hadamard inequality (1.2) in a natural way.
Namely, for a given g ∈ C (2) (E) define an auxiliary function h by h(t) := g(t) − m 2 t 2 /2. Since h ′′ (t) = g ′′ (t)−m 2 ≥ 0, we see that h is a convex function on E. Therefore, applying Hermite-Hadamard inequality, we get that is, On the other hand, taking the auxiliary function h to be h(t) = M 2 t 2 /2 − g(t), we see that it is also convex on E.
Applying Hermite-Hadamard inequality again, we get Apart from the fact that those inequalities improves HH inequality in the cases if g is convex (m 2 ≥ 0) or concave (M 2 ≤ 0) function on E, they also lead to an estimation of T g (a, b), as follows.
Inequalities (1.5) and (1.6) give Hence, Also, adjusting the left-hand sides of (1.5) and (1.6), we get and we finally obtain that Another and more efficient method to approximate T g (a, b) is to use its integral representations in the cases when g ∈ C (1) (E), g ∈ C (2) (E) or g ∈ C (3) (E). In this way we obtain the following estimations: Remark 1.8. A challenging task and an open problem is to improve the constants 5/72 and 1/162, if possible. We shall prove that 1/1152 is the best possible constant in part 3.
In the sequel we sharply refine Simpson's Rule by assuming that f ′′ (·) is a convex function on E. Then, ( Theorem 2.8, below).
Finally, applying the method described above, we shall give tight bounds for an improved form of Simpson's Rule of the fourth order.

Results and Proofs
We begin with refinements of Simpson's Rule in non-standard cases.
For this cause, the following integral representation of T φ (a, b) in the case φ ∈ C (1) (E) is of crucial value.
Lemma 2.1. The identity Proof. In the well-known formula Our first contribution is the following where m 1 := min t∈E φ ′ (t) and M 1 := max t∈E φ ′ (t).
Proof. In this case, the integral representation of T φ (a, b) has the following form.
Lemma 2.4. The identity holds for any φ ∈ C (2) (E), where u and v are defined as in Lemma 2.1.
Proof. Indeed, applying partial integration on the assertion from Lemma 2.1, we get Hence, Analogously, and we get the desired result.
A challenging task is to determine the best possible constant A such that the relation holds for any φ ∈ C (2) (E).
Note that the function φ(·), defined on E = [−x, x] by Another important result concerns the functions which are only 3-times differentiable on E.
The constant C = 1/1152 is best possible.
Proof. For this case we need a new integral representation of T φ (a, b).
Indeed, by partial integration we get and, by Lemma 2.4, the proof follows. Therefore, To prove that the constant C = 1/1152 is best possible, we consider the function d(·) defined as: x ≥ 1. It is easy to confirm that this function is 3-times continuously differentiable on the real line.
Applying the form of Simpson's Rule for x ∈ [−a, a], a > 1, we obtain Since, x ≥ 1, we see that m 3 = −1, M 3 = 1. We shall give in the sequel precise estimation of an extended form of Simpson's rule under a smoothness condition posed on the target function.
For example, supposing that φ ′′ is convex on E, we obtain a clarification of the formula (1.4).
Theorem 2.8. For a φ ∈ C (4) (E), let φ ′′ (·) be convex on E. Then Proof. The left-hand side inequality follows from the convexity of φ ′′ and (1.4). For the second inequality we need an interesting assertion from Convexity Theory [S].

Now, Lemma 2.4 gives
Since u, v ∈ [a, b] and u + v = a + b, applying Lemma 2.9 to both integrals separately, we obtain Applying the method which was demonstrated in Introduction, we obtain an improved form of Simpson's Rule.

Conclusion
The results of this paper are of purely theoretical nature. Namely, we considered the cases when the classical Simpson's Rule is not applicable, although they are rare in practice. An open problem of determining best possible constants in Theorems 2.2 and 2.3 and our solution in Theorem 2.5 are of the same kind. Comparison of the classical form T φ (a, b) given in (1.4) and new form T ′ φ (a, b) from Theorem 2.10 clearly shows that the later is much more precise.