# 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

${x}_{k+1}={\displaystyle \frac{1}{3}}\left[2{x}_{k}+{\displaystyle \frac{\alpha}{{x}_{k}^{2}}}\right].$ | (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.

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 |