Accuracy of numerical calculation of the Gamma andBeta function

Hello Adam,

Thank you for your interest.

Yes, I can show more data concerning the accuracy and the time of computation we
need to get the numercial results of the Beta function. At first I have posted
some further data for the Gamma function.

Next, I will give more data for the Beta function. The work on the Beta function
is concerned to the implementation of the Incomplete Beta function. Here I have
choosen different methods to implement the bigfloat algorithm.

I do not know all about Maxima. But a lot of functions work with bigfloat
precision. So changing the value of fpprec Maxima gives results with any desired
accuracy. You have only a problem with computation time and perhaps memory to
finish the calculation.

For the functions I am working on I try to test the algorithm up to some
thousands of digits and I never had a problem to verify the results (after
finding the bugs).

Dieter Kaiser 

-----Urspr?ngliche Nachricht-----
Von: maxima-bounces at math.utexas.edu [mailto:maxima-bounces at math.utexas.edu] Im
Auftrag von Adam Majewski
Gesendet: Samstag, 11. Oktober 2008 09:21
An: maxima at math.utexas.edu
Betreff: Re: [Maxima] Accuracy of numerical calculation of the Gamma andBeta
function

Dear Dieter,

Thx for intresting results.

.....
> 
> Results for double float precision:
> 
> maxerr = 1.88e-13 (formula r2)
> maxerr = 2.29e-13 (formula r3)
> 
> Result for bigfloat precision:
> 
> fpprec  maxerr r2  maxerr r3
> 16      3.24b-14   3.58b-14
> 32      3.84b-30   3.99b-30
> 64      7.87b-62   9.48b-62

Can you also show time of computations ? ( How it grows with fpprec )

> 
> Conclusion:
> 
> There is no significant difference beetween the different algorithm. The
> algoritm using log_gamma seems to be sligthly less accurate. Perhaps something
> in the routines for log_gamma can be done better.
> 
> In bigfloat precision we can achieve any desired accuracy. We have a
systematic
> difference of 2 to 4 digits in accuracy. 


Does it mean that Maxima can also be used not only as a CAS but also for 

numerical calculations ?

BTW :
Sometimes I (in my opinion) it's hard to change fpprec in Maxima, see 
for example:
http://commons.wikimedia.org/w/index.php?title=Image:Lemniscates5.png

Is it possible to change it ? ( to do it with simple one command fpprec)


Adam

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