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Nodal second-derivative perturbations bound the natural spline error

Statement

Let $x_{1} < \dots < x_{r}$ be natural interpolation knots, $r \geq 2$ , and put $h_{i} = x_{i + 1} - x_{i}$ and $H = \max_{i} h_{i}$ . Let $s$ be the natural cubic interpolant with data $y_{i}$ and nodal second derivatives $M_{i}$ , with $M_{1} = M_{r} = 0$ . Let ${\hat{M}}_{1} = {\hat{M}}_{r} = 0$ and let $\hat{s}$ be the piecewise cubic formed from the same values $y_{i}$ and the second derivatives ${\hat{M}}_{i}$ using the natural cubic piece formula of Natural cubic interpolation at distinct knots. On the knot hull $[x_{1}, x_{r}]$ , if $E = \max_{1 \leq i \leq r} ∣ {\hat{M}}_{i} - M_{i} ∣,$ then $∥ \hat{s} - s ∥_{L^{\infty} ([x_{1}, x_{r}])} \leq \frac{H^{2}}{8} E .$

Facts & Assumptions

Given: Strictly increasing knots $x_{1} < \dots < x_{r}$ , data $y_{i}$ , the natural interpolant $s$ , and arbitrary replacement nodal second derivatives ${\hat{M}}_{i}$ with the same zero endpoint values.

[F1]

Natural cubic interpolation at distinct knots: on $[x_{i}, x_{i + 1}]$ the spline is the cubic with endpoint values $y_{i}, y_{i + 1}$ and endpoint second derivatives $M_{i}, M_{i + 1}$ .

[A1]

${\hat{M}}_{1} = M_{1} = 0$ and ${\hat{M}}_{r} = M_{r} = 0$ .

Proof

1.1givenF1A1algebra

Fix $i$ and write $h = h_{i}$ , $t = (x - x_{i}) / h \in [0, 1]$ , and $Δ_{j} = {\hat{M}}_{j} - M_{j}$ . The cubic difference $q = \hat{s} - s$ vanishes at both endpoints. Its second derivative is affine with endpoint values $Δ_{i}, Δ_{i + 1}$ , so $q^{''} (x) = (1 - t) Δ_{i} + t Δ_{i + 1}$ . Integrating this polynomial twice and imposing the zero endpoint values gives $\hat{s} (x) - s (x) = \frac{h^{2}}{6} (Δ_{i} ((1 - t)^{3} - (1 - t)) + Δ_{i + 1} (t^{3} - t))$ .

2.1givenstep 1.1algebra∎

On $0 \leq t \leq 1$ , $(1 - t) - (1 - t)^{3} = t (1 - t) (2 - t) \geq 0$ and $t - t^{3} = t (1 - t) (1 + t) \geq 0$ . Their sum is $3 t (1 - t) \leq 3 / 4$ . Therefore the absolute value in step 1.1 is at most $h^{2} E (3 / 4) / 6 = h^{2} E / 8 \leq H^{2} E / 8$ . Taking the maximum over the finitely many knot intervals proves the claim on $[x_{1}, x_{r}]$ .

Depends on

Used by

A natural-spline tridiagonal residual bounds the spline value error Theorem

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

Carl de Boor, A Practical Guide to Splines, revised edition, Chapter IV, pp. 42–45 (standard reference, not scraped)