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Discrete derative?

  1. Apr 28, 2006 #1
    What types of discrete derative are there?

    Thanks in advance.
     
  2. jcsd
  3. Apr 29, 2006 #2

    arildno

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    Well, given a sequence [itex]a_{n}[/itex] the expression [itex]\frac{a_{n+1}-a_{n}}{(n+1)-n}=a_{n+1}-a_{n}[/itex] ought to count as one, agreed?
     
  4. Apr 29, 2006 #3
    Yes, I agree.
    However, I want to know if there are more sophisticated discrete derative which are commonly used.
    Maybe:
    [itex](0.25*a_{n+2}+0.75*a_{n+1}-0.75*a_{n-1}-0.25*a_{n-1})/2[/itex]
    Are there any other models of discrete deratives?
     
    Last edited: Apr 29, 2006
  5. Apr 29, 2006 #4

    arildno

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    Well, if you are interested in discretization schemes like leap-frog and that sort of thing, you should look into numerical maths/computational mechanics books.

    Alternatively, I'm sure our army of PF'ers will come along soon enough to supply you with more info.
     
  6. Apr 29, 2006 #5
    What is Differential Quadratures?
     
  7. Apr 29, 2006 #6

    arildno

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    It is a numerical method I haven't learnt about. :frown:
     
  8. Apr 29, 2006 #7
    Well, by comparing Taylor expansions you can prove that the sequence you posted is an approximation to the first derivative that is correct to at least second order, contrary to [itex](a_{n+1}-a_{n})/h[/itex], which is only first-order.

    With a little more insight you can convince yourself that any derivative of a smooth function can be approximated to any order, if only you have access to the values of the function at sufficiently many points.
     
  9. Apr 30, 2006 #8

    benorin

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    Do a search for "finite calculus" also called "umbral calculus" the forward/backward difference operators and rising/falling factorial powers are also akin to the topic, an excellect reference is the book "Concrete Mathematics" by Graham, Knuth, and Patashnik.
     
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