FACTORIALRATIO

Ratio of factorials, as in (a!b!c!...) / (d!e!f!g!...) (version 2.0, oct 2012)

FACTORIALRATIO - ratio of factorial numbers

FR = FACTORIALRATIO (A,B) calculates the ratio between products of
factorials specified by A and B. A and B are lists of positive
integers, with N and M elements, respectively. The number of values in
each list may differ. The factorial ratio is based on the following formula:

(A1! * A2! * ... * AN!)
FR = -------------------------
(B1! * B2! * ... * BM!)

However, an approach is taken that increases the accuracy when A or B
contain larger values, as mentioned in the help of FACTORIAL.

Examples:

f1 = factorialratio([10 8 6],[9 6 5]) ;
f2 = 10 * 8 * 7 * 6 ; % by hand
f3 = (factorial(10) * factorial(8) * factorial(6)) ./ ...
(factorial(9) * factorial(6) * factorial(5)) ;
disp('f1, f2, f3 = ') ; disp([f1 f2 f3])


% Also for larger values FACTORIALRATIO works correctly
f1 = factorialratio(150,143) ;
f2 = prod(144:150) ; % by hand
% where FACTORIAL lacks precision
f3 = factorial(150) ./ factorial(143) ;
disp('f1, f2, f3 = ') ; disp([f1 f2 f3])
disp(' Differences with f2 = ') ; disp([f1-f2 f2-f2 f3-f2])

% and even for extreme numbers things turn out well
f1 = factorialratio(30000,29998) ;
f2 = (30000 * 29999) ; % by hand
% whereas calls to FACTORIAL fail miserably ...
f3 = factorial(30000) ./ factorial(29998) ;
disp('f1, f2, f3 = ') ; disp([f1 f2 f3])

Note that, for instance, factorialratio(200,1) also fails, of course.
For such occasions other engines have to be used.

See also prod, factorial, gammaln