A. Gil

T. M. Dunster, A. Gil, J. Segura

Linear second order differential equations having a large real parameter and turning point in the complex plane are considered. Classical asymptotic expansions for solutions involve the Airy function and its derivative, along with two infinite series, the coefficients of which are usually difficult to compute. By considering the series as asymptotic expansions for two explicitly defined analytic functions, Cauchy's integral formula is employed to compute the coefficient functions to high order of accuracy. The method employs a certain exponential form of Liouville-Green expansions for solutions of the differential equation, as well as for the Airy function. We illustrate the use of the method with the high accuracy computation of Airy-type expansions of Bessel functions of complex argument.

A. Gil, J. Segura, N. M. Temme

A Fortran 90 module (GammaCHI) for computing and inverting the gamma and chi-square cumulative distribution functions (central and noncentral) is presented. The main novelty of this package are the reliable and accurate inversion routines for the noncentral cumulative distribution functions. Additionally, the package also provides routines for computing the gamma function, the error function and other functions related to the gamma function. The module includes the routines cdfgamC, invcdfgamC, cdfgamNC, invcdfgamNC, errorfunction, inverfc, gamma, loggam, gamstar and quotgamm for the computation of the central gamma distribution function (and its complementary function), the inversion of the central gamma distribution function, the computation of the noncentral gamma distribution function (and its complementary function), the inversion of the noncentral gamma distribution function, the computation of the error function and its complementary function, the inversion of the complementary error function, the computation of: the gamma function, the logarithm of the gamma function, the regulated gamma function and the ratio of two gamma functions, respectively.

J. J. Cabrera, V. Román, A. Gil et al.

The objective of this paper is to address the localization problem using omnidirectional images captured by a catadioptric vision system mounted on the robot. For this purpose, we explore the potential of Siamese Neural Networks for modeling indoor environments using panoramic images as the unique source of information. Siamese Neural Networks are characterized by their ability to generate a similarity function between two input data, in this case, between two panoramic images. In this study, Siamese Neural Networks composed of two Convolutional Neural Networks (CNNs) are used. The output of each CNN is a descriptor which is used to characterize each image. The dissimilarity of the images is computed by measuring the distance between these descriptors. This fact makes Siamese Neural Networks particularly suitable to perform image retrieval tasks. First, we evaluate an initial task strongly related to localization that consists in detecting whether two images have been captured in the same or in different rooms. Next, we assess Siamese Neural Networks in the context of a global localization problem. The results outperform previous techniques for solving the localization task using the COLD-Freiburg dataset, in a variety of lighting conditions, specially when using images captured in cloudy and night conditions.

J. J. Cabrera, A. Santo, A. Gil et al.

This paper presents MinkUNeXt, an effective and efficient architecture for place-recognition from point clouds entirely based on the new 3D MinkNeXt Block, a residual block composed of 3D sparse convolutions that follows the philosophy established by recent Transformers but purely using simple 3D convolutions. Feature extraction is performed at different scales by a U-Net encoder-decoder network and the feature aggregation of those features into a single descriptor is carried out by a Generalized Mean Pooling (GeM). The proposed architecture demonstrates that it is possible to surpass the current state-of-the-art by only relying on conventional 3D sparse convolutions without making use of more complex and sophisticated proposals such as Transformers, Attention-Layers or Deformable Convolutions. A thorough assessment of the proposal has been carried out using the Oxford RobotCar and the In-house datasets. As a result, MinkUNeXt proves to outperform other methods in the state-of-the-art.

A. Gil, J. Segura, N. M. Temme

The computation and inversion of the noncentral beta distribution $B_{p,q}(x,y)$ (or the noncentral $F$-distribution, a particular case of $B_{p,q}(x,y)$) play an important role in different applications. In this paper we study the stability of recursions satisfied by $B_{p,q}(x,y)$ 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 $B_{p,q}(x,y)$ 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.

A. Gil, J. Segura, N. M. Temme

An efficient algorithm and a Fortran 90 module (LaguerrePol) for computing Laguerre polynomials $L^{(α)}_n(z)$ are presented. The standard three-term recurrence relation satisfied by the polynomials and different types of asymptotic expansions valid for $n$ large and $α$ small, are used depending on the parameter region. Based on tests of contiguous relations in the parameter $α$ and the degree $n$ satisfied by the polynomials, we claim that a relative accuracy close or better than $10^{-12}$ can be obtained using the module LaguerrePol for computing the functions $L^{(α)}_n(z)$ in the parameter range $z \ge 0$, $-1 < α\le 5$, $n \ge 0$.

A. Wawrzynczak, R. Modzelewska, A. Gil

We present the stochastic model of the galactic cosmic ray (GCR) particles transport in the heliosphere. Based on the solution of the Parker transport equation we developed models of the short-time variation of the GCR intensity, i.e. the Forbush decrease (Fd) and the 27-day variation of the GCR intensity. Parker transport equation being the Fokker-Planck type equation delineates non-stationary transport of charged particles in the turbulent medium. The presented approach of the numerical solution is grounded on solving of the set of equivalent stochastic differential equations (SDEs). We demonstrate the method of deriving from Parker transport equation the corresponding SDEs in the heliocentric spherical coordinate system for the backward approach. Features indicative the preeminence of the backward approach over the forward is stressed. We compare the outcomes of the stochastic model of the Fd and 27-day variation of the GCR intensity with our former models established by the finite difference method. Both models are in an agreement with the experimental data.

A. Wawrzynczak, R. Modzelewska, A. Gil

We present the newly developed stochastic model of the galactic cosmic ray (GCR) particles transport in the heliosphere. Mathematically Parker transport equation (PTE) describing non-stationary transport of charged particles in the turbulent medium is the Fokker-Planck type. It is the second order parabolic time-dependent 4-dimensional (3 spatial coordinates and particles energy/rigidity) partial differential equation. It is worth to mention that, if we assume the stationary case it remains as the 3-D parabolic type problem with respect to the particles rigidity R. If we fix the energy it still remains as the 3-D parabolic type problem with respect to time. The proposed method of numerical solution is based on the solution of the system of stochastic differential equations (SDEs) being equivalent to the Parker's transport equation. We present the method of deriving from PTE the equivalent SDEs in the heliocentric spherical coordinate system for the backward approach. The obtained stochastic model of the Forbush decrease of the GCR intensity is in an agreement with the experimental data. The advantages and disadvantages of the forward and the backward solution of the PTE are discussed.

A. Gil, J. Segura, N. M. Temme

We describe methods for computing the Kummer function $U(a,b,z)$ for small values of $z$, with special attention to small values of $b$. For these values of $b$ the connection formula that represents $U(a,b,z)$ as a linear combination of two ${}_1F_1$-functions needs a limiting procedure. We use the power series of the ${}_1F_1$-functions and consider the terms for which this limiting procedure is needed. We give recursion relations for higher terms in the expansion, and we consider the derivative $U^\prime(a,b,z)$ as well. We also discuss the performance for small $\vert z\vert$ of an asymptotic approximation of the Kummer function in terms of modified Bessel functions.