| 释义 |
Archimedes AlgorithmSuccessive application of Archimedes' Recurrence Formula gives the Archimedes algorithm, which can be used toprovide successive approximations to  (Pi). The algorithm is also called the Borchardt-PfaffAlgorithm. Archimedes obtained the first rigorous approximation of by Circumscribing and Inscribing  -gons on a Circle. From Archimedes'Recurrence Formula, the Circumferences  and  of the circumscribed and inscribed Polygons are
where
 | (3) |
 and
where  . The first iteration of Archimedes' Recurrence Formula then gives
Additional iterations do not have simple closed forms, but the numerical approximations for  , 1, 2, 3, 4 (corresponding to6-, 12-, 24-, 48-, and 96-gons) are
 | (8) |
 | (9) |
 | (10) |
 | (11) |
 | (12) |
By taking  (a 96-gon) and using strict inequalities to convert irrational bounds to rational bounds at each step,Archimedes obtained the slightly looser result
 | (13) |
References
Miel, G. ``Of Calculations Past and Present: The Archimedean Algorithm.'' Amer. Math. Monthly 90, 17-35, 1983.Phillips, G. M. ``Archimedes in the Complex Plane.'' Amer. Math. Monthly 91, 108-114, 1984. |
