Essays/Stirling's Approximation
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Stirling's formulae provide approximations to the gamma or factorial functions. They are given in Abramowitz and Stegun Section 6.1.
sbase =: %:@(2p1&%) * %&1x1 ^ ] scorr =: 1 1r12 1r288 _139r51840 _571r2488320&p.@% stirlg=: sbase * scorr NB. 6.1.37 stirlf=: ^@(1r12&%) * %:@(2p1&*) * %&1x1 ^ ] NB. 6.1.38
stirlf approximates the factorial function while the more accurate stirlg approximates the gamma function.
g =: stirlg |@-.@% !@<: NB. relative difference between stirlg and !@<: f =: stirlf |@-.@% ! NB. relative difference between stirlf and ! (stirlf , stirlg@>: , !) 10 3.62881e6 3.6288e6 3.6288e6 f 10 2.7699e_6 g 11 4.84055e_9 (stirlf , stirlg@>: , !) 5j4 20.7062j23.4239 20.7061j23.4242 20.7061j23.4242 f 5j4 1.05637e_5 g 6j4 4.00025e_8
See also
Contributed by Roger Hui.