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Approximating derivative.

  1. Nov 14, 2013 #1
    Hi,
    1. The problem statement, all variables and given/known data
    I am asked to show that the formula:
    f'(x)≈Ʃ[i=0,n] Aif(xi)
    which is derived from differentiating the interpolation polynomial
    is similar to that derived from checking the order of accuracy formulae and similar to that derived through Taylor expansion.


    2. Relevant equations



    3. The attempt at a solution
    I have found the error in all these cases to be of the form:
    E(x)=C*f(k)*hn
    which means that the error becomes zero for any polynomial of degree ≤ k-1.
    Hence, for any such polynomial of degree ≤ k-1, the above formula could be rewritten thus:
    f'(x)=Ʃ[i=0,n] Aif(xi)
    Now, as the error in the Taylor expansion would also be zero for such polynomial, f(x) could be accurately replaced with:
    Ʃ[i=0,k-1] [f(i)(x0)]*(x-x0)i/i!
    Thus, the formula would hold and be similar.
    Is this attempt correct? Am I missing something? Would appreciate some advice.
     
  2. jcsd
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