Spouge's approximation

Last updated December 16, 2020

In mathematics, Spouge's approximation is a formula for computing an approximation of the gamma function. It was named after John L. Spouge, who defined the formula in a 1994 paper.^[1] The formula is a modification of Stirling's approximation, and has the form

\Gamma (z+1)=(z+a)^{z+{\frac {1}{2}}}e^{-z-a}\left(c_{0}+\sum _{k=1}^{a-1}{\frac {c_{k}}{z+k}}+\varepsilon _{a}(z)\right)

where a is an arbitrary positive integer and the coefficients are given by

{\begin{aligned}c_{0}&={\sqrt {2\pi }}\\c_{k}&={\frac {(-1)^{k-1}}{(k-1)!}}(-k+a)^{k-{\frac {1}{2}}}e^{-k+a}\qquad k\in \{1,2,\dots ,a-1\}.\end{aligned}}

Spouge has proved that, if Re(z) > 0 and a > 2, the relative error in discarding ε_a(z) is bounded by

a^{-{\frac {1}{2}}}(2\pi )^{-a-{\frac {1}{2}}}.

The formula is similar to the Lanczos approximation, but has some distinct features^[2]. Whereas the Lanczos formula exhibits faster convergence, Spouge's coefficients are much easier to calculate and the error can be set arbitrarily low. The formula is therefore feasible for arbitrary-precision evaluation of the gamma function. However, special care must be taken to use sufficient precision when computing the sum due to the large size of the coefficients c_k, as well as their alternating sign. For example, for a = 49, one must compute the sum using about 65 decimal digits of precision in order to obtain the promised 40 decimal digits of accuracy.

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References

↑ Spouge, John L. (1994). "Computation of the Gamma, Digamma, and Trigamma Functions" (PDF). SIAM Journal on Numerical Analysis. 31 (3): 931–000. doi:10.1137/0731050. JSTOR 2158038.
↑
- Pugh, Glendon (2004). An analysis of the Lanczos Gamma approximation (PDF) (PhD thesis).

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[1] Spouge, John L. (1994). "Computation of the Gamma, Digamma, and Trigamma Functions" (PDF). SIAM Journal on Numerical Analysis. 31 (3): 931–000. doi:10.1137/0731050. JSTOR 2158038.

[2] 
Pugh, Glendon (2004). An analysis of the Lanczos Gamma approximation (PDF) (PhD thesis).

[mwRA] Pugh, Glendon (2004). An analysis of the Lanczos Gamma approximation (PDF) (PhD thesis).

[1]

[2]

Spouge's approximation

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References