  I present and analyze a quadratically convergent algorithm for computing the
infinite product \prod_{n=1}^\infty (1 - tx^n) for arbitrary complex t and x
satisfying |x| < 1, based on the identity \prod_{n=1}^\infty (1 - tx^n)
  = \sum_{m=0}^\infty {(-t)^m x^{m(m+1)/2} \over (1-x)(1-x^2) ... (1-x^m)} due
to Euler. The efficiency of the algorithm deteriorates as |x| \uparrow 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, \eta(iy), that
is sharp at the two endpoints y=0,\infty and is accurate to within 9.1% over
the entire interval 0 < y < \infty.

Numerical Analysis