Abstract
In this article, we prove the Brent–Salamin algorithm for \(\pi \) using Jacobi’s theta functions and use this approach to derive new analogues of the Brent–Salamin algorithm for elliptic functions to alternative bases.
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Acknowledgments
This article is written during my stay at the University of Hong Kong. I would like to thank K. M. Tsang and Y. K. Lau for their warm hospitality. I would also like to thank B. C. Berndt, J. M. Borwein, R. Brent, Y. Tanigawa and K. M. Tsang for their encouragement and discussions during the preparation of this article. Finally, I thank S. Cooper for many of his excellent suggestions which have significantly improved an earlier version of this work.
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Dedicated to Siew Lian, Si Min, Si Ya, and Si En.
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Chan, H.H. Analogues of the Brent–Salamin algorithm for evaluating \(\pi \) . Ramanujan J 38, 75–100 (2015). https://doi.org/10.1007/s11139-014-9560-0
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DOI: https://doi.org/10.1007/s11139-014-9560-0