Sphaleron

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 $\mathrm{\Delta }{x}_k=\frac{1}{n}[\frac{A}{x_k^{n-1}}-x_k];x_{k+1}=x_k+\mathrm{\Delta }{x}_k$.
3. Repeat step 2 until the desired precision is reached, i.e. $\left|\mathrm{\Delta }{x}_k\right|<ϵ$ .

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:

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

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 $x_0$
2. Set $x_{k+1}=x_k-\frac{f(x_k)}{f^{\prime }(x_k)}$
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

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

So the derivative is

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

and the iteration rule is

$x_{k+1}=x_k-\frac{f(x_k)}{f^{\prime }(x_k)}$

$=x_k-\frac{x_k^n-A}{n{x}_k^{n-1}}$

$=x_k-\frac{x_k}{n}+\frac{A}{n{x}_k^{n-1}}$

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

leading to the general n^th root algorithm.

References

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