# Spline Derivation

1. Jul 22, 2009

### Bazman

Hi there,

I have a big problem following the derivation in the following paper:

http://www3.iam.metu.edu.tr/iam/images/e/e1/Isabakithesis.pdf [/URL]

If you look on page 18 it says that $$B_{-3}^{''}(k_0) = 1$$

Howeverlooking at the form of $$B_{i-3}(x)$$ on page 17 I cannot see how they get to this result?

Assumig each knot span $$[k_i-k_{i-1}]$$ is the same length: I get

$$B_{i-3}(x) = \frac{(k_{i+1}-x)^3}{(k_{i+1}-k_{i-2})(k_{i+1}-k_ {i-1})(k_{i+1}-k_i)}$$

$$B_{i-3}^{'}(x) = \frac{-2(k_{i+1}-x)^2}{(k_{i+1}-k_{i-2})(k_{i+1}-k_ {i-1})(k_{i+1}-k_i)}$$

$$B_{i-3}^{''}(x) = \frac{6(k_{i+1}-x)}{(k_{i+1}-k_{i-2})(k_{i+1}-k_ {i-1})(k_{i+1}-k_i)}$$

$$= \frac{6h}{3h*2h*h}$$

$$= \frac{1}{h^2}$$

I'm sure the result in the paper is correct I've seen the same result in several other sources I'm just not sure how they got to this answer?

Last edited by a moderator: Apr 24, 2017
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