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fr:Algorithme de calcul de la racine n-ième The principal n^th root <math>\sqrt[n]{A}</math> of a positive real number A, is the positive real solution of the equation

<math>x^n = A</math>

(for integer n there are n distinct complex solutions to this equation if <math>A > 0</math>, but only one is positive and real).

There is a very fast-converging n^th root algorithm for finding <math>\sqrt[n]{A}</math>:

1. Make an initial guess <math>x_0</math>
2. Set <math>x_{k+1} = \frac{1}{n} \left[{(n-1)x_k +\frac{A}{x_k^{n-1}}}\right]</math>
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:

<math>x_{k+1} = \frac{1}{2}\left(x_k + \frac{A}{x_k}\right)</math>

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 <math>f(x)</math> 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 <math>x_k^{n-1}</math> at each step, but can be efficiently implemented with a good exponentiation algorithm.

[edit] 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 <math>x_0</math>
2. Set <math>x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}</math>
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

<math>f(x) = x^n - A</math>

So the derivative is

<math>f^\prime(x) = n x^{n-1}</math>

and the iteration rule is

<math>x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}</math>

<math> = x_k - \frac{x_k^n - A}{n x_k^{n-1}}</math>

<math> = x_k - \frac{x_k}{n}+\frac{A}{n x_k^{n-1}}</math>

<math> = \frac{1}{n} \left[{(n-1)x_k +\frac{A}{x_k^{n-1}}}\right]</math>

leading to the general n^th root algorithm.