The [[Principal branch|principal]] [[nth root|''n''th root]] <math>\sqrt[n]{A}</math> of a [[negative and positive numbers|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 number|complex]] solutions to this equation if <math>A > 0</math>, but only one is positive and real).
There is a very fast-[[Limit of a sequence|converging]] ''' ''n''th root algorithm''' for finding <math>\sqrt[n]{A}</math>:
#Make an initial guess <math>x_0</math>
#Set <math>x_{k+1} = \frac{1}{n} \left[{(n-1)x_k +\frac{A}{x_k^{n-1}}}\right]</math>. In practice we do <math>\Delta x_k = \frac{1}{n} \left[{\frac{A}{x_k^{n-1}}} - x_k\right]; x_{k+1} = x_{k} + \Delta x_k </math>.
#Repeat step 2 until the desired precision is reached, i.e. <math> | \Delta x_k | < \epsilon</math> .
A special case is the familiar [[Methods of computing square roots#Babylonian method|square-root algorithm]]. By setting ''n'' = 2, the ''iteration rule'' in step 2 becomes the square root iteration rule:
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''<sup>th</sup> 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.
