Methods and an algorithm for computing the generalized Marcum Q−function
(Qμ(x,y)) and the complementary function (Pμ(x,y)) are described.
These functions appear in problems of different technical and scientific areas
such as, for example, radar detection and communications, statistics and
probability theory, where they are called the non-central chi-square or the non
central gamma cumulative distribution functions.
The algorithm for computing the Marcum functions combines different methods
of evaluation in different regions: series expansions, integral
representations, asymptotic expansions, and use of three-term homogeneous
recurrence relations. A relative accuracy close to 10−12 can be obtained
in the parameter region (x,y,μ)∈[0,A]×[0,A]×[1,A],
A=200, while for larger parameters the accuracy decreases (close to
10−11 for A=1000 and close to 5×10−11 for A=10000).Comment: Accepted for publication in ACM Trans. Math. Soft