Gamma Function by Stieltjes Continued Fraction

09132015, 01:55 PM
(This post was last modified: 09132015 02:09 PM by Dieter.)
Post: #15




RE: Gamma Function by Stieltjes Continued Fraction
(09132015 02:06 AM)lcwright1964 Wrote: Hey, Dieter, for HP41 work with 10 digits I think I am sticking with the refinement mentioned above. Expressing the coefficient that is around 5 to a few more digits doesn't seem to confer extra benefit on the HP41, and indeed when I plot the error curve in an arbitrary precision environment I really see no big improvement at all. The additional digits only help squeeze the error a tiny bit below the 2 E–12 mark. ;) You won't notice much of a difference even in a higher precision environment. For the '41 and other 10digit machines the earlier methods are perfectly OK. BTW, with a=4,01 and b=5,00008 you can lower the threshold to 8 and still get similar accuracy (error < 5 E–12), while for small x one loop less is required. (09132015 02:06 AM)lcwright1964 Wrote: I should play around with your four term versions, but I suspect that the advantages of them are lost in a 10 digit environment and, if anything, the extra computations due to the extra term produce their own round off issues. Sure, all this was done with higherprecision environments in mind. The best I got so far is an error < 3,7 E–14 (threshold = 8), and I tried this on a WP34s in SP mode (16 digits). It's nice to see the 12 digit display show the same result as the builtin Gamma function. ;) (09132015 02:06 AM)lcwright1964 Wrote: Of course, all of these great refinements are good candidates for MCODE programming, but I think none of them can top some of the optimized Lanczos approximations we were discussing elsewhere. Getting rid of the shiftdivide step for smaller arguments really goes far to speeding things up across the board. Right, this shiftanddivide thing slows down everything, especially on older 10digit calculators like the '41 or 67/97. But in MCode speed should be no big issue, and with some careful, accuracypreserving coding one may expect something like 11 valid digits. Finally I would like to sum up some results for threeterm Stieltjes approximations in the attached graphics. The yaxis is the EDD figure, i.e. the base10 log of the relative error, or roughly the number of exact digits (±1 unit). The peaks are the points where the error passes through zero and accuracy becomes infinite. The sawtooth pattern for small x is due to the shiftanddivide method applied there. You can see the original Stieltjes method in the black graph, while the red one shows the improvement due to the a=4 simplification. I especially like the blue version as it reaches an error level slightly below 1 E–11 down to t=8. Yes, you may omit the ...04 at the end of the b coefficient. ;) Dieter 

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