# nth root algorithm

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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 ${\sqrt[{n}]{A}}$ :

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 nth 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:

The nth 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 nth root algorithm.