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Peano remainder bound for the eight-panel rule

Statement

If $f \in C^{10} [- 1, 1]$ , then the rule $Q_{8}$ of Closed Newton–Cotes rule on eight equal panels satisfies $∣ \int_{- 1}^{1} f (x) d x - Q_{8} (f) ∣ \leq (\int_{- 1}^{1} ∣ K_{10} (t) ∣ d t) ∥ f^{(10)} ∥_{\infty},$ where $K_{10} (t) = \frac{(1 - t)^{10}}{10!} - \sum_{j = 0}^{8} w_{j} \frac{(x_{j} - t)_{+}^{9}}{9!}, x_{j} = - 1 + j / 4,$ and $u_{+} : = \max (u, 0)$ .

Facts & Assumptions

Given: A function $f \in C^{10} [- 1, 1]$ and the nine-node quadrature $Q_{8}$ .

[A1]

$f$ has continuous derivatives through order $10$ on $[- 1, 1]$ .

[F1]

Closed Newton–Cotes rule on eight equal panels: $Q_{8} (f) = \sum_{j = 0}^{8} w_{j} f (x_{j})$ on the nodes $x_{j} = - 1 + j / 4$ .

[F2]

Exact weights and degree-nine exactness of the eight-panel rule: $Q_{8}$ is exact for the zero polynomial and every nonzero polynomial of degree at most $9$ .

[F3]

Peano kernel of a finite quadrature functional: for $r = 10$ , $K_{10} (t) = L (g_{10, t})$ , where $g_{10, t} (x) = (x - t)_{+}^{9} / 9!$ and $L (u) = \int_{- 1}^{1} u - Q_{8} (u)$ .

[F4]

Peano representation and L1 error bound: exactness through degree $9$ and $f \in C^{10}$ imply $∣ L (f) ∣ \leq (\int_{- 1}^{1} ∣ K_{10} (t) ∣ d t) ∥ f^{(10)} ∥_{\infty}$ .

Proof

1.1givenF1F3algebra

By [F1] and [F3], $K_{10} (t) = \int_{- 1}^{1} (x - t)_{+}^{9} / 9! d x - \sum_{j = 0}^{8} w_{j} (x_{j} - t)_{+}^{9} / 9!$ . For $- 1 \leq t \leq 1$ , the integral is $\int_{t}^{1} (x - t)^{9} / 9! d x = (1 - t)^{10} / 10!$ , by the power antiderivative and endpoint evaluation. This gives the displayed kernel.

2.1givenA1F2F4step 1.1∎

The exactness condition of [F4] holds by [F2], and its smoothness condition is [A1]. Applying [F4] to this $Q_{8}$ and substituting the kernel from step 1.1 proves the bound.

Depends on

Used by

Negative weights amplify sample perturbations in the nine-point Newton–Cotes rule Example

Dependency tree · two levels

38 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

L. Ridgway Scott, Numerical Analysis, 2nd ed., §13.2 (standard reference, not scraped)