Algorithm to compute the fifth root of a positive real number

Question

I'm trying to solve this problem:

Devise a Newton algorithm for computing the fifth root of any positive real number.

What I'm trying to do is to make an analogy with the algorithm to compute the square root of a positive real number. We now that if $x = \sqrt{\alpha}$ then $x^2 = \alpha$, and $x = \dfrac{\alpha}{x}$. Having this in mind, the Newton algorithm for computing the square root of $\alpha$ is

$$ x_{n+1} = \dfrac{1}{2}\left( x_n + \dfrac{\alpha}{x_n} \right) $$

so, I have tried to find a recursion formula beginnig at $x = \sqrt[5]{\alpha}$, but I can't find this formula yet.

Thanks for your advices!

Answer 1

The find the $n^{th}$ root of $a$, we want to solve $x^n = a$, or $x^n-a = 0$.

Let $f(x) = x^n-a$. Then $f'(x) =nx^{n-1} $.

Since Newton's iteration is $x \leftarrow x-\frac{f(x)}{f'(x)} $, this becomes $x \leftarrow x-\frac{x^n-a}{nx^{n-1}} =x-(\frac{x}{n}-\frac{a}{nx^{n-1}}) =\frac1{n}((n-1)x+\frac{a}{x^{n-1}}) $.

Answer 2

Try this weighted mean; $$x_{n+1}=\frac15\Bigl(4x_n+\frac{\alpha}{x_n^4}\Bigr).$$