# Determine stability of an algorithm?

This is related to a question I answered on Stack Overflow regarding calculating the square root of a number. I was thinking about it and realized that the formula is just the first in a family of algorithms to calculate the $$p$$th root of a number (ie solve $$y=x^p$$ for $$x$$).

$$x_n = \left( \frac{p}{2} \right) \frac{\displaystyle y + \prod_{j=n-p}^{n-1} x_j}{ \displaystyle\sum_{j=n-p}^{n-1} \prod_{\substack{k=n-p \\ k\ne j}}^{n-1} x_k}$$

We can show that as $$n\rightarrow\infty$$ the result converges to the correct answer, but I'm not sure how to go about finding other important characteristics like a rate of convergence and stability requirements. Any advice on resources?

• Is the product taken over the previous approximations of the root? Can $y$ be replaced by $x^p$ in the formula? – nicoguaro Mar 4 '20 at 11:49
• Sorry, $x_j$ is the $j$th approximation of $x$. $y$ is an input the function – user1543042 Mar 4 '20 at 12:08
• For the convergence, maybe you can divide two consecutive iterates and check how the norm changes. – nicoguaro Mar 5 '20 at 18:32

• I know that this is a terrible algorithm, it requires keeping $p$ previous estimates. – user1543042 Mar 4 '20 at 12:14