math lessons - N-th root algorithm

The principal n-th root $\sqrt[n]{A}$ of a positive real number A, is the positive real solution of the equation

x n = A

(For integer n there are n distinct complex solutions to this equation if $A > 0$ , but only one is positive).

There is a very fast-converging n-th root algorithm for finding $\sqrt[n]{A}$ :

Make an initial guess $x 0$
Set $x_{k+1} = \frac{1}{n} \left[{(n-1)x_k +\frac{A}{x_k^{n-1}}}\right]$
Repeat step 2 until the desired precision is reached.

This is a generalization of the well-known square-root algorithm. By setting n = 2, the iteration rule in step 2 becomes the more familiar square root iteration rule:

$x_{k+1} = \frac{1}{2}\left(x_k + \frac{A}{x_k}\right)$

Several different derivations of this algorithm are possible. One derivation shows it is a special case of Newton's method (also called the Newton-Raphson method) for finding zeros of a function $f (x)$ beginning with an initial guess. Although Newton's method is iterative, meaning it approaches the solution through a series of increasingly-accurate guesses, it converges very quickly. The rate of convergence is quadratic, meaning roughly that the number of bits of accuracy doubles on each iteration (so improving a guess from 1 bit to 64 bits of precision requires only 6 iterations). For this reason, this algorithm is often used in computers as a very fast method to calculate square roots.

For large n, the n-th root algorithm is somewhat less efficient since it requires the computation of $x_k^n$ at each step, but can be efficiently implemented with a good exponentiation algorithm.