Suggest an iterative method to approximate $\sqrt[5]{a}$. What is it's rate of convergence?
So obviously Newton's method would be a good choice.
If we define $f(x) = x^5 - a$ then we want to find what is the solution for $f(x) = 0$ is.
$$x_{n+1} = x_n - \frac{x^5 -a}{5x_n^4}$$
We also know that the rate of convergence is squared.
Last thing is.. We want to validate the method does converges to the root.
We shall expect $|f'(r)| < 1$
$$f'(r) = f'(\sqrt[5] a) = 5\cdot a^{4/5}$$
but, $|f'(r)|$ can be greater than $1$ for a suitable $a$, and I was asked to suggest a bullet-proofed method (I guess) no matter what $a$ is.
What is your opinion regarding the choice of $a$ and the convergence of the method?
You could also use $f(x)=x^3-\frac{a}{x^2}$ as equation to solve. Miraculously it will give you even order 3 convergence.
For the fixed point condition, you have to compute the derivative and contractivity of the fixed point function $g(x)=x-\frac{f(x)}{f'(x)}$, not of the original function $f$ as $f(x)=0$ is no fixed-point equation. As one can readily find or compute, $$ g'(x)=\frac{f(x)f''(x)}{f'(x)^2} $$ which is zero at $x=r$ as $f(r)=0$.