Abstract
The Euler–Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum \({\sum \nolimits _{k=0}^{n-1} f(k)}\) of values of a function f by a linear combination of a corresponding integral of f and values of its higher-order derivatives \(f^{(j)}\). An alternative (Alt) summation formula is proposed, which approximates the sum by a linear combination of integrals only, without using high-order derivatives of f. Explicit and rather easy to use bounds on the remainder are given. Extensions to multi-index summation and to sums over lattice polytopes are indicated. Applications to summing possibly divergent series are presented. The Alt formula will in most cases outperform, or greatly outperform, the EM summation formula in terms of the execution time and memory use. One of the advantages of the Alt calculations is that, in contrast with the EM ones, they can be almost completely parallelized. Illustrative examples are given. In one of the examples, where an array of values of the Hurwitz generalized zeta function is computed with high accuracy, it is shown that both our implementation of the EM summation formula and, especially, the Alt formula perform much faster than the built-in Mathematica command HurwitzZeta[].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bailey, D.H., Borwein, J.M.: Experimental mathematics: recent developments and future outlook. In: Engquist, B., Schmid, W. (eds.) Mathematics Unlimited–2001 and Beyond, pp. 51–66. Springer, Berlin (2001)
Bornemann, F.: The SIAM 100-digit challenge: a decade later. Inspirations, ramifications, and other eddies left in its wake. Jahresber. Dtsch. Math.-Ver. 118(2), 87–123 (2016). https://doi.org/10.1365/s13291-016-0137-2
Bornemann, F., Laurie, D., Wagon, S., Waldvogel, J.: The SIAM 100-Digit Challenge. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2004). https://doi.org/10.1137/1.9780898717969. (A study in high-accuracy numerical computing, With a foreword by David H. Bailey)
Brent, R.P.: Multiple-precision zero-finding methods and the complexity of elementary function evaluation (2010). arXiv:1004.3412, see also http://maths-people.anu.edu.au/~brent/pub/pub028.html
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). https://doi.org/10.1007/s00025-010-0083-8
Candelpergher, B., Gadiyar, H.G., Padma, R.: Ramanujan summation and the exponential generating function \(\sum ^\infty_{k=0}{z^k\over k!}\zeta ^{\prime }(-k)\). Ramanujan J. 21(1), 99–122 (2010). https://doi.org/10.1007/s11139-009-9166-0
Fillebrown, S.: Faster computation of Bernoulli numbers. J. Algorithms 13(3), 431–445 (1992). https://doi.org/10.1016/0196-6774(92)90048-H
Gould, H.W.: Explicit formulas for Bernoulli numbers. Am. Math. Mon. 79, 44–51 (1972). https://doi.org/10.2307/2978125
Harvey, D.: A multimodular algorithm for computing Bernoulli numbers. Math. Comput. 79(272), 2361–2370 (2010). https://doi.org/10.1090/S0025-5718-2010-02367-1
Ierley, G.: Private communication (2017)
Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory Graduate Texts in Mathematics, vol. 84, 2nd edn. Springer, New York (1990). https://doi.org/10.1007/978-1-4757-2103-4
Kleinert, H.: Path integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th edn. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2009). https://doi.org/10.1142/9789814273572
Knopp, K.: Theory and Application of Infinite Series. Blackie, London (1951)
Knuth, D.E.: Euler’s constant to \(1271\) places. Math. Comput. 16, 275–281 (1962)
MathOverflow: a representation of the Bernoulli numbers. http://mathoverflow.net/questions/252404/a-representation-of-the-bernoulli-numbers#252410 (2016)
Pinelis, I.: An alternative to the Euler–Maclaurin formula: approximating sums by integrals only (2015). arXiv:1511.03247 [math.CA]
Pinelis, I.: Approximating sums by integrals only: multiple sums and sums over lattice polytopes (2017). arXiv:1705.09159 [math.CA]
Sadovskii, M.V.: Quantum Field Theory, De Gruyter Studies in Mathematical Physics, vol. 17. De Gruyter, Berlin (2013). https://doi.org/10.1515/9783110270358
Titchmarsh, E.C.: The Theory of Functions. Oxford University Press, Oxford (1958). (Reprint of the second (1939) edition)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pinelis, I. An alternative to the Euler–Maclaurin summation formula: approximating sums by integrals only. Numer. Math. 140, 755–790 (2018). https://doi.org/10.1007/s00211-018-0978-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-018-0978-y