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The arithmetic-geometric mean and fast computation of elementary functions (1984)

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Abstract

Paper 4: J. M. Borwein and P. B. Borwein, “The arithmetic-geometric mean and fast computation of elementary functions,” SIAM Review, vol. 26 (1984), p. 351–366. Copyright ©1984 Society for Industrial and Applied Mathematics. Reprinted with permission. All rights reserved.

Synopsis: In this paper, brothers Jonathan and Peter Borwein present a review of the recently discovered (as of 1984) quadratically convergent formulas for π and elementary functions (including some new formulas of their own), complete with a rigorous derivation of all the requisite mathematics. The paper is thus an excellent self-contained tutorial on the theory of the arithmetic-geometric mean, quadratically convergent algorithms (including Newton’s algorithm for computing square roots and roots of polynomials), and how these concepts lead to fast algorithms for π and elementary functions. They do this without needing to venture into incomplete elliptic integrals and Landen transforms, which were used to various degrees by earlier writers.

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Authors and Affiliations

  1. University of Newcastle, Newcastle, New South Wales, Australia

    J. M. Borwein

  2. Simon Fraser University, Burnaby, British Columbia, Canada

    P. B. Borwein

Authors
  1. J. M. Borwein
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  2. P. B. Borwein
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Borwein, J.M., Borwein, P.B. (2016). The arithmetic-geometric mean and fast computation of elementary functions (1984). In: Pi: The Next Generation. Springer, Cham. https://doi.org/10.1007/978-3-319-32377-0_4

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