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I Generating polynomials for a multistep method

  1. Jul 30, 2016 #1
    Hi, I'm struggling to understand how the generating polynomials work and are implemented in the difference equation for a general ODE y' = f(t,y)
    Difference Equation
    D%20h%20%5Csum_%7Bj%3D0%7D%5E%7Bk%7D%20b_%7Bj%7D%20f%28t_%7Bn+j%7D%2C%20y_%7Bn+j%7D%29.gif
    Generating polynomials
    %7D%20%5C%5C%20%5Csigma%20%28w%29%20%3D%20%5Csum%5E%7Bk%7D_%7Bj%3D0%7D%20b_%7Bj%7D%20w%5E%7Bj%7D.gif
    "Coefficients are normalized either by a_k = 1 or sigma(1) = 1
     
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  3. Jul 31, 2016 #2

    Simon Bridge

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    You can gain understanding by doing examples: lots and lots of examples.
    Do you have a specific question?
     
  4. Jul 31, 2016 #3
    Sorry if I wasn't clear enough, I don't understand the concept behind the generating polynomials. My notes state the examples of the three theta methods but I can't understand how they are obtained (e.g. implicit euler is sigma(w) = w )
     
  5. Jul 31, 2016 #4

    Simon Bridge

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    You notes just say there are these things called generating polynomials that can be used t help solve ODE's but does not tell you what they generate or how the method is motivated?

    Have you seen:
    https://en.wikipedia.org/wiki/Generating_function
     
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