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Construct extrapolation table with optimal convergence

  1. Mar 23, 2013 #1
    Construct extrapolation table with optimal rates of convergence

    1. The problem statement, all variables and given/known data

    Let [itex]S[/itex] be a cubic spline interpolant that approximates a function [itex]f[/itex] on the given nodes [itex]x_{0},x_{1},...,x_{n}[/itex] with the boundary conditions: [itex]S''(x_{0})=0[/itex] and [itex]S'(x_{n})=f'(x_{n})[/itex]. Use [itex]S[/itex] to estimate [itex]f(0.1234567)[/itex] where [itex]f(x)=xe^{x}[/itex] and the nodes are [itex]n+1[/itex] uniformly distributed points on [itex][-1;1][/itex] for [itex]n=20, 40, 80, 160, 320[/itex]. Construct an extrapolation table with optimal rates of convergence using these estimates.

    2. Relevant equations



    3. The attempt at a solution

    I've already computed the approximation to [itex]f(0.1234567)[/itex] using various choices of nodes, the results are listed below:
    0.139678933961527 0.139678959983576 0.139679035306608 0.139679035488050 0.139679035532356

    I just don't know how to do the last part, that is, find the extrapolation table with optimal convergence. I'm supposed to use Richardson's extrapolation to generate such a table but what hinders me is the truncation error involved in the cubic spline approximation. From what I've learned, extrapolation is applied only when the truncation error has a predictable form, like [itex]M = {N_1}(h) + {K_1}h + {K_2}{h^2} + {K_3}{h^3} + ...[/itex].

    edit:
    Well after some searches (http://www.alglib.net/interpolation/spline3.php#header4) I've found out that the errors for clamped and natural cubic splines are [itex]O({h^4})[/itex] and [itex]O({h^2})[/itex] respectively, but have no idea how they are derived. My cubic spline is something kind of in-between, like half-clamped and half-natural.
     
    Last edited: Mar 23, 2013
  2. jcsd
  3. Mar 23, 2013 #2

    mfb

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    Did you plot (interpolation value)-(real value) as function of your node distances? A logarithmic scale can be useful.
     
  4. Mar 23, 2013 #3
    Yes. Here it is:
    JaVcHbW.png

    E is the absolute error and h is the distance between nodes (h=2/n for n=20,40,80,160,320)

    The slope (and maybe the rate of convergence?) is 1.9993.
     
  5. Mar 23, 2013 #4

    mfb

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    Exactly (well, the exact value is 2 and not 1.9993).
     
  6. Mar 23, 2013 #5
    Thanks. So is it safe to say that the error for my cubic spline is [itex]O({h^2})[/itex] because the rate of convergence is 2? If it is, then how the error form would look like, is it [itex]M = {N_1}(h) + {K_1}{h^2} + {K_2}{h^3} + {K_3}{h^4} + ...[/itex] or something else?
     
  7. Mar 24, 2013 #6

    mfb

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    I don't know what that formula expresses, but if N1 has no linear term, this looks reasonable.
     
  8. Mar 24, 2013 #7
  9. Mar 24, 2013 #8

    mfb

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    I cannot access the link, and I don't know about Richardson's extrapolation.
     
  10. Mar 24, 2013 #9
    Well, then hopefully you can access this: https://docs.google.com/viewer?a=v&...ymKG3t&sig=AHIEtbR1oV9rtANIhNQFrQOQFusrck0_rA . It's pretty much the same as the previous one, both serve as introductions to Richardson's extrapolation.

    What I need is the behavior of truncation error, something like Eq.(24.1) in the article, without which I cannot apply the formulas to construct an extrapolation table.
     
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