Re: i^3 == -i
- From: lcw1964 <leslie.wright@xxxxxxxxxxxxx>
- Date: Sat, 30 Jun 2007 05:50:43 -0700
My favourite recent treatment of the Lanczos gamma approximation is at
http://bh0.physics.ubc.ca/People/matt/Doc/ThesesOthers/Phd/pugh.pdf
You can use Pugh's recommended constants from the appendices of his
thesis with the method of Godfrey summarized here
http://www.rskey.org/gamma.htm
to generate Lanczos approximations even better than the one's
originally proposed by Cornelius Lanczos himself. (Note that the
constant's used in Numerical Recipes are a slight refinement of the
original 1964 work.)
Glen Pugh himself has sent me in the past year an update version of
the appendices to his thesis. I can share that with anyone interested.
Keep in mind it is not meaningful without looking at the thesis
itself, which I should say is actually pretty accessible to
intelligent non-mathematicians.
Despite the built-in Gamma on the HP49G+, I am presently working on a
Lanczos approximation version myself in SysRPL that returns real
results in extended precision (I am not that worried about complex
arguments right now). I am relying on Pugh's work and generating my
own constants with the help of Maple since the commonly published
version doesn't have small enough erro. I want to use this internally
as a subroutine in computing the complete and incomplete beta
functions. My hope is that the final 12 digit output will enjoy the
best possible accuracy if I do the internal work in extended precision
as much as possible. Relying on the calculator's own GAMMA, which
returns only real results, may compromise this goal. I will share how
I make out, if anyone is interested.
BTW, does anyone know what the 49G+ or 50G use for their GAMMA? It
handles complex arguments, so I am guessing something more
sophisticated than the trusty Stirling asymptotic series.
Les
.
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