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| Format: | Preprint |
| Veröffentlicht: |
2002
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| Online-Zugang: | https://arxiv.org/abs/math/0212035 |
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| _version_ | 1866912655400239104 |
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| author | Sokal, Alan D. |
| author_facet | Sokal, Alan D. |
| contents | 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, η(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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0212035 |
| institution | arXiv |
| publishDate | 2002 |
| record_format | arxiv |
| spellingShingle | Numerical Computation of \prod_{n=1}^\infty (1 - tx^n) Sokal, Alan D. Numerical Analysis 33F05 (Primary) 05A30, 11F20, 11P82, 33D99, 65D20, 82B23 (Secondary) 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, η(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. |
| title | Numerical Computation of \prod_{n=1}^\infty (1 - tx^n) |
| topic | Numerical Analysis 33F05 (Primary) 05A30, 11F20, 11P82, 33D99, 65D20, 82B23 (Secondary) |
| url | https://arxiv.org/abs/math/0212035 |