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)

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 stand-alone 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.The authors acknowledge financial support from Ministerio de Economía y Competitividad, project MTM2015-67142-P (MINECO/FEDER, UE)

Asymptotic Expansions of Jacobi Polynomials for Large Values of Beta

Asymptotic approximations of Jacobi polynomials are given for large values of the Beta-parameter and of their zeros. The expansions are given in terms of Laguerre polynomials and of their zeros. The levels of accuracy of the approximations are verified by numerical examples.This work was supported by Ministerio de Econom a y Competitividad, project MTM2015-67142-P (MINECO/FEDER, UE). NMT thanks CWI, Amsterdam, for scienti c support

Exposición mediante realidad virtual para el TOC: ¿Es factible?

Virtual reality exposure therapy (VRET) is receiving increased attention, especially in the fields of anxiety and eating disorders. This study is the first trial examining the utility of VRET from the perspective of OCD patients. Four OCD women assessed the sense of presence, emotional engagement, and reality judgment, and the anxiety and disgust levels they experimented in four scenarios, called the Contaminated Virtual Environment (COVE), in which they had to perform several activities. The COVE scenarios were presented on a Full HD 46” TV connected to a laptop and to a Kinect device. Results indicate that the COVE scenarios generated a good sense of presence. The anxiety and disgust levels increased as the virtual contamination increased, and the anxiety produced was related to the emotional engagement and sense of presence.La Exposición mediante Realidad Virtual (ERV) está recibiendo una atención cada vez mayor, especialmente para los trastornos de ansiedad y los alimentarios. Este estudio es el primer ensayo que evalúa la utilidad de la ERV desde la propia perspectiva de pacientes con Trastorno Obsesivo-Compulsivo (TOC). Cuatro mujeres con TOC evaluaron la sensación de presencia, implicación emocional, el juicio de realidad, y los niveles de ansiedad y asco que experimentaban en cuatro escenarios virtuales, que denominamos Entorno Virtual Contaminado (EVCO), en los que debían realizar varias actividades. Los escenarios se presentaron en una TV Full HD de 46’’, conectada a un ordenador y a un dispositivo Kinect. Los resultados indican que EVCO produjo una buena sensación de presencia. Los niveles de ansiedad y asco aumentaron a medida que aumentaba la “contaminación” de los escenarios, y la ansiedad se asoció con la sensación de presencia y la implicación emocional

On the computation and inversion of the cumulative noncentral beta distribution function

The computation and inversion of the noncentral beta distribution Bp,q(x, y) (or the noncentral F-distribution, a particular case of Bp,q(x, y)) play an important role in different applications. In this paper we study the stability of recursions satisfied by Bp,q(x, y) and its complementary function and describe asymptotic expansions useful for computing the function when the parameters are large. We also consider the inversion problem of finding x or y when a value of Bp,q(x, y) is given. We provide approximations to x and y which can be used as starting values of methods for solving nonlinear equations (such as Newton) if higher accuracy is needed

Fast, reliable and unrestricted iterative computation of Gauss-Hermite and Gauss-Laguerre quadratures

Methods for the computation of classical Gaussian quadrature rules are described which are effective both for small and large degree. These methods are reliable because the iterative computation of the nodes has guaranteed convergence, and they are fast due to their fourth-order convergence and its asymptotic exactness for an appropriate selection of the variables. For Gauss?Hermite and Gauss?Laguerre quadratures, local Taylor series can be used for computing efficiently the orthogonal polynomials involved, with exact initial values for the Hermite case and first values computed with a continued fraction for the Laguerre case. The resulting algorithms have almost unrestricted validity with respect to the parameters. Full relative precision is reached for the Hermite nodes, without any accuracy loss and for any degree, and a mild accuracy loss occurs for the Hermite and Laguerre weights as well as for the Laguerre nodes. These fast methods are exclusively based on convergent processes, which, together with the high order of convergence of the underlying iterative method, makes them particularly useful for high accuracy computations. We show examples of very high accuracy computations (of up to 1000 digits of accuracy)

Efficient and accurate algorithms for the computation and inversion of the incomplete gamma function ratios

Algorithms for the numerical evaluation of the incomplete gamma function ratios $P(a,x)=\gamma(a,x)/\Gamma(a)$ and $Q(a,x)=\Gamma(a,x)/\Gamma(a)$ are described for positive values of $a$ and $x$. Also, inversion methods are given for solving the equations $P(a,x)=p$, $Q(a,x)=q$, with $0<p,q<1$. Both the direct computation and the inversion of the incomplete gamma function ratios are used in many problems in statistics and applied probability. The analytical approach from earlier literature is summarized, and new initial estimates are derived for starting the inversion algorithms. The performance of the associated software to our algorithms (the Fortran 90 module IncgamFI) is analyzed and compared with earlier published algorithms

Efficient algorithms for the inversion of the cumulative central beta distribution

Accurate and efficient algorithms for the inversion of the cumulative central beta distribution are described. The algorithms are based on the combination of a fourth-order fixed point method with good non-local convergence properties (the Schwarzian-Newton method), asymptotic inversion methods and sharp bounds in the tails of the distribution function.The authors acknowledge financial support from Ministerio de Economía y Competitividad, project MTM2012-34787. NMT thanks CWI, Amsterdam, for scientific support

Asymptotic inversion of the binomial and negative binomial cumulative distribution functions

ABSTRACT: The computation and inversion of the binomial and negative binomial cumulative distribution functions play a key role in many applications. In this paper, we explain how methods used for the central beta distribution function (described in Gil, Segura, and Temme, [Numer. Algorithms, 74 (2017), pp. 77?91]) can be utilized to obtain asymptotic representations of these functions and also for their inversion. The performance of the asymptotic inversion methods is illustrated with numerical examples.Acknowledgments. The authors thank the anonymous referees for their constructive comments and suggestions. This work was supported by Ministerio de Ciencia e Innovación, Spain, projects MTM2015-67142-P (MINECO/FEDER, UE) and PGC2018-098279-B-I00 (MCIU/AEI/FEDER, UE). NMT thanks CWI, Amsterdam, for scientific support

The Asymptotic and Numerical Inversion of the Marcum Q-Function

The generalized Marcum functions appear in problems of technical and scientific areas such as, for example, radar detection and communications. In mathematical statistics and probability theory these functions are called the noncentral gamma or the noncentral chi-squared cumulative distribution functions. In this paper, we describe a new asymptotic method for inverting the generalized Marcum Q-function and for the complementary Marcum P-function. Also, we show how monotonicity and convexity properties of these functions can be used to find initial values for reliable Newton or secant methods to invert the function. We present details of numerical computations that show the reliability of the asymptotic approximations