Riemann's Zeta Function - another approach (RPL)

[Gerson W. Barbosa]

(07-30-2017 10:18 PM)Dieter Wrote:  

(07-30-2017 04:17 PM)Gerson W. Barbosa Wrote:  Yes, that's what ABS in line 19 is for.

I see, but the HP-41 version has is another RCL 00 without an ABS in line 24. ;-)

Yes, but it's located after a call to your "GAM+1" program, which incidentally saves the argument in register 0 also. But you're not supposed to have to remember that after 3+ years. I'd failed to notice what your concern was about, sorry!

(07-30-2017 10:18 PM)Dieter Wrote:  

(07-30-2017 04:17 PM)Gerson W. Barbosa Wrote:  A few guard digits (perhaps just a couple of them) combined with built-in Gamma might give perfect 10-digit results most always, even when using 10-digits constants, which is quite impressive.

To assess the final accuracy it might be helpful to know the error of the two polynomial approxmations for x between 0 and 2. If evaluated with sufficient precision and using the coefficients given in the HP-41 program (note that c0 effectively has 12 digits since 0,57 is added later) the one for 0≤x<1 has a largest error of ~3,5 units in the 12th significant digit, while for 1<x≤2 it's less than 0,7 units.

(07-30-2017 04:17 PM)Gerson W. Barbosa Wrote:  On Free42:

I tried Free42 BCD where Gamma seems to be good for 30+ digits. So the results should only be limited by the approximation error. However, for x>2 the number of terms (cf. line 38 ff. in the 42s Zeta program) of course has to be adjusted in a higher precision environment. ;-) This should improve the results for x>2 resp. x<–1.

This was just a quick test on Free42. I have yet to try it on the 42S, where some accuracy loss is expected, but not so much, I hope. Yes, the number of terms will have to be increased accordingly. Despite that, the running times will be better, since the 42S is about five times faster than the 41.

Gerson.