||It has been suggested that this article be merged into arithmetic–geometric mean. (Discuss) Proposed since September 2012.|
In mathematics, the AGM method (for arithmetic–geometric mean) makes it possible to construct fast algorithms for calculation of exponential and trigonometric functions, and some mathematical constants and in particular, to quickly compute .
have the same limit:
The number π
which can be computed without loss of precision using
Complete elliptic integral K(α)
At the same time, if we take
where K(α) is a complete elliptic integral
Using this property of the AGM and also the ascending transformations of Landen, Richard Brent suggested the first AGM algorithms for fast evaluation of elementary transcendental functions (ex, cos x, sin x). Subsequently, many authors went on to study the use of the AGM algorithms, see, for example, the book Pi and the AGM by Jonathan and Peter Borwein.
- B. C. Carlson (1971). "Algorithms involving arithmetic and geometric means". Amer. Math. Monthly 78: 496–505. doi:10.2307/2317754. MR 0283246.
- B. C. Carlson (1972). "An algorithm for computing logarithms and arctangents". Math.Comp. 26 (118): 543–549. doi:10.2307/2005182. MR 0307438.
- E. Salamin (1976). "Computation of using arithmetic-geometric mean". Math. Comp. 30 (135): 565–570. doi:10.2307/2005327. MR 0404124.
- J. Landen (1775). "An investigation of a general theorem for finding the length of any arc of any conic hyperbola, by means of two elliptic arcs, with some other new and useful theorems deduced therefrom". Philosophical Transactions of the Royal Society 65: 283–289. doi:10.1098/rstl.1775.0028.
- R.P. Brent (1976). "Fast Multiple-Precision Evaluation of Elementary Functions". J. Assoc. Comput. Mach. 23 (2): 242–251. doi:10.1145/321941.321944. MR 0395314.
- Borwein, J.M.; Borwein, P.B. (1987). Pi and the AGM. New York: Wiley. ISBN 0-471-83138-7. MR 0877728.