arXiv subjects

N. M. Temme

Publications and source records attributed to N. M. Temme.

22 records · Page 2Linked to original sources

arXiv2014

Computation of a numerically satisfactory pair of solutions of the differential equation for conical functions of non-negative integer orders

We consider the problem of computing satisfactory pairs of solutions of the differential equation for Legendre functions of non-negative integer order $μ$ and degree $-\frac12+iτ$, where $τ$ is a non-negative real parameter. Solutions of this equation are the conical functions ${\rm{P}}^μ_{-\frac12+iτ}(x)$ and ${Q}^μ_{-\frac12+iτ}(x)$, $x>-1$. An algorithm for computing a numerically satisfactory pair of solutions is already available when $-1 1$, the function $\Re\left\{e^{-iπμ} {Q}^μ_{-\frac{1}{2}+iτ}(x) \right\}$. The proposed algorithm allows the computation of the function on a large parameter domain without requiring the use of extended precision arithmetic.

math.CA↗

arXiv2014

Recent software developments for special functions in the Santander-Amsterdam project

We give an overview of published algorithms by our group and of current activities and future plans. In particular, we give details on methods for computing special functions and discuss in detail two current lines of research. Firstly, we describe the recent developments for the computation of central and non-central chi-square cumulative distributions (also called Marcum Q-functions), and we present a new quadrature method for computing them. Secondly, we describe the fourth-order methods for computing zeros of special functions recently developed, and we provide an explicit example for the computation of complex zeros of Bessel functions. We end with an overview of published software by our group for computing special functions.

math.CA↗

arXiv2013

Computation of the Marcum Q-function

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 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 $(x,y,μ) \in [0,\,A]\times [0,\,A]\times [1,\,A]$, $A=200$, while for larger parameters the accuracy decreases (close to $10^{-11}$ for $A=1000$ and close to $5\times 10^{-11}$ for $A=10000$).

cs.MS↗

arXiv2006

On the Temporal Order of First-Passage Times in One-Dimensional Lattice Random Walks

A random walk problem with particles on discrete double infinite linear grids is discussed. The model is based on the work of Montroll and others. A probability connected with the problem is given in the form of integrals containing modified Bessel functions of the first kind. By using several transformations simpler integrals are obtained from which for two and three particles asymptotic approximations are derived for large values of the parameters. Expressions of the probability for $n$ particles are also derived.

math.CA↗