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How statement and proof provenance work

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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.
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The natural interpolant of x squared has a second-order endpoint error

Example

For $H > 0$ , interpolate $f (x) = x^{2}$ naturally at $0, H, 2 H$ . The error on the first interval is $H^{2} / 16$ at its midpoint, even though $f^{(4)} = 0$ . The clamped interpolant with exact endpoint slopes is $f$ itself. This displays why the fourth-order bound for the clamped method requires compatible boundary data.

Facts & Assumptions

Given: $H > 0$ , the data $(0, 0)$ , $(H, H^{2})$ , $(2 H, 4 H^{2})$ for $x^{2}$ , and the natural endpoint conditions.

[F1]

Natural cubic interpolation at distinct knots gives the natural second-derivative system and the cubic piece formula.

[F2]

Clamped cubic spline interpolation exists uniquely: for strictly increasing knots and prescribed endpoint slopes there is a unique clamped cubic interpolant.

[F3]

Boundary-compatible clamped cubic interpolation is fourth order in the maximum norm gives the fourth-order bound when the endpoint slopes equal those of the target function.

[A1]

The knots are $0, H, 2 H$ and the target function is $f (x) = x^{2}$ .

Verification

1.1A1F1algebra

The interval lengths are both $H$ , and the two secant slopes are $H$ and $3 H$ . The natural endpoint second derivatives are zero. The single interior equation from [F1] is $4 H M_{1} = 6 (3 H - H) = 12 H$ , hence $M_{1} = 3$ .

2.1A1F1step 1.1algebra

Substituting $M_{0} = 0$ , $M_{1} = 3$ , $y_{0} = 0$ , and $y_{1} = H^{2}$ into the first-interval formula in [F1] gives $s (x) = x^{3} / (2 H) + H x / 2$ . Thus $s (H / 2) = 5 H^{2} / 16$ whereas $f (H / 2) = H^{2} / 4 = 4 H^{2} / 16$ , so the absolute error is $H^{2} / 16 > 0$ .

3.1A1F2F3step 2.1algebra∎

Since $f^{(4)} = 0$ , a general natural-boundary estimate of the form $C H^{4} ∥ f^{(4)} ∥_{\infty}$ would force zero error, contradicting step 2.1. The polynomial $f$ has the prescribed clamped slopes $f^{'} (0) = 0$ and $f^{'} (2 H) = 4 H$ , so [F2] makes it the unique clamped interpolant; [F3] also gives zero error.

Depends on

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Sources

Carl de Boor, A Practical Guide to Splines, revised edition, Chapter IV, pp. 43–45 (standard reference, not scraped)