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Conditional asymptotic validity of the Simpson /15 indicator

Statement

Let $a < b$ , $f \in C^{4} ([a, b])$ , $n_{k} = 2^{k}$ for $k \geq 1$ , $h_{k} = (b - a) / n_{k}$ , $Q_{k} = Q_{S} (f; a, b, n_{k})$ , and $c = - (1 / 180) \int_{a}^{b} f^{(4)} (x) d x$ . Let $I = \int_{a}^{b} f$ . Then the refinement indicator $η_{k} = \frac{∣ Q_{k + 1} - Q_{k} ∣}{15}$ satisfies $η_{k} = ∣ c ∣ h_{k}^{4} / 16 + o (h_{k}^{4})$ . If $c \neq 0$ , then $η_{k} / ∣ I - Q_{k + 1} ∣ \to 1$ . The Richardson-corrected value $R_{k + 1} = Q_{k + 1} + (Q_{k + 1} - Q_{k}) / 15$ satisfies $I - R_{k + 1} = o (h_{k}^{4})$ . When $c = 0$ , these statements give no relative error estimate for $η_{k}$ ; at no finite level does $η_{k}$ alone certify the true error.

Facts & Assumptions

Given: A nondegenerate interval, a $C^{4}$ integrand, nested composite Simpson grids with mesh widths $h_{k + 1} = h_{k} / 2$ , and exact arithmetic.

[F1]

Leading composite Simpson error coefficient: $I - Q_{k} = c h_{k}^{4} + o (h_{k}^{4})$ with the stated coefficient $c$ .

[F2]

Embedded Simpson refinement indicator: the magnitude of the embedded indicator is $∣ Q_{k + 1} - Q_{k} ∣ / 15$ ; a Richardson correction adds the signed quantity $(Q_{k + 1} - Q_{k}) / 15$ to the refined value.

[F3]

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ : $C^{4}$ specifies continuous derivatives through order four on the closed interval.

[F4]

The reverse triangle inequality: for reals $x, y$ , $∣ ∣ x ∣ - ∣ y ∣ ∣ \leq ∣ x - y ∣$ .

Proof

1.1givenF1F3algebra

By [F3], the regularity hypothesis is $f \in C^{4} ([a, b])$ . By [F1], write $E_{k} : = I - Q_{k} = c h_{k}^{4} + r_{k}$ where $r_{k} = o (h_{k}^{4})$ . Since $h_{k + 1} = h_{k} / 2$ , $E_{k + 1} = c h_{k}^{4} / 16 + o (h_{k}^{4})$ .

2.1givenF2F4step 1.1algebra

The signed difference $Q_{k + 1} - Q_{k} = E_{k} - E_{k + 1}$ . Therefore [F2] and step 1.1 give $(Q_{k + 1} - Q_{k}) / 15 = c h_{k}^{4} / 16 + s_{k}$ for some $s_{k} = o (h_{k}^{4})$ . By [F4], the difference between $η_{k} = ∣ (Q_{k + 1} - Q_{k}) / 15 ∣$ and $∣ c ∣ h_{k}^{4} / 16$ is at most $∣ s_{k} ∣ = o (h_{k}^{4})$ , so $η_{k} = ∣ c ∣ h_{k}^{4} / 16 + o (h_{k}^{4})$ .

3.1givenF4step 1.1step 2.1algebra∎

If $c \neq 0$ , step 1.1 and [F4] give $∣ E_{k + 1} ∣ / h_{k}^{4} \to ∣ c ∣ / 16 > 0$ , so $E_{k + 1} \neq 0$ for all sufficiently large $k$ ; dividing the step 2.1 relation by $∣ E_{k + 1} ∣$ then gives $η_{k} / ∣ E_{k + 1} ∣ \to 1$ . Also $I - R_{k + 1} = (16 E_{k + 1} - E_{k}) / 15$ , whose order- $h_{k}^{4}$ terms cancel by step 1.1, leaving $o (h_{k}^{4})$ . If $c = 0$ , both leading terms vanish, and these asymptotic formulas do not give a relative-error conclusion. The limit statements do not bound any finite-level error.

Depends on

Used by

Cancellation of the leading Simpson error coefficient Example

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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, §§5.6–5.7 (standard reference, not scraped)
L. Ridgway Scott, Numerical Analysis, 2nd ed., Chapter 13, §13.2 (standard reference, not scraped)