PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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 quotients are defined and interpolate at support nodes

Statement

For the data and nonzero weights in Barycentric rational quotient with support values and reduced poles, $Q$ is not the zero polynomial, the rational function $P / Q$ is well-defined, and $r (z_{j}) = f_{j}$ for each support node $z_{j}$ .

Facts & Assumptions

Given: Distinct complex nodes $z_{1}, \dots, z_{k}$ , complex data $f_{j}$ , and nonzero complex weights $w_{j}$ .

[F1]

Barycentric rational quotient with support values and reduced poles: $P (z) = \sum_{j} w_{j} f_{j} \prod_{ℓ \neq j} (z - z_{ℓ})$ and $Q (z) = \sum_{j} w_{j} \prod_{ℓ \neq j} (z - z_{ℓ})$ .

[F2]

A barycentric denominator polynomial with nonzero weights is nonzero: for distinct nodes and nonzero weights, $Q (z_{j}) = w_{j} \prod_{ℓ \neq j} (z_{j} - z_{ℓ}) \neq 0$ at each support node, so $Q$ is not the zero polynomial.

Proof

1.1givenF1F2

By [F2], $Q$ is not the zero polynomial. The definition [F1] therefore forms a rational function $P / Q$ for these data.

2.1F1F2step 1.1algebra∎

At $z_{j}$ , all summands of $P$ and $Q$ except the $j$ th vanish. Thus $P (z_{j}) = w_{j} f_{j} \prod_{ℓ \neq j} (z_{j} - z_{ℓ}) = f_{j} Q (z_{j})$ , and [F2] gives $Q (z_{j}) \neq 0$ . Hence $r (z_{j}) = f_{j}$ . Canceling a common polynomial factor cannot introduce a zero into the reduced denominator at $z_{j}$ , since the original $Q (z_{j})$ is nonzero. This proves the support convention.

Depends on

Used by

Dependency tree · one level

2 results 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)