Log of factorial of large numbers

computes log10 of the factorial

This function computes log10 of the factorial. Unlike FACTORIAL, it is not limited to arguments less than 171.

Two methods are implemented, one based on the gamma function (thank you, John), as well as a direct sum. The user can choose between the two methods. The gamma method is superior and is therefore the default choice.

The function returns the log10 of the factorial, the mantissa and the exponent of the factorial, as well as a string representing the factorial.

For instance
[L,M,X,S] = logfactorial(1E6)
yields
L = 5.5657e+006 % log10(N!)
M = 8.2639 % mantissa
X = 5565708 % exponent
S = '8.26393e+5565708' % N! as a string

The function accepts scalars or arrays as inputs.

LOGFACTORIAL is the result of a seqeunce of improvements that were suggested by John D'Errico and Urs Schwarz in their discussion of another FEX file, LARGEFACTORIAL. LOGFACTORIAL is preferrable for two reasons:

1. The gamma-based code is much faster than the direct sum code, which is the only one implemented in LARGEFACTORIAL.

2. LARGEFACTORIAL returns a string as the first argument, which is not very convenient to perform further computations. LOGFACTORIAL returns the log10 as the first argument, so for instance 10.^logfactorial(...) does make sense.