How to compute the "factorial root"?

[David Wilson]

----- Original Message ----- 
From: "Alonso Del Arte" <alonso.delarte at gmail.com>
To: <seqfan at ext.jussieu.fr>
Sent: Tuesday, June 03, 2008 7:27 PM
Subject: How to compute the "factorial root"?

> So is there a formula [for the inverse of the gamma function], or will I
> just have to settle for a program that merely automates my trial and 
> error?
>
> Alonso del Arte

No, there is no formula for inverse gamma. Your best bets are probably 
bisection or the secant method, depending on how much work you want to put 
into this. If you have a mathematical language on hand, you could probably 
use a numerical root finder on Gamma(x)-y = 0.

But even if there were, it would not necessarily help you evaluate inverse 
gamma any more easily or accurately. For example, there are formulae for the 
roots of cubics and quartics, but they involve so many roots and divisions 
that it's computationally faster and more accurate to use Newton's method to 
compute them. And in some cases, a formula introduces numerical 
instabilities that are avoided by "trial and error" methods.

Even numerical evaluation of functions as simple as 1/x and sqrt(x) 
ultimately involve "trial and error" methods of varying sophistication. A 
formula just hides the work involved.