Analogues of the Brent-Salamin algorithm for evaluating \(\pi\)
type: Item
instance of: scholarly article
title: Analogues of the Brent-Salamin algorithm for evaluating \(\pi\)
zbMATH Open document ID: 1369.11108
author: Heng Huat Chan
published in: The Ramanujan Journal
publication date: 2015-10-16T00:00:00Z
review text: The Brent-Salamin algorithm is a method to compute the digits of \(\pi\). The algorithm involves the Gauss Arithmetic-Geometric Mean (AGM) iteration and was proved by Brent and Salamin using identities of elliptic integrals. The author gives an alternative proof of the Brent-Salamin algorithm using identities of the Jacobi theta functions, which are known to be related to elliptic integrals. The author's proof is somewhat involved but as a by-product several new AGM iterations for computing \(\pi\) are derived. Also, some comparisons are made with AGM iterations for computing \(\pi\) due to the Borweins.
Mathematics Subject Classification ID: 11Y60
Mathematics Subject Classification ID: 33E05
Mathematics Subject Classification ID: 11Y16
zbMATH DE Number: 6495133
zbMATH Keywords: Brent-Salamin algorithm
zbMATH Keywords: elliptic integrals
zbMATH Keywords: hypergeometric series
zbMATH Keywords: theta series
zbMATH Keywords: modular equations
reviewed by: Ghaith A. Hiary
MaRDI profile type: Publication
full work available at URL: s11139-014-9560-0
OpenAlex ID: W2064638369
cites work: Q3995284
cites work: Q4000402
cites work: Ramanujan's Theories of Elliptic Functions to Alternative Bases
cites work: On Ramanujan's quartic theory of elliptic functions
cites work: Q4377085
cites work: Q3751634
cites work: A Cubic Counterpart of Jacobi's Identity and the AGM
cites work: Some Cubic Modular Identities of Ramanujan
cites work: Fast Multiple-Precision Evaluation of Elementary Functions
cites work: Q4432302
cites work: On Ramanujan's cubic transformation formula for 2 F 1(1/3, 2/3; 1; z)
cites work: Computation of π Using Arithmetic-Geometric Mean
cites work: A simple proof of η(— 1/τ)= η(τ)√τ/i
DOI: 10.1007/S11139-014-9560-0
description: scientific article; zbMATH DE number 6495133