Gamma Function Using Spouge's Method
08-20-2015, 12:36 PM
Post: #10
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: Gamma Function Using Spouge's Methjod
(08-19-2015 05:32 PM)lcwright1964 Wrote:  I think I was once drawn to this pseudocode (edited by me to include Luschny's suggested simplification in the formula) which implements the Stieltjes continued fraction:

Stieltjes3Factorial(x)
y = x+1; p = 1;
while y < 7 do p = p*y; y = y+1; od;
r = exp(y*(log(y)-1)+1/(2*(6*y+1/(5*(y+1/(4*y))))));
if x < 7 then r = r/p fi;
return r*sqrt(2*Pi / y) end;

As far as I can see the last x<7 test can be omitted. I tried this in VBA for Excel:

Code:
Function Gamma(x) y = x p = 1 Do While y < 7    p = p * y    y = y + 1 Loop r = Exp(y * (Log(y) - 1) + 1 / (2 * (6 * y + 1 / (5 * (y + 1 / (4 * y)))))) Gamma = 2.506628274631 * r / p / Sqr(y) End Function

The constant in the last line is sqrt(2*pi). It turned out that the results seem to have not just eight but 10 valid digits, at least when evaluated with 15-digit precision. Only in some close cases I noticed that the 10th digit may be rounded down instead of up (e.g. for x=1,7 or 7,8). I'll try this one on my '41. ;-)

Thank you very much for reminding me of Peter Luschny's excellent Gamma/factorial pages.

Dieter
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 Messages In This Thread Gamma Function Using Spouge's Method - Namir - 12-18-2013, 12:55 AM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-07-2015, 07:23 PM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-12-2015, 04:03 AM RE: Gamma Function Using Spouge's Methjod - Dieter - 08-12-2015, 05:39 PM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-12-2015, 06:03 PM RE: Gamma Function Using Spouge's Methjod - Dieter - 08-12-2015, 09:11 PM RE: Gamma Function Using Spouge's Methjod - Ángel Martin - 08-19-2015, 08:56 PM RE: Gamma Function Using Spouge's Methjod - Dieter - 08-20-2015, 11:36 AM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-20-2015, 07:49 PM RE: Gamma Function Using Spouge's Methjod - Ángel Martin - 08-21-2015, 09:48 AM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-22-2015, 02:09 AM RE: Gamma Function Using Spouge's Methjod - Ángel Martin - 08-22-2015, 08:35 AM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-22-2015, 08:58 AM RE: Gamma Function Using Spouge's Methjod - Ángel Martin - 08-22-2015, 10:45 AM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-19-2015, 05:32 PM RE: Gamma Function Using Spouge's Methjod - Dieter - 08-20-2015 12:36 PM RE: Gamma Function Using Spouge's Methjod - Thomas Klemm - 08-20-2015, 08:22 PM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-22-2015, 09:57 AM RE: Gamma Function Using Spouge's Methjod - Ángel Martin - 08-22-2015, 01:17 PM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-22-2015, 11:51 PM RE: Gamma Function Using Spouge's Methjod - Dieter - 08-23-2015, 09:01 PM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-24-2015, 08:12 AM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-24-2015, 09:23 AM RE: Gamma Function Using Spouge's Methjod - Dieter - 08-24-2015, 07:32 PM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-24-2015, 08:40 PM RE: Gamma Function Using Spouge's Methjod - Dieter - 08-24-2015, 11:14 PM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-25-2015, 01:56 AM RE: Gamma Function Using Spouge's Methjod - Dieter - 08-25-2015, 06:48 PM RE: Gamma Function Using Spouge's Methjod - Dieter - 08-27-2015, 11:21 PM RE: Gamma Function Using Spouge's Methjod - Dieter - 09-15-2015, 06:43 PM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-25-2015, 11:29 PM RE: Gamma Function Using Spouge's Methjod - Dieter - 08-26-2015, 09:50 AM RE: Gamma Function Using Spouge's Methjod - lcwright1964 - 08-26-2015, 12:50 PM RE: Gamma Function Using Spouge's Methjod - Dieter - 08-26-2015, 05:08 PM

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