Short addition sequences for theta functions

Andreas Enge 1 William Hart 2 Fredrik Johansson 1
1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : The main step in numerical evaluation of classical Sl2 (Z) modular forms and elliptic functions is to compute the sum of the first N nonzero terms in the sparse q-series belonging to the Dedekind eta function or the Jacobi theta constants. We construct short addition sequences to perform this task using N + o(N) multiplications. Our constructions rely on the representability of specific quadratic progressions of integers as sums of smaller numbers of the same kind. For example, we show that every generalised pentagonal number c 5 can be written as c = 2a + b where a, b are smaller generalised pentagonal numbers. We also give a baby-step giant-step algorithm that uses O(N/ log r N) multiplications for any r > 0, beating the lower bound of N multiplications required when computing the terms explicitly. These results lead to speed-ups in practice.
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Submitted on : Wednesday, March 7, 2018 - 4:34:46 PM
Last modification on : Monday, May 20, 2019 - 2:30:24 PM
Long-term archiving on : Friday, June 8, 2018 - 2:44:04 PM


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  • HAL Id : hal-01355926, version 2
  • ARXIV : 1608.06810



Andreas Enge, William Hart, Fredrik Johansson. Short addition sequences for theta functions. Journal of Integer Sequences, University of Waterloo, 2018, 18 (2), pp.1-34. ⟨hal-01355926v2⟩



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