nth root algorithm

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

xⁿ = 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 x₀
2. Set $x_{k+1} = \frac{1}{n} \left[{(n-1)x_k +\frac{A}{x_k^{n-1}}}\right]$
3. Repeat step 2 until the desired precision is reached.

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 n^th 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 x₀
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 n^th root problem can be viewed as searching for a zero of the function

f(x) = xⁿ − 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 n^th root algorithm.

References

• Atkinson, Kendall E. (1989), An introduction to numerical analysis (2nd ed.), New York: Wiley, ISBN 0471624896 .