The procedure _ZetaX[1] evaluates Riemann's Zeta function ζ(x) for real arguments using different algorithms and internal tables[2], utilizing the x87 FPU's 80-bit registers.[3] Evaluation time varies between 0.25 µs (for fetching a value from the table) and 95 µs (for calculating the function at x ≈ 1). The relative error ε is kept below 10-18.ζ(|x| ≤ 16)
The characteristics for negative arguments is emphasized in the following graph:ζ(x < 0)
At x = 0 the value of ζ is exactly -½. Near zero the procedure converges relatively slowly (83 µs) but the error can be kept small, nonetheless:
ζ(x ≈ 0)
| 1 |
The proprietary procedure _ZetaX can be called, through appropriate interfaces, by any standard X86/87 compiler (e.g. FORTRAN, C, etc). |
| 2 |
Handbook of Mathematical Functions, Abramowitz & Stegun, Dover Publications, fifth printing, p807 |
| 3 |
Calculations were performed on an AMD Athlon XP 1700+, 1.47 GHz, microprocessor. Computer programs were written in Assembly (MASM 5.1). |
Copyright Dan Baruth © 2006.
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