Amparo Gil

Amparo Gil, Javier Segura, Nico M. Temme

Asymptotic approximations to the zeros of Jacobi polynomials are given, with methods to obtain the coefficients in the expansions. These approximations can be used as standalone methods for the non-iterative computation of the nodes of Gauss--Jacobi quadratures of high degree ($n\ge 100$). We also provide asymptotic approximations for functions related to the first order derivative of Jacobi polynomials which are used for computing the weights of the Gauss--Jacobi quadrature. The performance of the asymptotic approximations is illustrated with numerical examples, and it is shown that nearly double precision relative accuracy is obtained both for the nodes and the weights when $n\ge 100$ and $-1< α, β\le 5$. For smaller degrees the approximations are also useful as they provide $10^{-12}$ relative accuracy for the nodes when $n\ge 20$, and just one Newton step would be sufficient to guarantee double precision accuracy in that cases.

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.

Alvaro Francisco Gil, Walther Litteri, Victor Rodriguez-Fernandez et al.

The Three-Body Problem has fascinated scientists for centuries and it has been crucial in the design of modern space missions. Recent developments in Generative Artificial Intelligence hold transformative promise for addressing this longstanding problem. This work investigates the use of Variational Autoencoder (VAE) and its internal representation to generate periodic orbits. We utilize a comprehensive dataset of periodic orbits in the Circular Restricted Three-Body Problem (CR3BP) to train deep-learning architectures that capture key orbital characteristics, and we set up physical evaluation metrics for the generated trajectories. Through this investigation, we seek to enhance the understanding of how Generative AI can improve space mission planning and astrodynamics research, leading to novel, data-driven approaches in the field.

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

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.

Amparo Gil, Javier Segura, Nico M. Temme

Each family of Gauss hypergeometric functions $$ f_n={}_2F_1(a+ε_1n, b+ε_2n ;c+ε_3n; z), $$ for fixed $ε_j=0,\pm1$ (not all $ε_j$ equal to zero) satisfies a second order linear difference equation of the form $$ A_nf_{n-1}+B_nf_n+C_nf_{n+1}=0. $$ Because of symmetry relations and functional relations for the Gauss functions, many of the 26 cases (for different $ε_j$ values) can be transformed into each other. We give a set of basic equations from which all other equations can be obtained. For each basic equation, we study the existence of minimal solutions and the character of $f_n$ (minimal or dominant) as $n\to \pm\infty$. A second independent solution is given in each basic case which is dominant when $f_n$ is minimal and vice-versa. In this way, satisfactory pairs of linearly independent solutions for each of the 26 second order linear difference equations can be obtained.