Advertisement

The SIAM 100-Digit Challenge: A Decade Later

Inspirations, Ramifications, and Other Eddies Left in Its Wake
  • Folkmar Bornemann
Survey Article

Abstract

In 2002, L.N. Trefethen of Oxford University challenged the scientific community by ten intriguing mathematical problems to be solved, numerically, to ten digit accuracy each (I was one of the successful contestants; in 2004, jointly with three others of them, I published a book—Bornemann et al.: The SIAM 100-Digit Challenge, SIAM, Philadelphia, 2004—on the manifold ways of solving those problems). In this paper, I collect some new and noteworthy insights and developments that have evolved around those problems in the last decade or so. In the course of my tales, I will touch mathematical topics as diverse as divergent series, Ramanujan summation, low-rank approximations of functions, hybrid numeric-symbolic computation, singular moduli, self-avoiding random walks, the Riemann prime counting function, and winding numbers of planar Brownian motion. As was already the intention of the book, I hope to encourage the reader to take a broad view of mathematics, since one lasting moral of Trefethen’s contest is that overspecialization will provide too narrow a view for one with a serious interest in computation.

Keywords

Divergent series Numeric-symbolic methods Low-rank approximation Singular moduli Self-avoiding random walk Riemann \(R\) function 

Notes

Acknowledgements

I thank Martin Hanke-Bourgeois for having solicited this paper; it was presented first at the conference “New Directions in Numerical Computing: In celebration Nick Trefethen’s 60th birthday”, Oxford, August 27, 2015. Nick Trefethen has kindly ironed out some of my worst English language peculiarities.

References

  1. 1.
    Battles, Z., Trefethen, L.N.: An extension of MATLAB to continuous functions and operators. SIAM J. Sci. Comput. 25(5), 1743–1770 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berndt, B.C.: Ramanujan’s Notebooks. Part I. Springer, New York (1985) CrossRefzbMATHGoogle Scholar
  3. 3.
    Berndt, B.C.: Ramanujan’s Notebooks. Part V. Springer, New York (1998) CrossRefzbMATHGoogle Scholar
  4. 4.
    Berndt, B.C., Rankin, R.A.: Ramanujan: Letters and Commentary. Am. Math. Soc., Providence (1995) zbMATHGoogle Scholar
  5. 5.
    Bornemann, F.: Solution of a problem posed by Jörg Waldvogel. Unpublished note, www-m3.ma.tum.de/bornemann/RiemannRZero.pdf (2003)
  6. 6.
    Bornemann, F., Laurie, D., Wagon, S., Waldvogel, J.: The SIAM 100-Digit Challenge. Society for Industrial and Applied Mathematics, Philadelphia (2004) CrossRefzbMATHGoogle Scholar
  7. 7.
    Bornemann, F., Laurie, D., Wagon, S., Waldvogel, J.: Vom Lösen numerischer Probleme. Springer, Berlin (2006). (German edition of [6]) Google Scholar
  8. 8.
    Borwein, J.M.: Book review: the SIAM 100-digit challenge. Math. Intell. 27(4), 40–48 (2005) CrossRefGoogle Scholar
  9. 9.
    Borwein, J.M., Borwein, P.B.: Pi and the AGM. Wiley, New York (1998) zbMATHGoogle Scholar
  10. 10.
    Brassesco, S., García Pire, S.C.: On the density of the winding number of planar Brownian motion. J. Theor. Probab. 27(3), 899–914 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Butzer, P.L., Ferreira, P.J.S.G., Schmeisser, G., Stens, R.L.: The summation formulae of Euler–Maclaurin, Abel–Plana, Poisson, and their interconnections with the approximate sampling formula of signal analysis. Results Math. 59(3–4), 359–400 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Candelpergher, B., Coppo, M.A., Delabaere, E.: La sommation de Ramanujan. Enseign. Math. 43(1–2), 93–132 (1997) MathSciNetGoogle Scholar
  13. 13.
    Clisby, N.: Efficient implementation of the pivot algorithm for self-avoiding walks. J. Stat. Phys. 140(2), 349–392 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert \(W\) function. Adv. Comput. Math. 5(4), 329–359 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Costin, O.: Asymptotics and Borel Summability. CRC Press, Boca Raton (2009) zbMATHGoogle Scholar
  16. 16.
    Dixon, J.D.: Exact solution of linear equations using \(p\)-adic expansions. Numer. Math. 40(1), 137–141 (1982) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dumas, J.-G., Turner, W., Wan, Z.: Exact solution to large sparse integer linear systems. Abstract for ECCAD’2002 (2002) Google Scholar
  18. 18.
    Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. II, 2nd edn. Wiley, New York (1971) zbMATHGoogle Scholar
  19. 19.
    Gauss, C.F.: Werke. Band III. Georg Olms Verlag, Hildesheim (1973). Reprint of the 1866 original Google Scholar
  20. 20.
    Gautschi, W.: The numerical evaluation of a challenging integral. Numer. Algorithms 49(1–4), 187–194 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Guttmann, A.J., Kennedy, T.: Self-avoiding walks in a rectangle. J. Eng. Math. 84, 201–208 (2014) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hale, N., Trefethen, L.N.: Chebfun and numerical quadrature. Sci. China Math. 55(9), 1749–1760 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hardy, G.H.: Ramanujan. Twelve Lectures on Subjects Suggested by His Life and Work. Cambridge University Press, Cambridge (1940) zbMATHGoogle Scholar
  24. 24.
    Hardy, G.H.: Divergent Series. Oxford University Press, Oxford (1949) zbMATHGoogle Scholar
  25. 25.
    Havil, J.: Gamma. Princeton University Press, Princeton (2003) zbMATHGoogle Scholar
  26. 26.
    Jagger, G.: The making of logarithm tables. In: The History of Mathematical Tables, pp. 49–77. Oxford University Press, Oxford (2003) CrossRefGoogle Scholar
  27. 27.
    Kalugin, G.A., Jeffrey, D.J., Corless, R.M.: Bernstein, Pick, Poisson and related integral expressions for Lambert \(W\). Integral Transforms Spec. Funct. 23(11), 817–829 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kleinert, H.: Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th edn. World Scientific, Hackensack (2009) CrossRefzbMATHGoogle Scholar
  29. 29.
    Lawler, G.F., Schramm, O., Werner, W.: On the scaling limit of planar self-avoiding walk. In: Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2, pp. 339–364. Am. Math. Soc., Providence (2004) CrossRefGoogle Scholar
  30. 30.
    Lenstra, A.K., Lenstra, H.W. Jr., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261(4), 515–534 (1982) MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Lindelöf, E.: Le calcul des résidus. Gauthier–Villars, Paris (1905) zbMATHGoogle Scholar
  32. 32.
    Longman, I.M.: Note on a method for computing infinite integrals of oscillatory functions. Proc. Camb. Philos. Soc. 52, 764–768 (1956) MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mansuy, R., Yor, M.: Aspects of Brownian Motion. Springer, Berlin (2008) CrossRefzbMATHGoogle Scholar
  34. 34.
    Mörters, P., Peres, Y.: Brownian Motion. Cambridge University Press, Cambridge (2010) CrossRefzbMATHGoogle Scholar
  35. 35.
    Nienhuis, B., Kager, W.: Stochastic Löwner evolution and the scaling limit of critical models. In: Polygons, Polyominoes and Polycubes, pp. 425–467. Springer, Dordrecht (2009) CrossRefGoogle Scholar
  36. 36.
    Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974) zbMATHGoogle Scholar
  37. 37.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010) zbMATHGoogle Scholar
  38. 38.
    Ramanujan, S.: Notebooks, Vol. II. Tata Institute of Fundamental Research, Bombay (1957) zbMATHGoogle Scholar
  39. 39.
    Ramis, J.-P.: Séries divergentes et théories asymptotiques. Bull. Soc. Math. Fr. 121(suppl.), 74 (1993) MathSciNetGoogle Scholar
  40. 40.
    Riesel, H., Göhl, G.: Some calculations related to Riemann’s prime number formula. Math. Comput. 24, 969–983 (1970) zbMATHGoogle Scholar
  41. 41.
    Runge, C., König, H.: Vorlesungen über numerisches Rechnen. Springer, Berlin (1924) CrossRefzbMATHGoogle Scholar
  42. 42.
    Sadovskii, M.V.: Quantum Field Theory. De Gruyter, Berlin (2013) CrossRefzbMATHGoogle Scholar
  43. 43.
    Schmelzer, T., Baillie, R.: Summing a curious, slowly convergent series. Am. Math. Mon. 115(6), 525–540 (2008) MathSciNetzbMATHGoogle Scholar
  44. 44.
    Shi, Z.: Liminf behaviours of the windings and Lévy’s stochastic areas of planar Brownian motion. In: Séminaire de Probabilités, XXVIII, pp. 122–137. Springer, Berlin (1994) CrossRefGoogle Scholar
  45. 45.
    Strang, G.: Learning from 100 numbers. Science 307(5709), 521–522 (2005) CrossRefGoogle Scholar
  46. 46.
    Townsend, A., Trefethen, L.N.: An extension of Chebfun to two dimensions. SIAM J. Sci. Comput. 35(6), C495–C518 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Trefethen, L.N.: Ten digit algorithms. A.R. Mitchell Lecture, 2005 Dundee Biennial Conf. on Numer. Anal., people.maths.ox.ac.uk/trefethen/essays.html (2005)
  48. 48.
    Trefethen, L.N.: Computing numerically with functions instead of numbers. Math. Comput. Sci. 1(1), 9–19 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Trefethen, L.N., Weideman, J.A.C.: The exponentially convergent trapezoidal rule. SIAM Rev. 56(3), 385–458 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Van Deun, J., Cools, R.: Algorithm 858: computing infinite range integrals of an arbitrary product of Bessel functions. ACM Trans. Math. Softw. 32(4), 580–596 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Wan, Z.: An algorithm to solve integer linear systems exactly using numerical methods. J. Symb. Comput. 41, 621–632 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Watson, G.N.: Theorems stated by Ramanujan. VIII: theorems on divergent series. J. Lond. Math. Soc. 4, 82–86 (1929) zbMATHGoogle Scholar
  53. 53.
    Watson, G.N.: Theorems stated by Ramanujan. XII: a singular modulus. J. Lond. Math. Soc. 6, 65–70 (1931) MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Weber, H.: Elliptische Functionen und algebraische Zahlen. Vieweg, Braunschweig (1891) zbMATHGoogle Scholar
  55. 55.
    Wiedemann, D.H.: Solving sparse linear equations over finite fields. IEEE Trans. Inf. Theory 32(1), 54–62 (1986) MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Wittgenstein, L.: On Certainty. Blackwell Sci., Oxford (1969) Google Scholar

Copyright information

© Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Zentrum Mathematik—M3Technische Universität MünchenMunichGermany

Personalised recommendations