Numerical Computations

Abstract

I present and analyze a quadratically convergent algorithm for computing the infinite product ∞ n=1 (1 − tx n) for arbitrary complex t and x satisfying |x| < 1, based on the identity ∞ n=1 (1 − tx n) = ∞ m=0 (−t) m x m(m+1)/2 (1 − x)(1 − x 2) · · · (1 − x m) due to Euler. The efficiency of the algorithm deteriorates as |x| ↑ 1, but much more slowly than in previous algorithms. The key lemma is a two-sided bound on the Dedekind eta function at pure imaginary argument, η(iy), that is sharp at the two endpoints y = 0, ∞ and is accurate to within 9.1% over the entire interval 0 < y < ∞.