Find the value $p,q,r$ that make the sequence converge with highest order as possible When using iteration $x_{k+1}=g(x_k),\displaystyle{g(x)=px+\frac{qa}{x^2}+\frac{ra^2}{x^5}}$ in order to find the value of $a^\frac{1}{3}$ 
What is value of $p,q,r$ that make the sequence converge with highest order as possible?
 A: Assume $a\neq 0$ because otherwise $g(x)=px$ which we cannot tweak much.
Let $x_\infty$ be the desired limit, so $x_\infty^3=a$.
The goal is to maximize the order $m$ in
$$\frac{1}{x_\infty}\,g\left(x_\infty (1+\varepsilon)\right) =
1 + \mathrm{O}(\varepsilon^m)$$
Substituting the given Ansatz for $g$ und expanding into power series of $\varepsilon$,
we get
$$\begin{aligned}
 \frac{1}{x_\infty}\,g\left(x_\infty (1+\varepsilon)\right) &=
 p\,(1+\varepsilon) + q\,(1+\varepsilon)^{-2} + r\,(1+\varepsilon)^{-5}
\\ &= p\,(1+\varepsilon) +
\\ &\quad q \left(1 - 2\varepsilon + 3\varepsilon^2 -4\varepsilon^3 +
 \mathrm{O}(\varepsilon^4)\right) +
\\ &\quad r \left(1 - 5\varepsilon + 15\varepsilon^2 -35\varepsilon^3 +
 \mathrm{O}(\varepsilon^4)\right)
\\ &= \underbrace{(p+q+r)}_1 +
 \underbrace{(p-2q-5r)}_0\,\varepsilon +
 \underbrace{(3q+15r)}_0\,\varepsilon^2 +
 (-4q-35r)\,\varepsilon^3 + \mathrm{O}(\varepsilon^4)
\end{aligned}$$
The values under the horizontal braces indicate what we are trying to achieve:
We have three parameters $p,q,r$ available for tweaking, therefore we try to
set the first three coefficients of the above series to the desired values
$1,0,0$. This results in a system of three linear equations in $p,q,r$
which turns out to have the unique solution
$$p=q=\frac{5}{9};\quad r=-\frac{1}{9}$$
(If more than one solution had been possible, we would have tried to zero
more coefficients.)
With those parameters our relative error estimate becomes
$$\frac{1}{x_\infty}\,g\left(x_\infty (1+\varepsilon)\right) =
(-4q-35r)\,\varepsilon^3 + \mathrm{O}(\varepsilon^4) =
\frac{5}{3}\varepsilon^3 + \mathrm{O}(\varepsilon^4)$$
As demonstrated, any other setting for $p,q,r$ would decrease the order of convergence.
Edit: Typesetting and minor wording corrections.
