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 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 $x_{k+1}=x_{k}-{\frac {f(x_{k})}{f'(x_{k})}}$
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

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

and the iteration rule is

$x_{k+1}=x_{k}-{\frac {f(x_{k})}{f'(x_{k})}}$

$=x_{k}-{\frac {x_{k}^{n}-A}{nx_{k}^{n-1}}}$

$=x_{k}-{\frac {x_{k}}{n}}+{\frac {A}{nx_{k}^{n-1}}}$

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

leading to the general n^th root algorithm.

See also

• Recurrence relation

References

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