Fastest nth root algorithm to a lot of digits?

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?

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

An iterative method may be as follows:

$x_2=\frac{x_1(n-1)+\frac{x}{x_1^{n-1}}}{n}$

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.

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