# 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

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

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

For example, if one wants to compute $\sqrt[3]{2}$ and uses  $x_{0}=1$, already the fifth step gives

 $x_{5}\;=\;1.259921049894873$

which decimals.

Title nth root by Newton’s method NthRootByNewtonsMethod 2013-03-22 19:09:38 2013-03-22 19:09:38 pahio (2872) pahio (2872) 10 pahio (2872) Example msc 49M15 msc 65H05 msc 26A06 cube root of 2 NthRoot