RE: [Axiom-math] special functions

[Bill Page]

> On January 26, 2006 3:32 PM Yigal Weinstein wrote:
> >
> > Is there a way in Axiom >= 3.9 to get a numerical approximation
> > for Gamma(x,y)- without the use of NAG?  I know there is for 
> > Gamma(x) but for incomplete gamma there seems no straightforward
> > way.
>
On January 26, 2006 5:09 PM Vanuxem Grégory wrote:
> 
> No :-(
> 

I suggest that someone tackle this problem based on the following
article:

http://www.theorie.physik.uni-muenchen.de/~serge/papers/Winitzki_2003_Comput
ing_the_incomplete_Gamma_function_to_arbitrary_precision_LNCS_2667.pdf

Computing the incomplete Gamma function to arbitrary precision

Serge Winitzki1
Department of Physics, Ludwig-Maximilians University, Theresienstr. 37,
80333
Munich, Germany (address@hidden)

Abstract. I consider an arbitrary-precision computation of the
incomplete Gamma function from the Legendre continued fraction.
Using the method of generating functions, I compute the convergence
rate of the continued fraction and find a direct estimate of the
necessary number of terms. This allows to compare the performance
of the continued fraction and of the power series methods. As an
application, I show that the incomplete Gamma function Gamma(a, z)
can be computed to P digits in at most O (P) long multiplications
uniformly in z for Re z > 0. The error function of the real argument,
erf x, requires at most O(P2/3) long multiplications.

--------

I would be glad to help someone with the SPAD coding.

Regards,
Bill Page.