CryptRndTest (version 1.2.7)

Strlng2: Stirling Number of The Second Kind

Description

Asymptotically computes natural logarithm of Stirling numbers of the second kind for large values of inputs by the approach of Bleick and Wang (1954) and Temme (1993). For small or moderate values of inputs, this function is not as precise as available functions.

Usage

Strlng2(n, k, log = TRUE)

Arguments

n

positive integer greater than zero.

k

positive integer between 1 and n.

log

if TRUE, natural logarithm of the Stirling numbers of the second kind is returned.

Value

Stirling.num

the corresponding Stirling number of the second kind to the pair (n,k).

Details

Due to the overflows in the calculation of large factorials, an asymptotic calculation of the Stirling numbers of the second kind is required. This function makes use of Lambert W function to calculate the Stirling numbers of the second kind with large values of n and k.

References

Bleick, W.W., Wang, P.C.C., Asymptotics of Stirling Numbers of the Second Kind. Proceedings of the American Mathematical Society (1974), 42(2), 575--580.

Temme, N.M., Asymptotic estimates of Stirling numbers. Studies in Applied Mathematics (1993), 89, 233--243.

See Also

See also Stirling2 function from the package copula.

Examples

# NOT RUN {
# When n = 10 and k = 4, exact value is 34105
gmp::Stirling2(10,4)           
Strlng2(10,4,log=FALSE)
# ---- Moderate values of n and k ----
# When n = 30 and k = 20, exact value is 581535955088511150
log(581535955088511150)-log(gmp::Stirling2(30,20))
log(581535955088511150)-Strlng2(30,20,log=TRUE)  
# ---- Large values of n and k ----
gmp::Stirling2(50,10)
Strlng2(50,10,log=FALSE) 
# }