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Re: Stirling Approx to N! for large Number?

by tall_man (Parson)
on Mar 25, 2005 at 15:46 UTC ( #442358=note: print w/ replies, xml ) Need Help??


in reply to Stirling Approx to N! for large Number?

There was a more accurate formula than Stirling's in tilly's post. Wolfram's Mathematica attributes it to Gosper:

sub factorial_gosper { # sqrt(2*n*PI + 1/3)*n**n/exp(n) my $n = shift; return exp($n*log($n) - $n + 0.5*log(2*$n*PI + 1.0/3.0)); }

This gives better accuracy for small numbers. For 8!, I get 40034.48, compared to the correct 40320.

There are several problems with your implemenation of Stirling's formula. For one thing, log(e^(-n)), which you give as -e*log(n). Since the log is just the exponent base e, the correct answer is -n.

Tilly's formula is closer to being right; there was only one parenthesis mistake. I corrected that and tried them all in a modified version of your code:

#!/usr/bin/perl -w use strict; + my $h =8; my $pure = factorial_pure($h); my $stirling = factorial_stirling($h); my $real_stirling = real_stirling($h); my $gosper = factorial_gosper($h); + print "PURE : $pure\n"; print "STIRLING : $stirling\n"; print "REAL : $real_stirling\n"; print "GOSPER : $gosper\n"; + #------Subroutines------------ + sub factorial_stirling{ my $n = shift; use constant PI => 4*atan2 (1,1); use constant e => exp(1); + my $log_nfact = $n * log ($n) - e* log($n) + 0.5 * (log(2*PI)+log($n)); + # Below is Tilly's suggestion, with less accuracy # $n * log ($n) - ($n) + (0.5 * (log(2+ log(PI))+log($n))); + return exp($log_nfact); } + + sub factorial_pure { + my ($n,$res) = (shift,1); return undef unless $n>=0 and $n == int($n); $res *= $n-- while $n>1; return $res; + } + sub real_stirling { my $n = shift; my $log_nfact = $n*log($n) - $n + 0.5*(log(2) + log(PI)+ log($n)); return exp($log_nfact); } + sub factorial_gosper { # sqrt(2*n*PI + 1/3)*n**n/exp(n) my $n = shift; return exp($n*log($n) - $n + 0.5*log(2*$n*PI + 1.0/3.0)); } +
I get the answers below:
PURE : 40320 STIRLING : 417351.253768542 REAL : 39902.3954526567 GOSPER : 40034.4823215513


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