;ò ¿£ðEc@sdZdZdkZeideiƒZd„Zeid„Zeieƒeid„Z eei d„Z dd d d d d gdeeid„Z ei eieidƒd„Z d„Zdei d„Zdeid„Z[[d„Zd„Zd„ZdZdS(sbApproximations to factorial (and its natural logarithm) Stirling's formula gives an approximation to factorial: log(n!) = (n+.5) * log(n) - n + .5 * log(2*pi) -1/12/n with errors of order 1/n/n. Lanczos's formula does pretty well too - see lngamma() below. Note: the gamma function attains a local minimum, 0.885603194411, at about 1.4616321 and takes the value sqrt(pi)/2 = .886 at 3/2. It is worth knowing that, once n! ← n is extended to the whole complex plane, we find: (-n)!.n!.sinc(n.pi) = 1 for each complex non-integer n (and for n=0). Thus factorial has a simple pole at each negative integer. The HAKMEM papers e.g. at http://www.inwap.com/pdp10/hbaker/hakmem/hakmem.html offer n! * (-n)! * sinc(pi*n) = 1 (2*pi)**(.5*(n-1)) * (n*z)! / n**(n*z+.5) = z!*(z-1./n)!*...*(z-(n-1)/z)! Stirling's formula gives an approximation to factorial: log(n!) = (n+.5) * log(n) - n + .5 * log(2*pi) -1/12/n with errors of order 1/n/n. I have another scribble elsewhere which claims n! = n**(n+.5) * exp(-n) * (2*pi)**.5 * (1 +1/12/n +1/288/n/n +...) which would appear to be disagreeing about the sign of the 1/12/n term. It also suggests the tail will have a factor of 12 for each factor of n. Lanczos's formula does pretty well too - see lngamma() below. Note: the gamma function attains a local minimum, 0.885603194411, at about 1.4616321 and takes the value sqrt(pi)/2 = .886 at 3/2. It is worth knowing that, once n! ← n is extended to the whole complex plane, we find: (-n)!.n!.sinc(n.pi) = 1 for each complex non-integer n (and for n=0). Thus factorial has a simple pole at each negative integer. For chose(N,m) = N!/m!/(N-m)! Stirling implies the approximation log(chose(n+m,m)) = (n+m+.5)*log(n+m) -(n+.5)*log(n) -(m+.5)*log(m) -.5*log(2*pi) +(1/n +1/m -1/(n+m))/12 = .5 * log((1/n +1/m)/2/pi) +n*log(1+m/n) +m*log(1+n/m) +(1+n/m+m/n)/(n+m)/12 s7 $Id: stirling.py,v 1.4 2007/03/09 00:01:03 eddy Exp $ NicCswd}xf|djoXy|||df\}}Wq tj o't|ƒ||df\}}q Xq W|SdS(Ni(sresultsns OverflowErrorslong(snsresult((s*/home/eddy/.sys/py/study/maths/stirling.pys factorial2s *cCsCd}x2|djo$|||ƒ|df\}}q W|SdS(Nf0.0i(sresultsnslog(snslogsresult((s*/home/eddy/.sys/py/study/maths/stirling.pys lnfactorial<s  %cCs*|d||ƒ||dd|SdS(Nf0.5f1.0i (snslogsbase(snsbaseslog((s*/home/eddy/.sys/py/study/maths/stirling.pys lnStirlingCscCs@|o |||dd|ƒ}nt|d|ƒ|SdS(Nf1.0i f0.5(sns__ssexpspow(sns__ssexp((s*/home/eddy/.sys/py/study/maths/stirling.pysStirlingFs f76.180091729471457f-86.505320329416776f24.014098240830911f-1.231739572450155f0.001208650973866179f5.3952393849530003e-06f1.0000000001900149cCsg|d}|d||ƒ|}x,|D]$}|||d|f\}}q)W||||ƒSdS(sFLanczos's approximation to log(Gamma). This works by taking Stirling's formula, with Gamma(1+n) = n!, and putting in corrections for the first few poles in Gamma: for some whole g, N: Gamma(z+1) = pow(z +g +.5, z +.5)*exp(-(z +g +.5)) *sqrt(2*pi) *series with series = a + b/(z+1) +... +c/(z+N) for suitable constants a, b, ..., c Substitute x = z+1 to turn this into Gamma(x) = pow(x +g -.5, x -.5) *exp(-(x +g -.5)) *sqrt(2*pi) *series with series = a + sum(: coefficients[i]/(x+i) ← i :) for suitable a. Interestingly, Numerical Recipies (without explaining itself) uses Gamma(z) = Gamma(z+1)/z rather than substitution, as here; it also asserts that the error in the above requires a correction of less than 2e-10 in the series, for z with positive real part. f4.5f0.5iN(sxsbaseslogs coefficientsscssumsscale(sxs coefficientsssumsscaleslogscsbase((s*/home/eddy/.sys/py/study/maths/stirling.pyslngammaLs "icCsd}y |i}Wntj o |}nXyAx:|djo,|d|d|d|f\}}}q5WWntj otd‚nXx6|djo(|d|df\}}||}q’W|ddfjo|Sn|djo ||Sn||t|ƒƒSdS(Nif1.0s9The Gamma function has poles at all non-positive integersif1.5( sresultsxsrealsrsAttributeErrorsZeroDivisionErrorsspecialsexpslngamma(sxsexpsspecialsrsresult((s*/home/eddy/.sys/py/study/maths/stirling.pysgammans(   1  cCs5yt|dƒSWntj otd‚nXdS(Nis:The factorial function has a pole at each negative integer(sgammasxsZeroDivisionError(sx((s*/home/eddy/.sys/py/study/maths/stirling.pys gactorialsicCs¢x4|djo&||d||df\}}qWx.|djo |d}||d|}q:W|djo|Sn||||td|ƒ ƒSdS(Nif1.0i(snsscalesxsexpslngamma(sxsnsscalesexp((s*/home/eddy/.sys/py/study/maths/stirling.pysexpterm‡s '   f1.0cCst||d|dƒSdS(Nif2.0(sexptermspisradiussdim(sdimsradiusspi((s*/home/eddy/.sys/py/study/maths/stirling.pyssphere‘scCs(t|ƒ}tt|ƒ|ƒ|SdS(N(s factorialsxsfsabssStirling(sxsf((s*/home/eddy/.sys/py/study/maths/stirling.pyserror—s cCs(t|ƒ}tt|ƒ|ƒ|SdS(N(s lnfactorialsxsfsabss lnStirling(sxsf((s*/home/eddy/.sys/py/study/maths/stirling.pyserrorln›s cCs,t|ƒ}ttd|ƒ|ƒ|SdS(Ni(s lnfactorialsxsfsabsslngamma(sxsf((s*/home/eddy/.sys/py/study/maths/stirling.pysgerrorŸs sî $Log: stirling.py,v $ Revision 1.4 2007/03/09 00:01:03 eddy Note contrary rumours. Revision 1.3 2005/01/17 22:27:48 eddy Added comment taken from HAKMEM note Revision 1.2 2003/10/11 15:15:38 eddy Made gamma cope better with complex (comparison has changed behaviour; it's now not OK to ask whether greater/less than one), told it about special case at 1.5 (since that happens to be analytically exact, and relevant to sphere). Revision 1.1 2003/09/21 16:29:14 eddy Initial revision (s__doc__s_rcs_id_smathssqrtspis roottwopis factorialslogs lnfactorials lnStirlingsexpsStirlingslngammasgammas gactorialsexptermssphereserrorserrorlnsgerrors _rcs_log_(s lnfactorials_rcs_id_s roottwopiserrorlns factorialsStirlings lnStirlingsgerrorsexptermsspheresgammaserrors _rcs_log_slngammas gactorialsmath((s*/home/eddy/.sys/py/study/maths/stirling.pys?)s"  +""