nth root by Newton’s method

The Newton’s method is very suitable for computing approximate values of higher $n^{\mathrm{th}}$ roots (http://planetmath.org/NthRoot) of positive numbers (and odd roots of negative numbers!).

The general recurrence formula

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

of the method for determining the zero of a function $f$, applied to

$$f(x):=x^n-\alpha$$

whose zero is $\sqrt[n]{\alpha }$, reads

$x_{k+1}={\displaystyle \frac{1}{n}}\left[(n-1)x_k+{\displaystyle \frac{\alpha }{x_k^{n-1}}}\right].$(1)

For a radicand $\alpha$, beginning from some initial value $x_0$ and using (1) repeatedly with successive values of$k$, one obtains after a few steps a sufficiently accurate value of $\sqrt[n]{\alpha }$ if $x_0$ was not very far from the searched root.

Especially for cube root $\sqrt[3]{\alpha }$, the formula (1) is

$x_{k+1}={\displaystyle \frac{1}{3}}\left[2x_k+{\displaystyle \frac{\alpha }{x_k^2}}\right].$(2)

For example, if one wants to compute $\sqrt[3]{2}$ and uses  $x_0=1$, already the fifth step gives

$$x_5=\mathrm{\hspace{0.33em}1.259921049894873}$$

which decimals.