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Homenth root by Newton's method

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# nth root by Newton’s method

The Newton’s method is very suitable for computing approximate values of higher $n^{{\mathrm{th}}}$ roots 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

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

Related:

NthRoot

Synonym:

cube root of 2

Type of Math Object:

Example

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

49M15*no label found*65H05

*no label found*26A06

*no label found*

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