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.

Barycentric rational quotient with support values and reduced poles

Definition

Let $k \geq 1$ , let $z_{1}, \dots, z_{k} \in C$ be distinct support nodes, let $f_{1}, \dots, f_{k} \in C$ be data, and let each weight $w_{j} \in C$ be nonzero. Define the cleared polynomials $P (z) = \sum_{j = 1}^{k} w_{j} f_{j} \prod_{ℓ \neq j} (z - z_{ℓ}), Q (z) = \sum_{j = 1}^{k} w_{j} \prod_{ℓ \neq j} (z - z_{ℓ}) .$ The barycentric rational quotient is the rational function $r = P / Q$ . Away from the support nodes this equals $r (z) = \frac{\sum_{j = 1}^{k} w_{j} f_{j} / (z - z_{j})}{\sum_{j = 1}^{k} w_{j} / (z - z_{j})} .$ At a support node its value is given by the cleared quotient $P (z_{j}) / Q (z_{j})$ . A pole means a zero of the denominator after all common polynomial factors of numerator and denominator have been canceled.

Depends on

A barycentric denominator polynomial with nonzero weights is nonzero

Used by

Dependency tree · one level

1 result within one dependency step 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

Yuji Nakatsukasa, Olivier Sète, and Lloyd N. Trefethen, The AAA algorithm for rational approximation, Section 2 (standard reference, not scraped)
Jean-Paul Berrut and Lloyd N. Trefethen, Barycentric Lagrange Interpolation, Section 9.2 (standard reference, not scraped)