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[robve]

(03-14-2022 09:17 PM)Thomas Klemm Wrote:  That's an alternating series so we can use Valentin's program:
(11C) Summation of infinite, alternating series

We use \(\text{PSum} = 10\), \(\text{NDif} = 7\) for maximum accuracy:

10 ENTER 7 A

After some time, we get the following result in the display:

0.9068996822

The correct result \(\frac{\pi}{\sqrt{12}}\) on the HP-11C is:

0.9068996821

The number of function evaluations with n=10 and d=7 is n+d+2=19, but only 18 function evaluations are needed with Sharp's series to get 10 decimal places. So I am not sure if this is the best choice of parameterization for this particular alternating series.

To demonstrate what I mean, I did a bit of experimentation using a BASIC version of the algorithm with the PC-1475 double precision float (up to 20 digits, see my prior post). I found the following results to compute 19 decimal places:

Direct computation of Sharp's series to get 19 decimal places of pi:
35 function evaluations

Sum of alternating series with Aitken extrapolation and summing in reverse order to get 19 decimal places of pi:
n=26 d=1: 29 function evaluations + 5 table operations
n=25 d=3: 30 function evaluations + 14 table operations
n=24 d=4: 30 function evaluations + 20 table operations
n=23 d=5: 30 function evaluations + 27 table operations
n=22 d=6: 30 function evaluations + 35 table operations
n=21 d=7: 30 function evaluations + 44 table operations

The table operations are cheaper to compute compared to the function evaluations, but the overhead of increasing d grows quadratically. These computations are not negligible, so the smallest d should be chosen even if that requires increasing n by the same amount or slightly more.

The n=26 d=1 choice is optimal for 19 decimal places and slightly better than direct computation of the series. Testing with fewer places of pi show that d=1 is still the best choice.

This result is only applicable to the Sharp's alternating series. Other series will give different results and require different n and d parameterizations.

- Rob