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Composite Simpson error bound for a four-times continuously differentiable function

Statement

Let $a < b$ , let $n \geq 2$ be an even integer, and put $h = (b - a) / n$ . If $f \in C^{4} ([a, b])$ , set $M_{4} = \max_{x \in [a, b]} ∣ f^{(4)} (x) ∣$ . Then the value $Q_{S} (f; a, b, n)$ of Composite Simpson rule on an even uniform grid satisfies

$∣ \int_{a}^{b} f (x) d x - Q_{S} (f; a, b, n) ∣ \leq \frac{(b - a) h^{4}}{180} M_{4} .$

The integral is the Riemann integral. This is an exact-arithmetic quadrature bound; evaluation and summation roundoff are not included.

Facts & Assumptions

Given: Reals $a < b$ , an even integer $n \geq 2$ , $h = (b - a) / n$ , and $f \in C^{4} ([a, b])$ .

[F1]

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ : $C^{4} ([a, b])$ makes $f^{(4)}$ continuous on the closed interval.

[F2]

Heine-Borel by bisection: every closed bounded interval $[a, b]$ is compact: every closed bounded interval $[a, b]$ is compact.

[F3]

Extreme value theorem: a continuous real function on a nonempty compact subset of $R$ attains a greatest and a least value: a continuous real function on a nonempty compact subset of $R$ attains its maximum.

[F4]

Composite Simpson rule on an even uniform grid: the composite value is the sum, over $j < n / 2$ , of $\frac{h}{3} (f (x_{2 j}) + 4 f (x_{2 j + 1}) + f (x_{2 j + 2}))$ .

[F5]

Peano kernel identity for one Simpson panel: on a panel of length $L$ , the Simpson error is $\int K f^{(4)}$ , where $K \leq 0$ and $\int ∣ K ∣ = L^{5} / 2880$ .

[F6]

For $a < c < b$ : $f$ is integrable on $[a, b]$ if and only if it is integrable on $[a, c]$ and on $[c, b]$ , and then $\int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f$ ; with the oriented form for arbitrary $a, b, c$ : the integral over a finite partition is the sum of its integrals over the adjacent panels.

[F7]

Triangle inequality for finite sums: the absolute value of a finite sum is at most the sum of the absolute values.

[F8]

If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ : pointwise order between integrable functions on an increasing interval gives the same order between their integrals.

[F9]

Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ : scalar multiples of integrable functions are integrable and their integrals scale by that scalar.

Proof

1.1givenF1F2F3F4F5F8F9construct

By [F1], $f^{(4)}$ is continuous. The interval is compact by [F2], so [F3] gives $M_{4} = \max_{[a, b]} ∣ f^{(4)} ∣ < \infty$ . For each $j = 0, \dots, n / 2 - 1$ , put $u = x_{2 j}$ and $L = 2 h$ . The local rule from [F4] is $S (f; u, u + L)$ ; by [F5], its error is $E_{j} = \int_{u}^{u + L} K_{u, u + L} f^{(4)}$ and $\int_{u}^{u + L} ∣ K_{u, u + L} ∣ = (2 h)^{5} / 2880$ . Since $K_{u, u + L} \leq 0$ , its absolute value equals its negative, hence is integrable by [F5] and [F9]. The integrable functions satisfy $- M_{4} ∣ K_{u, u + L} ∣ \leq K_{u, u + L} f^{(4)} \leq M_{4} ∣ K_{u, u + L} ∣$ . Monotonicity [F8] and scalar linearity [F9] therefore give $∣ E_{j} ∣ \leq M_{4} \int_{u}^{u + L} ∣ K_{u, u + L} ∣ = M_{4} (2 h)^{5} / 2880$ .

2.1step 1.1F4F6algebra

Additivity [F6] gives $\int_{a}^{b} f$ as the sum of the local panel integrals. Summing the local Simpson expressions in [F4], each interior even-indexed value appears in two adjacent blocks and each odd-indexed value appears once; hence their sum is exactly $Q_{S} (f; a, b, n)$ . Therefore the global error is $\sum_{j = 0}^{n / 2 - 1} E_{j}$ .

3.1step 1.1step 2.1F7algebra∎

The finite-sum triangle inequality [F7] yields $∣ \sum_{j} E_{j} ∣ \leq (n / 2) M_{4} (2 h)^{5} / 2880 = n M_{4} h^{5} / 180$ . Using $n h = b - a$ gives the stated bound.

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Sources

Art B. Owen, Monte Carlo Theory, Methods and Examples, Chapter 7, §7.2 (standard reference, not scraped)
Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, §5.6 (standard reference, not scraped)