|
|
| Line 1: |
Line 1: |
| In [[mathematics]], '''Spouge's approximation''' is a formula for the [[gamma function]] due to John L. Spouge in 1994.<ref>{{cite doi|10.1137/0731050}}</ref> The formula is a modification of [[Stirling's approximation]], and has the form
| | The author is called Araceli Gulledge. Managing people is what I do in my working day job. Alabama has always been his house. Bottle tops collecting is the only pastime his wife doesn't approve of.<br><br>Also visit my blog post :: [http://Www.Dienoobs-Css.de/index.php?mod=users&action=view&id=17012 http://Www.Dienoobs-Css.de] |
| | |
| :<math>\Gamma(z+1) = (z+a)^{z+1/2} e^{-(z+a)} \left[ c_0 + \sum_{k=1}^{a-1} \frac{c_k}{z+k} + \varepsilon_a(z) \right]</math>
| |
| | |
| where ''a'' is an arbitrary positive integer and the coefficients are given by
| |
| | |
| :<math>c_0 = \sqrt{2 \pi}\,</math> | |
| :<math>c_k = \frac{(-1)^{k-1}}{(k-1)!} (-k+a)^{k-1/2} e^{-k+a} \quad k\in\{1,2,\dots, a-1\}.</math> | |
| | |
| Spouge has proved that, if Re(''z'') > 0 and ''a'' > 2, the relative error in discarding ε<sub>''a''</sub>(''z'') is bounded by
| |
| | |
| :<math>\,a^{-1/2} (2 \pi)^{-(a+1/2)}.</math> | |
| | |
| 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 arithmetic|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 <math> c_k </math>, as well as their alternating sign. For example, for ''a''=49, you 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}}
| |
| | |
| ==External links==
| |
| * [http://en.literateprograms.org/Gamma_function_with_Spouge's_formula_(Mathematica) Gamma function with Spouge's formula] - [[Mathematica]] implementation at LiteratePrograms
| |
| | |
| [[Category:Gamma and related functions]]
| |
| [[Category:Computer arithmetic algorithms]]
| |
| | |
| {{numtheory-stub}}
| |
The author is called Araceli Gulledge. Managing people is what I do in my working day job. Alabama has always been his house. Bottle tops collecting is the only pastime his wife doesn't approve of.
Also visit my blog post :: http://Www.Dienoobs-Css.de