TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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.

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de Boor evaluation uses quadratic work in the degree

Statement

For a supplied nonempty knot span and degree $p$ , a successful call to the loop in de Boor coefficient evaluation on a specified knot span performs exactly $p (p + 1) / 2$ coefficient updates when $p \geq 1$ and none when $p = 0$ . A call that returns a range failure performs no more than this many updates. Each update uses at most a constant number of scalar arithmetic operations, so the arithmetic work is $O (p^{2})$ and the coefficient storage is $O (p)$ ; the caller's span search is excluded.

Facts & Assumptions

Given: The finite degree- $p$ loop and one already identified nonempty knot span.

[F1]

de Boor coefficient evaluation on a specified knot span runs, for each $r = 1, \dots, p$ , one update for each $j = p, p - 1, \dots, r$ and retains only $p + 1$ local coefficients.

[A1]

A supplied span index is used; finding the index is not charged to the coefficient evaluation.

Proof

1.1A1F1algebra

At loop level $r$ , there are $p - r + 1$ values of $j$ . Therefore a completed loop performs $\sum_{r = 1}^{p} (p - r + 1) = \sum_{s = 1}^{p} s = p (p + 1) / 2$ updates; for $p = 0$ the loop is empty. A range failure stops the loop early, so every call performs at most this many updates.

2.1A1F1step 1.1algebra∎

Each full update computes a fixed number of subtractions, one division, two multiplications and one addition, independent of $p$ ; an exact-zero numerator branch or a range failure can only reduce that count. Initializing and returning the local list takes at most a constant times $p + 1$ copies. Thus the scalar arithmetic count is bounded by a constant times $p^{2} + 1$ , while the list contains at most $p + 1$ entries. The excluded span search cannot change either bound.

Depends on

de Boor coefficient evaluation on a specified knot span

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Carl de Boor, A Practical Guide to Splines, revised edition, Chapter X, pp. 123–126 (standard reference, not scraped)