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.

Divided differences on distinct nodes

Definition

Let $q \geq 0$ , let $t_{0}, \dots, t_{q}$ be distinct real numbers, and let $f$ be real-valued at these nodes. Define the divided difference recursively by $f [t_{0}] : = f (t_{0})$ and, for $q \geq 1$ , $f [t_{0}, \dots, t_{q}] : = \frac{f [t_{1}, \dots, t_{q}] - f [t_{0}, \dots, t_{q - 1}]}{t_{q} - t_{0}} .$ All sublists in the recursion remain distinct. The denominator is nonzero because $t_{q} \neq t_{0}$ , so this recursion gives a finite real number from the node data.

Used by

Dependency tree · 0 levels

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

Sources

Lloyd N. Trefethen, Approximation Theory and Approximation Practice, Chapter 5 (standard reference, not scraped)
Jean-Paul Berrut and Lloyd N. Trefethen, Barycentric Lagrange Interpolation, Sections 2–3 (standard reference, not scraped)