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## Re: [Axiom-developer] Axiom and the Numerical Mathematics Consortium

 From: M. Edward (Ed) Borasky Subject: Re: [Axiom-developer] Axiom and the Numerical Mathematics Consortium Date: Thu, 17 Jan 2008 19:35:17 -0800 User-agent: Thunderbird 2.0.0.9 (X11/20071227)

Doug Stewart wrote:
>> I have been concentrating on Axiom's numerical capabilities. The last
>> patch is the beginnings of regression tests against
>> Abramowitz and Stegun (1985) and Zwillinger's CRC Standard (2003).
>> I've also created firefox hyperdoc pages for the gamma function
>> standard from the new DLMF. I plan to fill these pages out with
>> Spad code and test cases as time permits.
>>
>> I'm a member of the Numerical Mathematics Consortium
>> (http://www.nmconsortium.org).A recently published draft
>> standard, which I'm reviewing, is available at:
>> <http://www.nmconsortium.org/docs/NMC_Technical_Specification%20(9-24-2007).pdf>
>>
>>
>> The A&S handbook lists polynomial coefficients for approximation of E1,
>> the exponential integral. Does anyone know how these coefficients were
>> derived? Is it a chebyshev polynomial? I want to dynamically compute
>> these coefficients to the required precision.
>>
>> Tim
>>
>>
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>>
>>
>
>
> The exponential integral can be written as a special case of the
> incomplete gamma function
> <http://en.wikipedia.org/wiki/Incomplete_gamma_function>:
>
>    {\rm E}_n(x) =x^{n-1}\Gamma(1-n,x).\,
>
> The exponential integral may also be generalized to
>
>    E_n(x) = \int_1^\infty \frac{e^{-xt}}{t^n}\, dt
>
>
> this is from
>
> http://en.wikipedia.org/wiki/Exponential_integral
>
>
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>

P.S.: Another classic from those days was _Approximations for Digital
Computers_ by Cecil Hastings. That's on my list of collectibles to pick
up. :)