DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

The complex-step derivative estimator and its component-error model

Definition

Let $U \subseteq C$ be open, $x \in U \cap R$ , and $h > 0$ with $x + i h \in U$ . For a holomorphic $f : U \to C$ that is real-valued on some real interval $I \subseteq U \cap R$ containing $x$ , the complex-step estimate is

$C_{h} f (x) : = \frac{Im f (x + i h)}{h} .$

For an inexact complex evaluation $\hat{f} (x + i h)$ , define its imaginary-component error by $δ_{i m} : = ∣ Im \hat{f} (x + i h) - Im f (x + i h) ∣$ . The inexact estimate is ${\hat{C}}_{h} f (x) : = Im \hat{f} (x + i h) / h$ ; its evaluation-error contribution is controlled by $δ_{i m} / h$ . No claim is made that arbitrary code representing a real function has a holomorphic extension.

Depends on

Used by

Dependency tree · two levels

9 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

Nicholas J. Higham, What Is the Complex Step Approximation? (standard reference, not scraped)
Mike Giles, Monte Carlo methods, Lecture 6: Complex Variable Trick (standard reference, not scraped)
R. Boudjemaa, M. G. Cox, A. B. Forbes, and P. M. Harris, Automatic Differentiation Techniques and their Application in Metrology, NPL Report CMSC 26/03 (standard reference, not scraped)