- Title
- The Borwein brothers, Pi and the AGM
- Creator
- Brent, Richard P.
- Relation
- JBCC: Jonathan M. Borwein Commemorative Conference . Proceedings of the JBCC: Jonathan M. Borwein Commemorative Conference, Volume 313 (Callaghan, Australia 25-29 September, 2017) p. 323-347
- Relation
- ARC.DP140101417 http://purl.org/au-research/grants/arc/DP140101417
- Publisher Link
- http://dx.doi.org/10.1007/978-3-030-36568-4_21
- Publisher
- Springer
- Resource Type
- conference paper
- Date
- 2020
- Description
- We consider some of Jonathan and Peter Borweins’ contributions to the high-precision computation of π and the elementary functions, with particular reference to their book Pi and the AGM (Wiley, 1987). Here “AGM” is the arithmetic–geometric mean of Gauss and Legendre. Because the AGM converges quadratically, it can be combined with fast multiplication algorithms to give fast algorithms for the n-bit computation of π, and more generally the elementary functions. These algorithms run in “almost linear” time O(M(n)logn), where M(n) is the time for n-bit multiplication. We outline some of the results and algorithms given in Pi and the AGM, and present some related (but new) results. In particular, we improve the published error bounds for some quadratically and quartically convergent algorithms for π, such as the Gauss–Legendre algorithm. We show that an iteration of the Borwein-Borwein quartic algorithm for π is equivalent to two iterations of the Gauss–Legendre quadratic algorithm for π, in the sense that they produce exactly the same sequence of approximations to π if performed using exact arithmetic.
- Subject
- arrithmetic-geometric mean; Borwein-Borwein algorithm; Borwein-Borwein quartic algorithm; Brent-Salamin algorithm; Chudnovsky algorithm; computation of π
- Identifier
- http://hdl.handle.net/1959.13/1437744
- Identifier
- uon:40448
- Identifier
- ISBN:9783030365677
- Language
- eng
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