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The Borwein Brothers, Pi and the AGM

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From Analysis to Visualization (JBCC 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 313))

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Abstract

We consider some of Jonathan and Peter Borweins’ contributions to the high-precision computation of \(\pi \) 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 \(\pi \), and more generally the elementary functions. These algorithms run in “almost linear” time \(O(M(n)\log n)\), 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 \(\pi \), such as the Gauss–Legendre algorithm. We show that an iteration of the Borwein-Borwein quartic algorithm for \(\pi \) is equivalent to two iterations of the Gauss–Legendre quadratic algorithm for \(\pi \), in the sense that they produce exactly the same sequence of approximations to \(\pi \) if performed using exact arithmetic.

In fond memory of Jonathan M. Borwein 1951–2016

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Notes

  1. 1.

    The formula (1) is listed in Bailey’s compendium [5], and is attributed to Madhava of Sangamagramma (c.1340–c.1425). It follows from the Taylor series for \(\arctan (1/\sqrt{3})\).

  2. 2.

    Attributed to Hero of Alexandria (c.10–70 A.D.), though also called the Babylonian method.

  3. 3.

    Here and elsewhere, \(\log \) denotes the natural logarithm.

  4. 4.

    Salamin [42] defines \(c_n\) using the relation \(c_n^2 = a_n^2-b_n^2\). This has the advantage that \(c_0\) is defined naturally, and for \(n > 0\) it is equivalent to our definition. However, it is computationally more expensive to compute \((a_n^2-b_n^2)^{1/2}\) than \(a_n-a_{n+1}\).

  5. 5.

    In [10, §10], Jon Borwein says “It [Algorithm GL] is based on the arithmetic–geometric mean iteration (AGM) and some other ideas due to Gauss and Legendre around 1800, although neither Gauss, nor many after him, ever directly saw the connection to effectively computing \(\pi \)”.

  6. 6.

    Similarly, where we exchange the order of taking derivatives and limits elsewhere in this section, it is easy to justify.

  7. 7.

    For example, one might compute \(x^{1/4}\) using two inverse square roots, i.e. \((x^{-1/2})^{-1/2}\), which is possibly faster than two square roots, i.e. \((x^{1/2})^{1/2}\), see [23, §4.2.3].

  8. 8.

    For example, the equivalence is not mentioned in [4], [15], [29], [30] or [33].

  9. 9.

    In fact, this is how Algorithm BB4 was discovered, by doubling Algorithm BB2 and then making some straightforward program optimisations.

  10. 10.

    Somewhat more general, but based on the same idea, is E. Karatsuba’s FEE method [34].

  11. 11.

    Alternatively, we could drop the simplifying assumption that \(a_0, b_0 \in \mathcal {H}\) and use the “right choice” of Cox [25, pg. 284] to implement the AGM correctly.

  12. 12.

    Mahler’s result is sufficient for the usual elementary functions, whose zeros are rational multiples of \(\pi \), but it is not applicable to the problem of computing combinations of these functions, e.g. \(\exp (\sin x) + \cos (\log x)\), with small relative accuracy. In general, we do not know enough about the rational approximation of the zeros of such functions to guarantee a small relative error. However, the result that we stated for computing elementary functions with a small absolute error extends to finite combinations of elementary functions under the operations of addition, multiplication, composition, etc. Indeed, the set of elementary functions is usually considered to include such finite combinations, although precise definitions vary. See, for example, §7.3 of Pi and the AGM, Knopp [35, pp. 96–98], Liouville [37], Ritt [41], and Watson [46, pg. 111].

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Acknowledgements

I am grateful to Jon and Peter Borwein for becoming sufficiently interested in this subject to write their book Pi and the AGM only a few years after the publication of [18,19,20, 42]. Reading a copy of Pi and the AGM was my first introduction to the Borwein brothers, and was the start of my realisation that we shared many common interests, despite living in different hemispheres. Much later, after Jon and his family moved to Newcastle (NSW), I followed him, bringing our common interests closer together, and benefitting from frequent interaction with him.

      Thanks are also due to David H. Bailey for his assistance, and to the Magma group for their excellent software [16].

      The author was supported in part by an Australian Research Council grant DP140101417. Jon Borwein was the Principal Investigator on this grant, which was held by Borwein, Brent and Bailey.

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Authors and Affiliations

  1. Mathematical Sciences Institute, Australian National University, Canberra, Australia

    Richard P. Brent

  2. School of Mathematical and Physical Sciences, The University of Newcastle, Newcastle, Australia

    Richard P. Brent

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Editors and Affiliations

  1. Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    David H. Bailey

  2. Western University, London, ON, Canada

    Naomi Simone Borwein

  3. Mathematical Sciences Institute, Australian National University, Canberra, ACT, Australia

    Richard P. Brent

  4. School of IT and Mathematical Sciences, University of South Australia, Mawson Lakes, SA, Australia

    Regina S. Burachik

  5. School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW, Australia

    Judy-anne Heather Osborn

  6. School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW, Australia

    Brailey Sims

  7. Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA

    Qiji J. Zhu

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Brent, R.P. (2020). The Borwein Brothers, Pi and the AGM. In: Bailey, D., et al. From Analysis to Visualization. JBCC 2017. Springer Proceedings in Mathematics & Statistics, vol 313. Springer, Cham. https://doi.org/10.1007/978-3-030-36568-4_21

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