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Finite continued-fraction enclosure for erfcx

Statement

Fix real $x > 0$ and integer $N \geq 0$ . Set $I_{N} = [0, (N + 1) / (2 x)]$ , and for $k = N - 1, \dots, 0$ define

$I_{k} = [\frac{k + 1}{2 (x + \sup I_{k + 1})}, \frac{k + 1}{2 (x + \inf I_{k + 1})}] .$

Then $r_{k} \in I_{k}$ for every $0 \leq k \leq N$ , where $r_{k}$ is the moment ratio from Moment recurrence for the scaled complementary error function. Therefore

$erfcx (x) \in [\frac{1}{\sqrt{π} (x + \sup I_{0})}, \frac{1}{\sqrt{π} (x + \inf I_{0})}] .$

Each finite depth gives a valid enclosure; no convergence or shrinking-width claim as $N$ grows is made.

Facts & Assumptions

Given: $x > 0$ , integer $N \geq 0$ , and the intervals $I_{k}$ defined above.

[F1]

Moment recurrence for the scaled complementary error function: $r_{k} = (k + 1) / (2 (x + r_{k + 1}))$ and $0 < r_{N} < (N + 1) / (2 x)$ .

[F2]

Moment recurrence for the scaled complementary error function: $erfcx (x) = 1 / (\sqrt{π} (x + r_{0}))$ .

Proof

1.1givenF1algebrainduction

The upper bound in [F1] puts $r_{N}$ in $I_{N}$ . The map $f_{k} (t) = (k + 1) / (2 (x + t))$ is positive and decreasing for $t \geq 0$ . Hence, if $r_{k + 1} \in I_{k + 1}$ , then $f_{k} (\sup I_{k + 1}) \leq f_{k} (r_{k + 1}) \leq f_{k} (\inf I_{k + 1})$ ; by [F1] the middle term is $r_{k}$ . Backward induction gives $r_{k} \in I_{k}$ down to $k = 0$ .

2.1step 1.1F2algebra∎

The function $t \mapsto 1 / (\sqrt{π} (x + t))$ is positive and decreasing on $t \geq 0$ . Applying it to the interval $I_{0}$ and using [F2] gives the stated erfcx enclosure. Every step used a finite number of inclusions, so the enclosure is valid at this $N$ regardless of whether its width decreases with larger $N$ .

Depends on

Moment recurrence for the scaled complementary error function

Used by

Finite certified evaluation of the scaled complementary error function Algorithm
Soundness of finite erfcx enclosures Theorem

Dependency tree · two levels

12 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

DLMF, §7.9, continued fractions for erfc (standard reference, not scraped)
N. M. Temme, “Numerical aspects of special functions,” Acta Numerica 17 (2008), §§3 and 5 (standard reference, not scraped)