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

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A successful rounded de Boor evaluation has a degree-dependent forward bound

Statement

Use the operationwise model in Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add with unit roundoff $0 < u < 1$ . Suppose the floating-point mode of de Boor coefficient evaluation on a specified knot span completes successfully on stored coefficients $c_{k - p}, \dots, c_{k}$ and every exact intermediate is zero or normal. Put $η = \frac{(1 + u)^{2}}{1 - u} - 1, e = η + u (2 + η), W = 1 + e, E = e + u (2 + u) W .$ If $S (x) = \sum_{i} c_{i} B_{i, p} (x)$ is the exact B-spline value on the supplied span and $\hat{S} (x)$ is the returned value, then $∣ \hat{S} (x) - S (x) ∣ \leq ((1 + 2 E)^{p} - 1) \max_{k - p \leq i \leq k} ∣ c_{i} ∣ .$ For $p = 0$ the error is zero. The conclusion is conditional on success and does not certify that the intermediate range conditions will hold.

Facts & Assumptions

Given: A successful operationwise floating-point evaluation of the de Boor recurrence on a nonempty span, with $0 < u < 1$ .

[F1]

Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add states that every successful real addition, subtraction, multiplication, and division has relative error at most $u$ for a nonzero normal exact result, while exact zeros are returned exactly; other nonzero range cases return failure.

[F2]

de Boor coefficient evaluation on a specified knot span first computes the rounded numerator subtraction. If it is zero, the algorithm copies the left coefficient; otherwise it computes the denominator subtraction and division, complement subtraction, two multiplications, and addition in that order.

[F3]

de Boor evaluation equals the B-spline coefficient expansion identifies the exact recurrence output with $S (x)$ .

[A1]

Every operation in the floating-point run succeeds, as required in the Statement.

Proof

1.1A1F1F2algebra

At one blend let $α \in [0, 1]$ be the exact ratio. If its numerator is zero, the algorithm's exact-zero branch returns the left coefficient copy. Otherwise the two subtractions and division in [F2] give $\hat{α} = α (1 + δ_{1}) (1 + δ_{3}) / (1 + δ_{2})$ , with $∣ δ_{i} ∣ \leq u$ by [F1]. Hence $∣ \hat{α} - α ∣ \leq η$ . The complement subtraction gives $∣ \hat{β} - (1 - α) ∣ \leq η + u (2 + η) = e$ , because $∣ 1 - \hat{α} ∣ \leq 2 + η$ . Also $∣ \hat{α} ∣, ∣ \hat{β} ∣ \leq 1 + e = W$ .

1.2A1F1F2algebra

Write the two rounded products and final rounded sum as $\hat{y} = fl (fl (\hat{β} a) + fl (\hat{α} b))$ . Subtract the exact convex blend $(1 - α) a + α b$ . The coefficient differences contribute at most $e (∣ a ∣ + ∣ b ∣)$ ; the two product roundings contribute at most $u W (∣ a ∣ + ∣ b ∣)$ ; and the final addition contributes at most $u (1 + u) W (∣ a ∣ + ∣ b ∣)$ . Thus $∣ \hat{y} - ((1 - α) a + α b) ∣ \leq E (∣ a ∣ + ∣ b ∣)$ . If $M$ is the maximum magnitude in the preceding computed column, this is at most $2 E M$ . Since the exact blend is a convex combination, its magnitude is at most $M$ , so the next computed column has maximum at most $(1 + 2 E) M$ .

2.1step 1.2algebrainduction

Let $M_{r}$ be the maximum magnitude of the computed column after $r$ stages and let $ε_{r}$ be the maximum difference from the exact column. Step 1.2 gives $M_{r} \leq (1 + 2 E)^{r} M_{0}$ . Since the exact weights are nonnegative and sum to one, the previous-column discrepancy contributes at most $ε_{r - 1}$ , while the local rounding contributes at most $2 E M_{r - 1}$ . Therefore $ε_{r} \leq ε_{r - 1} + 2 E (1 + 2 E)^{r - 1} M_{0}$ . Starting with $ε_{0} = 0$ and summing this geometric series gives $ε_{p} \leq ((1 + 2 E)^{p} - 1) M_{0}$ .

3.1F3step 2.1algebra∎

By [F3], the final exact column is $S (x)$ ; the final computed column is $\hat{S} (x)$ , and $M_{0} = \max_{k - p \leq i \leq k} ∣ c_{i} ∣$ . Substitution in step 2.1 proves the bound.

Depends on

Used by

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Sources

Carl de Boor, A Practical Guide to Splines, revised edition, Chapter X, pp. 123–126 (standard reference, not scraped)
Stuart and Voss, Matrix Analysis and Algorithms, Warwick lecture notes, floating-point model (standard reference, not scraped)