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Second-derivative error bound for the composite trapezoidal rule

Statement

Let $a < b$ , let $N \geq 1$ be an integer, and put $h : = (b - a) / N$ . For $f \in C^{2} ([a, b])$ in the sense of Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ , the value $T_{N} (f)$ from Composite trapezoidal rule satisfies $∣ \int_{a}^{b} f (x) d x - T_{N} (f) ∣ \leq \frac{(b - a) h^{2}}{12} \sup_{x \in [a, b]} ∣ f^{''} (x) ∣ .$ The integral is the Riemann integral of The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ .

Facts & Assumptions

Given: Reals $a < b$ , an integer $N \geq 1$ , the mesh width $h = (b - a) / N$ , and $f \in C^{2} ([a, b])$ .

[F1]

Composite trapezoidal rule: $T_{N} (f)$ is $h$ times the sum of the endpoint values with half weights and the interior mesh values with unit weights.

[F2]

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ : $C^{2} ([a, b])$ means $f, f^{'}, f^{''}$ are continuous on $[a, b]$ , with derivatives at the endpoints understood in the one-sided sense.

[F3]

A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: every continuous real function on a closed bounded interval is bounded and Riemann integrable.

[F4]

Pointwise linear interpolation error from a second-derivative bound: if a function is $C^{2}$ on an open interval containing a panel $[u, v]$ , its endpoint affine interpolant $L$ satisfies $∣ f (x) - L (x) ∣ \leq M (x - u) (v - x) / 2$ , where $M : = \sup_{[u, v]} ∣ f^{''} ∣$ .

[F5]

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 implies the same order between their integrals.

[F6]

Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ : sums and real scalar multiples of integrable functions are integrable and the integral is linear.

[F7]

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 adjacent intervals is the sum of the two interval integrals.

[F8]

Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative: a continuous function with an integrable interior derivative has integral equal to the endpoint difference of an antiderivative.

Proof

1.1givenF2F3construct

Since $f^{''}$ is continuous on $[a, b]$ , [F3] shows it is bounded; set $M : = \sup_{[a, b]} ∣ f^{''} ∣ < \infty$ . The function $f$ is also Riemann integrable.

2.1givenF2F4step 1.1construct

By [F2], the endpoint derivatives $f^{'} (a), f^{''} (a), f^{'} (b), f^{''} (b)$ are the one-sided derivatives from within $[a, b]$ . Define $\tilde{f}$ on $(a - 1, b + 1)$ by $\tilde{f} (x) : = {\begin{cases} f (a) + f^{'} (a) (x - a) + \frac{1}{2} f^{''} (a) (x - a)^{2}, & x < a, \\ f (x), & a \leq x \leq b, \\ f (b) + f^{'} (b) (x - b) + \frac{1}{2} f^{''} (b) (x - b)^{2}, & x > b . \end{cases}$ At each join the polynomial piece matches the value and the first two one-sided derivatives of $f$ . Since $f, f^{'}, f^{''}$ are continuous on $[a, b]$ , these matches make $\tilde{f}$ $C^{2}$ on the open interval $(a - 1, b + 1)$ . It agrees with $f$ on $[a, b]$ and $\sup_{[a, b]} ∣ {\tilde{f}}^{''} ∣ = M$ . For $j = 0, \dots, N - 1$ , let $x_{j} = a + j h$ and let $L_{j}$ be the affine interpolant between $f (x_{j})$ and $f (x_{j + 1})$ . Put $q_{j} (x) : = \frac{M}{2} (x - x_{j}) (x_{j + 1} - x)$ on that panel. Applying [F4] to $\tilde{f}$ on each panel gives $∣ f (x) - L_{j} (x) ∣ \leq q_{j} (x)$ , and $q_{j} \geq 0$ .

3.1step 2.1construct

Let $P_{N}$ be the continuous piecewise-affine function equal to $L_{j}$ on each panel, and let $q_{N}$ be the continuous piecewise function equal to $q_{j}$ on each panel; continuity of $q_{N}$ at a knot holds because both adjacent formulas vanish there. Step 2.1 gives $- q_{N} \leq f - P_{N} \leq q_{N}$ throughout $[a, b]$ .

3.2F1F7F8step 2.1algebra

By [F8], $\int_{x_{j}}^{x_{j + 1}} L_{j} = \frac{h}{2} (f (x_{j}) + f (x_{j + 1}))$ and $\int_{x_{j}}^{x_{j + 1}} q_{j} = \frac{M h^{3}}{12}$ ; the second identity follows by integrating $\frac{M}{2} t (h - t)$ on $[0, h]$ . Repeated additivity [F7] therefore gives $\int_{a}^{b} P_{N} = T_{N} (f)$ , because each interior knot occurs twice with half weight, and $\int_{a}^{b} q_{N} = N M h^{3} / 12$ .

4.1step 3.1F3F5F6algebra

The functions $q_{N}$ and $f - P_{N}$ are continuous and hence integrable by [F3]. Applying [F5] to the two pointwise inequalities in step 3.1 and using [F6] yields $∣ \int_{a}^{b} f - \int_{a}^{b} P_{N} ∣ \leq \int_{a}^{b} q_{N}$ .

5.1step 4.1step 3.2algebra∎

Substituting the identities from step 3.2 into step 4.1 gives $∣ \int_{a}^{b} f - T_{N} (f) ∣ \leq N M h^{3} / 12 = (b - a) h^{2} M / 12$ , since $N h = b - a$ . This is the claimed bound.

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Sources

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, §5.6.1–§5.6.4 (standard reference, not scraped)
Endre Süli, Numerical Computation, course notes (standard reference, not scraped)