Spouge's Approximation

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. The formula is a modification of Stirling's approximation, and has the form

:$\Gamma(z+1) = (z+a)^ e^ \left( c_0 + \sum_^ \frac + \varepsilon_a(z) \right)$

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

:$\begin c_0 &= \sqrt\\ c_k &= \frac (-k+a)^ e^ \qquad k\in\. \end$

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

:$a^ (2 \pi)^.$

The formula is similar to the Lanczos approximation, but has some distinct features.* 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.

See also

* Stirling's approximation
* Lanczos approximation

References

{{reflist

Gamma and related functions
Computer arithmetic algorithms