A not so useful HP-16C program

[Gerson W. Barbosa]

(04-22-2018 01:37 AM)rprosperi Wrote:  So it's easy to see the input is the number of iterations, but not so clear to see how the resulting X and Y converge. Look forward to the insights once folks have suffered long enough.

Back to my good old desktop computer!

That's Archimedes' method to approximate Pi, except that instead of starting with the hexagon, I start with inscribed and circumscribed squares to a radius 1/2 circumference (thus saving a few steps as the initial constants, 2 and 4, don't involve surds). Without a positional number system and without a consistent math notation, that was such a feat, 0.6 digits per iteration! Not bad around 250 BC.

Pressing R/S accelerates the convergence by a factor of 3. Basically, I use formula 2.6 in this paper in terms of a and b (perimeters of the inscribed and circumscribed n-gons to a radius 1/2 circumference ). I came up with a similar precision formula seven years ago, albeit an empirically-obtained one. Currently, I have a method that yields 3 times as much digits per iteration, compared to the basic Archimedes' algorithms, but I think it can go up to more than 8 times as much (5 digits per iteration). No intention to compete with modern methods, though. Square root extractions are a time-consuming task, be it done my hand or by machine.

P.S.: BTW, the perimeter of the circumscribed 96-gon, Archimedes’ second bound, is about 21.9990021999/7 (well, actually 22.9990021975, but the former is nicer). This is also the place to discuss near integers like 2*(e - atan(e)) = 2.9999978 (that ‘s gonna be our 3 in that nerd’s clock!).