8,001 research outputs found
Computation of the Marcum Q-function
Methods and an algorithm for computing the generalized Marcum function
() and the complementary function () 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 can be obtained
in the parameter region ,
, while for larger parameters the accuracy decreases (close to
for and close to for ).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 the
asymptotic methods are enough for a double precision accuracy computation
(- 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 , a
real-valued numerically satisfactory companion of the function for . 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
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