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Spline Derivation

  1. Jul 22, 2009 #1
    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 [tex] B_{-3}^{''}(k_0) = 1 [/tex]

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

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

    [tex] 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)} [/tex]

    [tex] 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)} [/tex]


    [tex] 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)} [/tex]


    [tex] = \frac{6h}{3h*2h*h} [/tex]

    [tex] = \frac{1}{h^2} [/tex]

    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
  2. jcsd
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