What is that fastest algorithm that can calculate a lot of digits of a decimal root? For example: 10,000 digits of the 3.56th root of 60.1?

  • $\begingroup$ I don't know the algorithm... but this package might help. $\endgroup$ – WhatsUp Aug 2 '15 at 14:07
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    $\begingroup$ What did you try? $\endgroup$ – Sagnik Aug 2 '15 at 14:08
  • $\begingroup$ Another optimized package for arbitrary precision. $\endgroup$ – rcgldr Oct 3 '15 at 4:15

An iterative method may be as follows:


where $n$ is the root desired, $x_1$ is a guess of the root of $x$ and $x_2$ is a better guess.

The iterative method is given in the book The Art of Programming Embedded Systems by Jack G. Ganssle.

  • $\begingroup$ Perhaps "The normal procedure" could be fixed. ​ $\endgroup$ – user12859 Aug 2 '15 at 17:13
  • $\begingroup$ Sorry. Didn't notice at all. $\endgroup$ – Sagnik Aug 2 '15 at 18:01
  • $\begingroup$ Uh, I just meant replacing $ylnx$ with $(\ln(x))/y$. $\;$ $\endgroup$ – user12859 Aug 2 '15 at 18:18
  • $\begingroup$ Wouldn't that method have a problem with $n = 1 \pm \epsilon_{mach}$? $\endgroup$ – Francesco Gramano Aug 14 '15 at 23:55
  • $\begingroup$ That is Newton's method. $\endgroup$ – vonbrand Sep 1 '15 at 20:13

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