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Numerical analysis confusion

  1. Nov 9, 2008 #1
    I'm having trouble with a sample exam question. I don't really understand the question, don't know what section of the book it relates to, and don't have any idea on how to solve it. I might be in trouble :)

    Can anyone provide any suggestions or guidance on how I might go about solving this problem? Again, I'm not even really sure what's being asked.


    1. The problem statement, all variables and given/known data
    Suppose that [tex] L = \lim_{h \rightarrow 0} f(h)[/tex]
    and [tex] L -f(h) = c_6 h^6 + c_9 h^9 + \cdots [/tex]

    Find a combination of f(h) and f(h/2) that is a much better estimate of L
     
  2. jcsd
  3. Nov 9, 2008 #2

    gabbagabbahey

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    Well, if [itex] L -f(h) = c_6 h^6 + c_9 h^9 + \ldots[/itex] , then surely you can say that [itex]f(h) =L-( c_6 h^6 + c_9 h^9 + \ldots)[/itex] right?....What does that make [itex]f(h/2)[/itex]?....Basically you want to use this to find some linear combination of f(h) and f(h/2) that is closer to L than f(h) is.
     
  4. Nov 9, 2008 #3
    thanks for your reply. Let's see if I understand

    if [tex] f(h) = L - c_6 h^6 - c_9 h^9 - \cdots [/tex]
    and [tex] f(\frac{h}{2}) = L - \frac{c_6 h^6}{64} - \frac{c_9 h^9}{512}[/tex]

    what I'm trying to find is [tex] a_0, a_1 [/tex] such that [tex] a_0 f(h) + a_1 f(\frac{h}{2}) \approx L [/tex]. Is that right?

    I'm still somewhat confused as to what this problem relates to, ie what methods have we studied would this problem apply to
     
  5. Nov 9, 2008 #4

    gabbagabbahey

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    You're not even looking for something this restrictive, you just want [itex] a_0 f(h) + a_1 f(\frac{h}{2})[/itex]
    to be closer to L than f(h) was....so as long as [itex] a_0 f(h) + a_1 f(\frac{h}{2})-L< c_6 h^6 +c_9 h^9 + \cdots [/itex], then it is mission accomplished. ...what happens if you take [itex]a_0=1[/itex] and [itex]a_1=-2^6[/itex]?

    I'm not sure what methods you've studied :tongue:, but I think this is basically the beginning of an algorithm to numerically approximate the limit L.
     
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