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Even-power expansion of trapezoid levels with a uniform remainder

Statement

Let $a < b$ , let $j \in N$ with $j \geq 1$ , and let $f \in C^{2 j + 2} ([a, b])$ . For every integer $N \geq 1$ , set $h = (b - a) / N$ , $I = \int_{a}^{b} f$ , and $c_{r} : = \frac{B_{2 r}}{(2 r)!} (f^{(2 r - 1)} (b) - f^{(2 r - 1)} (a)), 1 \leq r \leq j + 1,$ using the Bernoulli numbers of Bernoulli polynomials by a normalized coefficient recursion. Then $T_{N} (f) = I + \sum_{r = 1}^{j} c_{r} h^{2 r} + ε_{N}, ∣ ε_{N} ∣ \leq C_{j} h^{2 j + 2},$ where the level-independent finite constant $C_{j} : = ∣ c_{j + 1} ∣ + \frac{(b - a) M_{2 j + 2}}{(2 j + 2)!} ∥ f^{(2 j + 2)} ∥_{\infty}, M_{2 j + 2} : = \max_{0 \leq u \leq 1} ∣ B_{2 j + 2} (u) ∣,$ does not depend on $N$ . Thus the remainder bound is uniform over every set of dyadic levels.

Facts & Assumptions

Given: $a < b$ , an integer $j \geq 1$ , $f \in C^{2 j + 2} ([a, b])$ , and an integer trapezoid panel count $N \geq 1$ .

[F1]

Finite Euler–Maclaurin formula for the composite trapezoid: for every $p \geq 1$ , the composite trapezoid equals the integral, the first $p$ endpoint Bernoulli corrections, and a remainder bounded by $(b - a) M_{2 p} h^{2 p} ∥ f^{(2 p)} ∥_{\infty} / (2 p)!$ .

[F2]

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ : $C^{2 j + 2}$ supplies all continuous endpoint derivatives through order $2 j + 2$ .

[F3]

Composite trapezoidal rule: $T_{N} (f)$ uses the uniform mesh of width $h = (b - a) / N$ .

[F4]

For a nonempty subset of $R^{n}$ with $n \geq 1$ , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent: continuous real functions on the compact interval $[a, b]$ are bounded and attain finite extrema.

Proof

1.1givenF1F2F3

Apply [F1] with $p = j + 1$ . The formula has the displayed coefficients $c_{1}, \dots, c_{j + 1}$ and a remainder $ρ_{N}$ satisfying $∣ ρ_{N} ∣ \leq (b - a) M_{2 j + 2} h^{2 j + 2} ∥ f^{(2 j + 2)} ∥_{\infty} / (2 j + 2)!$ .

2.1givenF1F2F4step 1.1algebra∎

By [F2] and [F4], each derivative norm in $c_{j + 1}$ and $C_{j}$ is finite; the Bernoulli maximum exists by the Euler–Maclaurin theorem's construction. Put $ε_{N} : = c_{j + 1} h^{2 j + 2} + ρ_{N}$ . The triangle inequality gives $∣ ε_{N} ∣ \leq (∣ c_{j + 1} ∣ + (b - a) M_{2 j + 2} ∥ f^{(2 j + 2)} ∥_{\infty} / (2 j + 2)!) h^{2 j + 2} = C_{j} h^{2 j + 2}$ . Every term in $C_{j}$ depends only on $f, a, b, j$ , not on $N$ , so this is the claimed uniform remainder.

Depends on

Used by

Romberg cancels a proved finite even-power expansion Theorem

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Sources

L. Ridgway Scott, Numerical Analysis, 2nd ed., §13.3 (standard reference, not scraped)
NIST Digital Library of Mathematical Functions, §3.5(iii), Romberg Integration (standard reference, not scraped)