Nth root algorithm: Difference between revisions

Revision as of 12:10, 29 July 2020

Revision as of 12:11, 29 July 2020

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{{DISPLAYTITLE:''n''th root algorithm}}

The [[Principal branch|principal]] [[nth root|''n''th root]] <math>\sqrt[n]{aA}</math> of a [[negative and positive numbers|positive]] [[real number]] ''aA'', is the positive real solution of the equation <math>x^n = aA</math>. For a positive integer ''n'' there are ''n'' distinct [[complex number|complex]] solutions to this equation if <math>aA > 0</math>, but only one is positive and real.

==Using Newton's method==

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The ''n''^th root problem can be viewed as searching for a zero of the function

:<math>f(x) = x^n - aA</math>

So the derivative is

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:<math>x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}</math>

:<math> = x_k - \frac{x_k^n - aA}{n x_k^{n-1}}</math>

:<math> = x_k + \frac{1}{n} \left[-x_k +\frac{aA}{x_k^{n-1}}\right]</math>

:<math> = \frac{1}{n} \left[{(n-1)x_k +\frac{aA}{x_k^{n-1}}}\right]\,.</math>

==See also==