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For an exercise I am working with the function $f(x) = e^{6x} + 3(ln2)^2e^{2x} - (ln8)e^{4x} - (ln2)^3$. First I need to use Newton's method with $p_0 = 0$ to approximate a zero of f, which I did not have a problem doing.

The second part of the questions asks me to construct the sequence $\{\hat p_n\}$ using Aitken's method. I am kind of confused as to what exactly I need to do here. I know for each iteration of Aitken's I need to have 3 values: $p_n, p_{n+1}, p_{n+2}$. How do I obtain these? Do I just plug in numbers like $0, 1, 2...$ into the function to obtain the p values? For example would $p_4 = f(4)$ give me what I need? I don't think so because I've tried this and it appears to diverge. I've also been trying to solve this using Python and my program currently doesn't converge. Do I need to somehow integrate Aitken's method with Newton's method?

Any help would be appreciated.

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    Don't you get the $p_i$s from Newton's method? – 2017-02-01
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    Yes you're right! It totally went over my head when I was trying to do it – 2017-02-01

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As Raziman commented, I just needed to use the points generated by Newton's method.

P.S. If anyone one has any suggestions for writing a Python program that could automate this I would appreciate it. I have a program for Newton's method. I assume that I would have to save the points generated there and use them in Aitken's. But that's another issue haha