A not so useful HP16C program

04222018, 02:11 AM
(This post was last modified: 04222018 02:55 AM by Gerson W. Barbosa.)
Post: #5




RE: A not so useful HP16C program
(04222018 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 ngons to a radius 1/2 circumference ). I came up with a similar precision formula seven years ago, albeit an empiricallyobtained 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 timeconsuming task, be it done my hand or by machine. P.S.: BTW, the perimeter of the circumscribed 96gon, 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!). 

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Messages In This Thread 
A not so useful HP16C program  Gerson W. Barbosa  04212018, 11:37 PM
RE: A not so useful HP16C program  Joe Horn  04222018, 01:08 AM
RE: A not so useful HP16C program  Gerson W. Barbosa  04222018, 01:14 AM
RE: A not so useful HP16C program  rprosperi  04222018, 01:37 AM
RE: A not so useful HP16C program  Gerson W. Barbosa  04222018 02:11 AM
RE: A not so useful HP16C program  Dieter  04222018, 01:08 PM
RE: A not so useful HP16C program  Gerson W. Barbosa  04222018, 05:59 PM
RE: A not so useful HP16C program  Gerson W. Barbosa  04222018, 09:31 PM

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