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

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

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

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 $\mu$ and degree $-\frac12+i\tau$, where $\tau$ is a non-negative real parameter. Solutions of this equation are the conical functions ${\rm{P}}^{\mu}_{-\frac12+i\tau}(x)$ and ${Q}^{\mu}_{-\frac12+i\tau}(x)$, $x>-1$. An algorithm for computing a numerically satisfactory pair of solutions is already available when $-1<x<1$ (see \cite{gil:2009:con}, \cite{gil:2012:cpc}).In this paper, we present a stable computational scheme for a real valued numerically satisfactory companion of the function ${\rm{P}}^{\mu}_{-\frac12+i\tau}(x)$ for $x>1$, the function $\Re\left\{e^{-i\pi \mu} {{Q}}^{\mu}_{-\frac{1}{2}+i\tau}(x) \right\}$. The proposed algorithm allows the computation of the function on a large parameter domain without requiring the use of extended precision arithmetic.Comment: To be published in Numerical Algoritm

The χ2\chi^2 - divergence and Mixing times of quantum Markov processes

We introduce quantum versions of the $\chi^2$-divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes. An approach similar to the one presented in [1-3] for classical Markov chains is taken to bound the trace-distance from the steady state of a quantum processes. A strict spectral bound to the convergence rate can be given for time-discrete as well as for time-continuous quantum Markov processes. Furthermore the contractive behavior of the $\chi^2$-divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm. In this context we analyse different versions of quantum detailed balance and, finally, give a geometric conductance bound to the convergence rate for unital quantum Markov processes

Evanescence in Coined Quantum Walks

In this paper we complete the analysis begun by two of the authors in a previous work on the discrete quantum walk on the line [J. Phys. A 36:8775-8795 (2003) quant-ph/0303105 ]. We obtain uniformly convergent asymptotics for the "exponential decay'' regions at the leading edges of the main peaks in the Schr{\"o}dinger (or wave-mechanics) picture. This calculation required us to generalise the method of stationary phase and we describe this extension in some detail, including self-contained proofs of all the technical lemmas required. We also rigorously establish the exact Feynman equivalence between the path-integral and wave-mechanics representations for this system using some techniques from the theory of special functions. Taken together with the previous work, we can now prove every theorem by both routes.Comment: 32 pages AMS LaTeX, 5 figures in .eps format. Rewritten in response to referee comments, including some additional references. v3: typos fixed in equations (131), (133) and (134). v5: published versio

Mass-Movement Causes: Changes in Slope Angle

This chapter discusses and illustrates how changes in slope angle can cause mass movement. Several processes can cause removal of lateral or underlying support of a slope, and most of the time multiple processes are acting together on a landscape. Slow and sudden processes causing changes in slope angle are differentiated, and several examples and illustrations of each are given. In addition, this chapter reviews current literature on landscape evolution modeling in which researchers try to incorporate these geomorphological processes in the analysis and simulation of current and future landscapes

Quantitative Modeling of Landscape Evolution

This chapter reviews quantitative modeling of landscape evolution – which means that not just model studies but also modeling concepts are discussed. Quantitative modeling is contrasted with conceptual or physical modeling, and four categories of model studies are presented. Procedural studies focus on model experimentation. Descriptive studies use models to learn about landscapes in general. Postdictive and predictive try to correctly simulate the evolution of real landscapes, respectively in the past (with calibration) or in the future (with calibrated models). The geomorphic process is a central concept in landscape evolution modeling. We discuss problems with the field-based definition of these processes from a modelling perspective. After the classification of 117 landscape evolution studies in these categories, we find that descriptive studies are most common, and predictive studies are least common. In the remainder of the chapter, we list and review the 117 studies. In procedural studies, attention has been focussed at production methods for digital landscapes, spatial resolution and the role of sinks and depressions. Descriptive studies focussed mainly on surface–tectonic interactions, sensitivity to external forcing, and the definition of crucial field observations from model results. Postdictive and predictive studies operate mainly in time-forward mode and are sometimes validated (postdictive studies of soil redistribution over centennial to millennial timescales). Finally, we look ahead to the future of landscape evolution modeling, arguing for a larger role for complexity research, predictive studies and uncertainty analysis, process definition and feedbacks to and from other fields (including ecology)