Calculating Pi - Brent-Salamin Algorithm
Date: 05/11/2000 at 11:47:24 From: Bill Smith Subject: Pi What are some methods for finding pi?
Date: 05/11/2000 at 13:40:44 From: Doctor Rob Subject: Re: Pi Thanks for writing to Ask Dr. Math, Bill. Start at the following frequently asked questions (FAQ) page on pi from Ask Dr. Math: http://mathforum.org/dr.math/faq/faq.pi.html Then look at the links near the bottom of the page entitled, "Pi on the Web." Some of those should help. I particularly like "Pi through the ages." As a follow-up, here is something you may find interesting. Probably the easiest way to get digits of Pi is this. Start with a = 1, b = sqrt(2)/2, r = 1, and s = 1/2. Use as much precision in calculation as you want in the answer. Then do these steps: 1. A = (a+b)/2 2. B = sqrt(a*b) 3. C = (A-b)^2 4. R = 2*r 5. S = s - C*R 6. P = 2*A^2/S 7. If C = 0 to the precision used, stop, and the answer is Pi = P to that precision. 8. Replace a by A, b by B, r by R, and s by S 9. Go back to step 1. This is called the Brent-Salamin Algorithm. Each calculation is pretty simple, and after just a few rounds it gives answers accurate to many decimal places. (Five rounds will give 42 decimal digits, if you use that much precision.) Explaining *why* this works is very hard. Please accept that it does. If we make our precision eight decimal places, the first few steps go like this: a = 1 b = 0.70710678 r = 0 s = 1/2 A = 0.85355339 B = 0.84089642 C = 0.02144661 S = 0.45710678 R = 2 P = 3.18767264 C is not small enough, A = 0.84722490 B = 0.84720127 C = 0.00004005 S = 0.45694658 R = 3 P = 3.14168029 C is not small enough, A = 0.84721308 B = 0.84721308 C = 0.00000000 S = 0.45694658 R = 4 P = 3.14159265 C is small enough. Then the answer is pi = 3.14159265, to eight decimal places. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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