Algorithm to calculate modified Bessel function of the first kind

Question

Could you tell me a fast and accurate method to calcuate BesselK(mu,z), BesselI(mu,z), where mu is real number?

Answer 1

There are several ways to represent the Bessel functions and none of them are easy to do mathematically or closed-form. However, this paper seems to be a recent approach to evaluating them efficiently:

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.120.6055

It seems to be a better way than the old-school method from the 1950s:

http://www.ams.org/journals/mcom/1959-13-066/S0025-5718-1959-0105794-5/S0025-5718-1959-0105794-5.pdf

Answer 2

I haven't had time to extensively and exhaustively test it myself apart from seeing that it works, but the Colt library for Java includes integer-order Bessel function in the cern.jet.math package.

https://dst.lbl.gov/ACSSoftware/colt/

I also highly endorse the Digital Library of Mathematical Functions:

http://dlmf.nist.gov/