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Does this method have a name? Function Approximation by Polynomial Sum

  1. May 7, 2012 #1
    I created a method for both approximating a function and extending a it's domain from a Natural to a Real Domain. Does this have a name already or any interesting application?
    Basically. Add polynomial of degree 0, 1, 2, 3, etc. Making at the same time the approximation function equal to f(0), f(1), f(2), f(3), etc.

    The approximation of function f is:
    App. of f http://latex.codecogs.com/examples/a0d1659d13fd1904ffe767edc6cab6e4.gif [Broken]
    n is level of the approximation, the bigger the n the bigger the approximation and the more the polynomia.
    http://latex.codecogs.com/examples/10d3e8addce0f1cea366f3eafcae03df.gif [Broken]=f(n)-App. of f at n of level (n-1)
    http://latex.codecogs.com/examples/00d8763b00aac3dd87168ec5039ec758.gif [Broken]
    http://latex.codecogs.com/examples/92cf5169e1dbb1a99d49de59187bc652.gif [Broken]
    etc.

    The idea is actually very simple! Just add a constant and then a line and then a parabola, etc to make it similar to the function.
    Also can anyone find a simpler way to find the coefficients. E? Maybe for f(x)=x! ?
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. May 7, 2012 #2
  4. May 7, 2012 #3

    AlephZero

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    The general idea behind this is a good one (and kudos if you thought it out for yourself) but it needs some more work to make it "useful".

    For example suppose you approximate the function ##f(x) = \sin \pi x##. That is 0 at every integer point x = 1, 2, 3 ... so your approximation will always to 0.

    If you plot out the successive approximations to functions like ##f(x) = 2^x## or ##f(x) = 2^{-x}## you will probably see something bad happening, as well.

    But a slightly different version of this idea IS used in digital signal processing, where you are only interested in a restricted set of functions that excludes the examples I just gave. This is much too big a topic to write a summary here, but you might like to explore it on the web.

    Similar ideas are also used for fitting curves through an arbitrary set of points, and in the numerical methods used in Finite Element structural mechanics, computational fluid dynamics, etc.
     
  5. May 8, 2012 #4
    I knew about it thanks. I think I thought of this when I heard about Taylor series and I was guessing what it was.


    AlephZero:

    It looks bad with functions like sin(πx) and x^n; it looks too symmetric; it's limit is almost always + or - infinity; it completely ignores the negative of the function.
    But some problems would maybe be partially solved if I could find a way to take the limit as n goes to infinity. Help would be appreciated if possible. This math is messy unfortunately.

    I'll have a look at those other methods. Maybe they'll be helpful, thank you.
     
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