Computation of the Marcum Q-function

Methods and an algorithm for computing the generalized Marcum $Q-$function ($Q_{\mu}(x,y)$) and the complementary function ($P_{\mu}(x,y)$) 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 $(x,y,\mu) \in [0,\,A]\times [0,\,A]\times [1,\,A]$, $A=200$, while for larger parameters the accuracy decreases (close to $10^{-11}$ for $A=1000$ and close to $5\times 10^{-11}$ for $A=10000$).Comment: Accepted for publication in ACM Trans. Math. Soft

Algorithm 939: Computation of the Marcum Q-function

Methods and an algorithm for computing the generalized Marcum Q.function (QƒÊ(x, y)) and the complementary function (PƒÊ(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, ƒÊ) ¸ [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)

On the computation of moments of the partial non-central chi-squared distribution function

Properties satisfied by the moments of the partial non-central chi-square distribution function, also known as Nuttall Q-functions, and methods for computing these moments are discussed in this paper. The Nuttall Q-function is involved in the study of a variety of problems in different fields, as for example digital communications.Comment: 6 page

Some Useful Collective Properties of Bessel, Marcum Q-Functions and Laguerre Polynomials

Special functions have been used widely in many problems of applied sciences. However, there are considerable numbers of problems in which exact solutions could not be achieved because of undefined sums or integrals involving special functions. These handicaps force researchers to seek new properties of special functions. Many problems that could not be solved so far would be solved by means of these efforts. Therefore in this article, we derived some useful properties and interrelations of each others of Bessel functions, Marcum Q-functions and Laguerre polynomials

Improved limits on dark matter annihilation in the Sun with the 79-string IceCube detector and implications for supersymmetry

We present an improved event-level likelihood formalism for including neutrino telescope data in global fits to new physics. We derive limits on spin-dependent dark matter-proton scattering by employing the new formalism in a re-analysis of data from the 79-string IceCube search for dark matter annihilation in the Sun, including explicit energy information for each event. The new analysis excludes a number of models in the weak-scale minimal supersymmetric standard model (MSSM) for the first time. This work is accompanied by the public release of the 79-string IceCube data, as well as an associated computer code for applying the new likelihood to arbitrary dark matter models.Comment: 24 pages, 8 figs, 1 table. Contact authors: Pat Scott & Matthias Danninger. Likelihood tool available at http://nulike.hepforge.org. v2: small updates to address JCAP referee repor

Improved limits on dark matter annihilation in the Sun with the 79-string IceCube detector and implications for supersymmetry

We present an improved event-level likelihood formalism for including neutrino telescope data in global fits to new physics. We derive limits on spin-dependent dark matter-proton scattering by employing the new formalism in a re-analysis of data from the 79-string IceCube search for dark matter annihilation in the Sun, including explicit energy information for each event. The new analysis excludes a number of models in the weak-scale minimal supersymmetric standard model (MSSM) for the rst time. This work is accompanied by the public release of the 79-string IceCube data, as well as an associated computer code for applying the new likelihood to arbitrary dark matter models.M.G. Aarten ... G.C. Hill ... S. Robertson ... B.J. Whelan ... et al

Exact Results for the Distribution of the Partial Busy Period for a Multi-Server Queue

Exact explicit results are derived for the distribution of the partial busy period of the M/M/c multi-server queue for a general number of servers. A rudimentary spectral method leads to a representation that is amenable to efficient numerical computation across the entire ergodic region. An alternative algebraic approach yields a representation as a finite sum of Marcum Q-functions depending on the roots of certain polynomials that are explicitly determined for an arbitrary number of servers. Asymptotic forms are derived in the limit of a large number of servers under two scaling regimes, and also for the large-time limit. Connections are made with previous work. The present work is the first to offer tangible exact results for the distribution when the number of servers is greater than two