Short addition sequences for theta functions

Article, Preprint English OPEN
Enge , Andreas ; Hart , William ; Johansson , Fredrik (2018)
  • Publisher: University of Waterloo
  • Subject: [ MATH.MATH-NT ] Mathematics [math]/Number Theory [math.NT] | Mathematics - Number Theory

International audience; 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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