# nth root algorithm

The principal nth 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 and real).

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

1. Make an initial guess $x_0$
2. Set $x_{k+1} = \frac{1}{n} \left[{(n-1)x_k +\frac{A}{x_k^{n-1}}}\right]$. In practice we do $\Delta x_k = \frac{1}{n} \left[{\frac{A}{x_k^{n-1}}} - x_k\right]; x_{k+1} = x_{k} + \Delta x_k$.
3. Repeat step 2 until the desired precision is reached, i.e. $| \Delta x_k | < \epsilon$ .

A special case is the familiar square-root algorithm. By setting n = 2, the iteration rule in step 2 becomes the 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 nth root algorithm is somewhat less efficient since it requires the computation of $x_k^{n-1}$ at each step, but can be efficiently implemented with a good exponentiation algorithm.

## Derivation from Newton's method

Newton's method is a method for finding a zero of a function f(x). The general iteration scheme is:

1. Make an initial guess $x_0$
2. Set $x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}$
3. Repeat step 2 until the desired precision is reached.

The nth root problem can be viewed as searching for a zero of the function

$f(x) = x^n - A$

So the derivative is

$f^\prime(x) = n x^{n-1}$

and the iteration rule is

$x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}$
$= x_k - \frac{x_k^n - A}{n x_k^{n-1}}$
$= x_k - \frac{x_k}{n}+\frac{A}{n x_k^{n-1}}$
$= \frac{1}{n} \left[{(n-1)x_k +\frac{A}{x_k^{n-1}}}\right]$

leading to the general nth root algorithm.

## References

• Atkinson, Kendall E. (1989), An introduction to numerical analysis (2nd ed.), New York: Wiley, ISBN 0-471-62489-6.