Algorithm 939

Abstract

Methods and an algorithm for computing the generalized Marcum Q –function ( Q _μ ( x , y )) and the complementary function ( P _μ ( 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 noncentral chi-square or the noncentral 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).