Talk:Spline interpolation

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1 Definition
2 Natural cubic spline oscillation
3 Interpolation using natural cubic spline
4 Graphs

[edit] Definition

i have some questions about this phrase:

Given n+1 distinct knots x_i such that

$x_0 < x_1 < ... < x_{n-1} < x_n, \,\!$

with n+1 knot values y_i we are trying to find a spline function of degree n

$S(x) := \left\{\begin{matrix} S_0(x) & x \in [x_0, x_1] \\ S_1(x) & x \in [x_1, x_2] \\ \vdots & \vdots \\ S_{k-1}(x) & x \in [x_{k-1}, x_k] \end{matrix}\right.$

with each S_i(x) a polynomial of degree n.

Is this not confusing, using n both for the degree of the polynomial, and for the number of points? --Anonymus, wiki nl

There's a k for that. Very well explained then ;) --217.136.81.22, 11:16, 13 Jun 2005 (UTC)

[edit] Natural cubic spline oscillation

Clamped and natural cubic splines yield the least oscillation about f than any other twice continuously differentiable function.

In the above sentence from the article, just what is f? Perhaps the article can be updated to clarify this. --Abelani, 19 November 2005

This is to confirm that someone has posted a clarification. --Abelani, 2:16, 27 November 2005 (UTC)

Amongst all twice continuously differentiable functions, clamped and natural cubic splines yield the least oscillation about the function f which is interpolated.

In the above sentence from the article, surely f itself is the function with the least oscillation about f. What is the restricted set of interpolation functions for which the statement is true and interesting? Harold f 03:27, 20 August 2006 (UTC)

[edit] Interpolation using natural cubic spline

In the formulas for interpolation using natural cubic spline, it seems that one could replace each $z i$ by $6 z i$ , enabling one to cancel 6s and obtain

$S_i(x) = \frac{z_{i+1} (x-x_i)^3 + z_i (x_{i+1}-x)^3}{h_i} + \left(\frac{y_{i+1}}{h_i} - h_i z_{i+1}\right)(x-x_i) + \left(\frac{y_{i}}{h_i} - h_i z_i\right) (x_{i+1}-x)$