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A finite double-exponential trapezoid has separate strip and tail errors

Statement

Use one of the modes of Finite truncated tanh–sinh, exp–sinh, and sinh–sinh trapezoid rules and let $Φ$ be its transformed real integrand. Suppose $Φ$ has an extension $F$ satisfying all hypotheses of An analytic-strip error bound for the infinite trapezoid sum for some $0 < a < d$ , with horizontal integrals $I_{\pm} = \int_{R} ∣ F (x \pm i a) ∣ d x$ . Also suppose $∣ Φ (t) ∣ \leq A e^{- B e^{κ ∣ t ∣}}$ for $∣ t ∣ \geq T$ , where $A, B, κ > 0$ , and $N_{\pm} h \geq T$ . If $I$ is the target improper integral and $T_{h, N_{-}, N_{+}}$ is the finite sum in the algorithm, then it is absolutely convergent as an improper integral and $∣ I - T_{h, N_{-}, N_{+}} ∣ \leq \frac{I_{+} + I_{-}}{e^{2 π a / h} - 1} + \frac{A}{κ B} (e^{- B e^{κ N_{-} h}} + e^{- B e^{κ N_{+} h}}) .$ The strip and tail hypotheses are separate assumptions; integrability alone implies neither estimate.

Facts & Assumptions

Given: A mode and transformed function $Φ$ from Finite truncated tanh–sinh, exp–sinh, and sinh–sinh trapezoid rules, its target $I$ , an analytic extension satisfying the strip hypotheses, and a double-exponential real-tail envelope with the stated cutoff conditions.

[F1]

Finite truncated tanh–sinh, exp–sinh, and sinh–sinh trapezoid rules: the output is $h \sum_{k = - N_{-}}^{N_{+}} Φ (k h)$ .

[F2]

The double-exponential maps and their change-of-variable identities: if $Φ$ is absolutely integrable as an improper Riemann integral, the target exists absolutely and equals $\int_{R} Φ (t) d t$ .

[F3]

An analytic-strip error bound for the infinite trapezoid sum: under the closed-substrip exponential bound and analyticity hypotheses, $∣ \int Φ - h \sum_{k \in Z} Φ (k h) ∣ \leq (I_{+} + I_{-}) / (e^{2 π a / h} - 1)$ and the infinite sum is absolutely convergent.

[F4]

A double-exponential envelope bounds the omitted trapezoid tails: the continuous and sampled omitted tails each have the stated one-sided envelope bound.

[F5]

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

Proof

1.1givenF2F4F5algebra

Put $T_{0} = \max (T, 0)$ . The envelope holds for $∣ t ∣ \geq T_{0}$ , and $N_{\pm} h \geq 0$ together with $N_{\pm} h \geq T$ gives $N_{\pm} h \geq T_{0}$ . Thus [F4] makes the two tails outside $[- N_{-} h, N_{+} h]$ absolutely integrable. Set $S = (N_{-} + N_{+} + 1) h > 0$ . The continuous function $∣ Φ ∣$ is Riemann integrable on $[- S, S]$ by [F5], and this interval contains both cutoffs, so $Φ$ is absolutely integrable on $R$ as an improper Riemann integral. Now [F2] identifies $I = \int_{R} Φ$ .

1.2F1F3given

By [F3], $h \sum_{k \in Z} Φ (k h)$ is absolutely convergent. Splitting its terms into $- N_{-} \leq k \leq N_{+}$ and the two omitted tails is therefore valid.

2.1step 1.1step 1.2algebra

Add and subtract the infinite trapezoid sum. The triangle inequality bounds the target-to-finite-sum error by the target-to-infinite-sum error plus the absolute omitted sample sum.

3.1F3F4step 1.1step 2.1∎

Apply [F3] to the first term in step 2.1 and the sampled-tail estimate [F4], with cutoff $T_{0}$ from step 1.1, to the second. Their sum is the displayed bound.

Depends on

Used by

The Gaussian integral on the line with a certified sinh-sinh truncation tail Example

Cited to discharge well-definedness by Finite truncated tanh–sinh, exp–sinh, and sinh–sinh trapezoid rules.

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Sources

Lloyd N. Trefethen and J. A. C. Weideman, The Exponentially Convergent Trapezoidal Rule, SIAM Review 56(3), 2014 (standard reference, not scraped)