Riemann's Zeta Function - another approach (RPL)

[Dieter]

(07-12-2017 01:42 PM)DavidM Wrote:  If I remove the final conversion step from extended real -> real, the extra digits are visible:

Yes, some results look familiar, e.g. Zeta(0,8) where the 13th digit is just below the threshold for getting rounded up. ;-)

Anyway, I now tried a slightly different approach and got a result which fits even better. Evaluated with sufficient precision the largest error even with the rouned coefficients below is quite exactly 1 unit in the 13th significant digit. This is very close to the optimium with exact coefficients (0,89 units).

Here are the optimized coefficients (with 12 digits or less):

c0 =  0,577215664895
c1 =  0,0728158454396
c2 = -0,0048451809848
c3 = -0,0003422986463
c4 =  0,0000969189036
c5 = -0,0000065530879
c6 = -0,0000002673884
c7 =  0,00000014172

These are the results with 16 digit standard precision on a WP34s.
No extended precision required (yes, I love it!). ;-)

Code:

0,0:  -0,5
0,1:  -0,603037519856 1557
0,2:  -0,733920924896 3916
0,3:  -0,904559257254 0113
0,4:  -1,13479778386 6858
0,5:  -1,46035450880 9872
0,6:  -1,95266144822 4910
0,7:  -2,77838844555 4192
0,8:  -4,43753841589 4835
0,9:  -9,43011401940 3022
1,0:   +∞ error
1,05:  20,5808443020 3088
1,1:   10,5844484649 5657

The largest error here is 0,91 units in the 13th digit.

Now, if the largest error is 1 unit in the 13th digit (if evaluated with, say, 15 digit precision), the displayed 12-digit result should be off by at most 0,6 ULP. Which, as far as I remember, is on par with the accuracy of the built-in transcendental functions.

Dieter