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

Asymptotic approximations to the nodes and weights of Gauss-Hermite and Gauss-Laguerre quadratures

Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a standalone method of computation of Gaussian quadratures for high enough degrees, with Gaussian weights computed from asymptotic approximations for the orthogonal polynomials. We provide numerical evidence showing that for degrees greater than $100$ the asymptotic methods are enough for a double precision accuracy computation ($15$-$16$ digits) of the nodes and weights of the Gauss--Hermite and Gauss--Laguerre quadratures.Comment: Submitted to Studies in Applied Mathematic

Conical: an extended module for computing a numerically satisfactory pair of solutions of the differential equation for conical functions

Conical functions appear in a large number of applications in physics and engineering. In this paper we describe an extension of our module CONICAL for the computation of conical functions. Specifically, the module includes now a routine for computing the function ${{\rm R}}^{m}_{-\frac{1}{2}+i\tau}(x)$, a real-valued numerically satisfactory companion of the function ${\rm P}^m_{-\tfrac12+i\tau}(x)$ for $x>1$. In this way, a natural basis for solving Dirichlet problems bounded by conical domains is provided.Comment: To appear in Computer Physics Communication

Coupled h-m fracture interaction using fem with zero-thickness interface elements

Intensive hydraulic fracturing is a procedure employed for low permeability reservoir stimulation. This technique consists of generating a sequence of regularly spaced parallel fractures (multi-stage fracturing). The generation of a fracture involves the modification of the local stress state, and therefore, in the case of multi-stage fracturing, the propagation of a certain fracture can be affected by the injection sequence, as it has been observed with microseismicity monitoring [1]. This paper describes a study of this technique by means of the Finite Element Method with zero-thickness interface elements for the geo-mechanical modelling of discontinuities [2]. The technique consists in inserting interface elements in between standard elements to allow jumps in the displacement solution fields. For the mechanical problem, their kinematic constitutive variables are relative displacements, and the corresponding static variables are stress tractions. The relationship between variables is controlled via a fracture-based constitutive law with elasto-plastic structure [3]. Concerning the hydraulic problem, the interface formulation includes both the longitudinal flow (with a longitudinal conductivity parameter strongly dependent on the fracture aperture), as well as and the transversal flow across the element [4]. Previous work by the authors focused on the validation of the method, the analysis a single fracture plane problem [5, 6]. In this case the method is extended to allow free propagation of fractures in any direction, by means of inserting interface elements between all continuum elements. The results presented in this paper analyse the effect of material properties, in particular fracture characterization, in the propagation and the effect of different major to minor principal horizontal stress ratio, on the trajectory and interaction of the fractures

Jets as diagnostics of the circumstellar medium and the explosion energetics of supernovae: the case of Cas A

We present hydrodynamical models for the Cassiopeia A (Cas A) supernova remnant and its observed jet / counter-jet system. We include the evolution of the progenitor's circumstellar medium, which is shaped by a slow red supergiant wind that is followed by a fast Wolf-Rayet (WR) wind. The main parameters of the simulations are the duration of the WR phase and the jet energy. We find that the jet is destroyed if the WR phase is sufficiently long and a massive circumstellar shell has formed. We therefore conclude that the WR phase must have been short (a few thousand yr), if present at all. Since the actual jet length of Cas A is not known we derive a lower limit for the jet energy, which is ~10^{48} erg. We discuss the implications for the progenitor of Cas A and the nature of its explosion.Comment: 9 pages, 5 figures, ApJ accepted. Version with high resolution figures available at http://www.phys.uu.nl/~schure/CasA_jet.pd