So I'm given the scheme for computing $\sqrt[3]{a}$ $$x_{k+1}=px_k + \frac{qa}{x_k^2} + \frac{ra^2}{x_k^5}$$ and I have to find the p, q, and r so that this scheme is as fast as possible.
Any hints please?
I'm thinking let $g(x) = px_k + \frac{qa}{x_k^2} + \frac{ra^2}{x_k^5}$ Then $$g'(x) = p - \frac{2qa}{x^3} - \frac{ra^2}{x^6}=0$$ Do this for $g''(x)$,$g^{(3)}(x)$ also and try to solve for something.
Your problem corresponds to the Halley's cubic iterative scheme (1694) for solving x^3 = a starting at x=x0. This is an extension of Newton's quadratic method. Have a look to
http://www.sztaki.hu/~bozoki/oktatas/nemlinearis/SebahGourdon-Newton.pdf
This method gives the following iterative scheme
$$x_{new} = x_{old} - f[x_{old}] / f'[x_{old}] ( 1 + f[x_{old}] f''[x_{old}] / (2 (f'[x_{old}])^2) ) $$ Try with $f[x] = x^3 - a$. You will be able to find your terms.