mandli
commented
9 years ago You are right that strictly speaking the term there should have $x^$ rather than $c$. Here the $c$ is meant to represent a value between $x^$ and $e_k$ which can minimize the error bound but is not critical.
That being said, I think your suggestion is more clear in the derivation so the change is worth making. Thanks for the feedback! I will make a PR and make sure you agree with the change.
In 5_root_finding_optimization.ipynb, Analysis of Fixed Point Iteration part, it writes:
Using a Taylor expansion we know
$$g(x^* + ek) = g(x^) + g'(x^_) e_k + \frac{g''(c) e_k^2}{2}$$
$$x^* + e{k+1} = g(x^) + g'(x^_) e_k + \frac{g''(c) e_k^2}{2}$$
Why it's $$\frac{g''(c) e_k^2}{2}$$, instead of $$\frac{g''(x^*) e_k^2}{2}$$? Where does the $$c$$ come from?