AlgorithmAlgorithm: AI-adaptedProof: Not applicablePipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Rational natural-logarithm enclosure search

Definition

Input. Rational numbers $x > 0$ and $τ > 0$ . Output. A positive integer $N$ and a rational interval $[L, U]$ .

Compute $y = (x - 1) / (x + 1)$ and initialize $N = 1$ . At each iteration compute, by exact rational arithmetic, $S_{N} = 2 \sum_{j = 0}^{N - 1} \frac{y^{2 j + 1}}{2 j + 1}, T_{N} = \frac{2 ∣ y ∣^{2 N + 1}}{(2 N + 1) (1 - y^{2})} .$ If $T_{N} \leq τ$ , return $N$ and $[L, U] = [S_{N} - T_{N}, S_{N} + T_{N}]$ . Otherwise replace $N$ by $N + 1$ and repeat. Each comparison is an exact rational comparison. Arithmetic and integer bit costs are charged separately; no running-time bound is part of this definition.

Used by

Variance-aware quantum Monte Carlo Algorithm
A terminating rational enclosure for the natural logarithm Lemma

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

J. Lebl, Basic Analysis I, Section 5.4.1 (analytic logarithm interface) (standard reference, not scraped)