nth root by Newton’s method

The Newton’s method is very suitable for computing approximate values of higher nth roots (http://planetmath.org/NthRoot) of positive numbers (and odd roots of negative numbers!).

The general recurrence formula


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


whose zero is αn, reads

xk+1=1n[(n-1)xk+αxkn-1]. (1)

For a radicand α, beginning from some initial value x0 and using (1) repeatedly with successive values of k, one obtains after a few steps a sufficiently accurate value of αn if x0 was not very far from the searched root.

Especially for cube root α3, the formula (1) is

xk+1=13[2xk+αxk2]. (2)

For example, if one wants to compute 23 and uses  x0=1, already the fifth step gives

x5= 1.259921049894873

which decimals.

Title nth root by Newton’s method
Canonical name NthRootByNewtonsMethod
Date of creation 2013-03-22 19:09:38
Last modified on 2013-03-22 19:09:38
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Example
Classification msc 49M15
Classification msc 65H05
Classification msc 26A06
Synonym cube root of 2
Related topic NthRoot