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Piecewise cubic Hermite interpolant

Definition

Let $N \geq 1$ be an integer, let $x_{0} < \dots < x_{N}$ be real nodes, let $y_{0}, \dots, y_{N} \in R$ be values, and let $m_{0}, \dots, m_{N} \in R$ be prescribed slopes. For $0 \leq i < N$ put $h_{i} = x_{i + 1} - x_{i} > 0$ and $t = (x - x_{i}) / h_{i}$ on $[x_{i}, x_{i + 1}]$ . Define $H_{00} (t) = 2 t^{3} - 3 t^{2} + 1, H_{01} (t) = - 2 t^{3} + 3 t^{2}, H_{10} (t) = t^{3} - 2 t^{2} + t, H_{11} (t) = t^{3} - t^{2} .$ The cubic Hermite piece on $[x_{i}, x_{i + 1}]$ is $P_{i} (x) = y_{i} H_{00} (t) + y_{i + 1} H_{01} (t) + h_{i} m_{i} H_{10} (t) + h_{i} m_{i + 1} H_{11} (t) .$ The piecewise cubic Hermite interpolant is the function whose restriction to each such interval is $P_{i}$ ; adjacent pieces use their common prescribed value $y_{i}$ at a shared node. The slopes $m_{i}$ are input data, not values inferred from neighboring pieces.

Depends on

Integer powers $a^m$

Used by

Dependency tree · two levels

11 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 R. de Boor, A Practical Guide to Splines, revised edition, Chapter IV, pp. 39–40 (standard reference, not scraped)