Improving Numerical Approximations of Limits: A Sample Exam Question

[JamesF]

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.

Homework Statement

Suppose that L = \lim_{h \rightarrow 0} f(h)
and L -f(h) = c_6 h^6 + c_9 h^9 + \cdots

Find a combination of f(h) and f(h/2) that is a much better estimate of L

 

[gabbagabbahey]

Well, if L -f(h) = c_6 h^6 + c_9 h^9 + \ldots , then surely you can say that f(h) =L-( c_6 h^6 + c_9 h^9 + \ldots) right?...What does that make f(h/2)?...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.

 

[JamesF]

Well, if L -f(h) = c_6 h^6 + c_9 h^9 + \ldots , then surely you can say that f(h) =L-( c_6 h^6 + c_9 h^9 + \ldots) right?...What does that make f(h/2)?...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.

thanks for your reply. Let's see if I understand

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

what I'm trying to find is a_0, a_1 such that a_0 f(h) + a_1 f(\frac{h}{2}) \approx L. 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

 

[gabbagabbahey]

thanks for your reply. Let's see if I understand

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

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

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

I'm still somewhat confused as to what this problem relates to, i.e. what methods have we studied would this problem apply to

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