Abstract
Methods and an algorithm for computing the generalized MarcumQ–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 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 10−12can 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 10−11forA= 1000 and close to 5 × 10−11forA= 10000).
Highlights
We define the generalized Marcum Q−function by using the integral representation Qμ(x, y) = x (1−μ) +∞ t (μ−1) e−t−x Iμ−1 √ 2 xt dt, (1) y
For μ = 1 this coincides with the original definition of the Marcum Q−function [Marcum 1960]
The generalized Marcum Q−function is an important function used in radar detection and communications, see [Marcum 1960; Rice 1968; Robertson 1976]
Summary
We define the generalized Marcum Q−function by using the integral representation. ACM Transactions on Mathematical Software, Vol , No , Article 0, Publication date: 0. For μ = 1 this coincides with the original definition of the Marcum Q−function [Marcum 1960] This is the notation used, for instance, in the MATLAB built-in function marcumq. The generalized Marcum Q−function is an important function used in radar detection and communications, see [Marcum 1960; Rice 1968; Robertson 1976]. In this field, μ is the number of independent samples of the output of a square-law detector. Our tests reveal some bugs in certain parameter regions when computing the Marcum Q−function using MATLAB
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