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Amparo Gil

Publications and source records attributed to Amparo Gil.

17 recordsLinked to original sources
arXiv2025

Uniform Asymptotic approximation and numerical evaluation of the Reverse Generalized Bessel Polynomial zeros

Uniform asymptotic expansions are derived for the zeros of the reverse generalized Bessel polynomials of large degree $n$ and real parameter $a$. It is assumed that $-\Delta_{1} n+\frac{3}{2} \leq a \leq \Delta_{2} n$ for fixed arbitrary $\Delta_{1} \in (0,1)$ and bounded positive $\Delta_{2}$. For this parameter range at most one of the zeros is real, with the rest being complex conjugates. The new expansions are uniformly valid for all the zeros, and are shown to be highly accurate for moderate or large values of $n$. They are consequently used as initial values in a very efficient numerical algorithm designed to obtain the remaining complex zeros using Taylor series.

math.CA↗
arXiv2024

McMahon-type asymptotic expansions of the zeros of the Coulomb wave functions

We derive asymptotic expansions of the large zeros of the Coulomb wave functions and for those of their derivatives. The new expansions have the same form as the McMahon expansions of the zeros of the Bessel functions and reduce to them when a parameter is equal to zero. Numerical tests are provided to demonstrate the accuracy of the expansions.

math.CA↗
arXiv2023

New asymptotic representations of the noncentral $t$-distribution

New asymptotic approximations of the non-central $t$ distribution are given, a generalization of the Student's $t$ distribution. Using new integral representations, we give new asymptotic expansions for large values of the noncentrality parameter but also for large values of the degrees of freedom parameter. In some case we accept more than one large parameter. These results are in terms of elementary functions, but also in terms of the complementary error function and the incomplete gamma function. A number of numerical tests demonstrate the performance of the asymptotic approximations.

math.PR↗
arXiv2020

A new asymptotic representation and inversion method for the Student's t distribution

Some special functions are particularly relevant in applied probability and statistics. For example, the incomplete beta function is the cumulative central beta distribution. In this paper, we consider the inversion of the central Student's-$t$ distribution which is a particular case of the central beta distribution. The inversion of this distribution functions is useful in hypothesis testing as well as for generating random samples distributed according to the corresponding probability density function. A new asymptotic representation in terms of the complementary error function, will be one of the important ingredients in our analysis. As we will show, this asymptotic representation is also useful in the computation of the distribution function. We illustrate the performance of all the obtained approximations with numerical examples.

math.CA↗
arXiv2020

Asymptotic expansions of Jacobi polynomials and of the nodes and weights of Gauss-Jacobi quadrature for large degree and parameters in terms of elementary functions

Asymptotic approximations of Jacobi polynomials are given in terms of elementary functions for large degree $n$ and parameters $α$ and $β$. From these new results, asymptotic expansions of the zeros are derived and methods are given to obtain the coefficients in the expansions. These approximations can be used as initial values in iterative methods for computing the nodes of Gauss--Jacobi quadrature for large degree and parameters. The performance of the asymptotic approximations for computing the nodes and weights of these Gaussian quadratures is illustrated with numerical examples.

math.CA↗
arXiv2020

Numerical evaluation of Airy-type integrals arising in uniform asymptotic analysis

We describe a method to evaluate integrals that arise in the asymptotic analysis when two saddle points may be close together. These integrals, which appear in problems from optics, acoustics or quantum mechanics as well as in a wide class of special functions, can be transformed into Airy-type integrals and we use the trapezoidal rule to compute these integrals numerically. The quadrature method, which remains valid when two saddle points coalesce, is illustrated with numerical examples.

math.NA↗
arXiv2018

Non-iterative computation of Gauss-Jacobi quadrature

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 $-1< \alpha, \beta\le 5$. 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.

math.NA↗
arXiv2013

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 $P(a,x)=γ(a,x)/Γ(a)$ and $Q(a,x)=Γ(a,x)/Γ(a)$ 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 {\bf IncgamFI}) is analyzed and compared with earlier published algorithms.

math.CA↗
arXiv2013

An improved algorithm and a Fortran 90 module for computing the conical function $P^m_{-1/2+iτ}(x)$

In this paper we describe an algorithm and a Fortran 90 module ({\bf Conical}) for the computation of the conical function $P^m_{-\tfrac12+iτ}(x)$ for $x>-1$, $m \ge 0$, $τ>0$. These functions appear in the solution of Dirichlet problems for domains bounded by cones; because of this, they are involved in a large number of applications in Engineering and Physics. In the Fortran 90 module, the admissible parameter ranges for computing the conical functions in standard IEEE double precision arithmetic are restricted to $(x,m,τ) \in (-1,1) \times [0,\,40] \times [0,\,100]$ and $(x,m,τ) \in (1,100) \times [0,\,100] \times [0,\,100]$. Based on tests of the three-term recurrence relation satisfied by these functions and direct comparison with Maple, we claim a relative accuracy close to $10^{-12}$ in the full parameter range, although a mild loss of accuracy can be found at some points of the oscillatory region of the conical functions. The relative accuracy increases to $10^{-13}\,-\,10^{-14}$ in the region of the monotonic regime of the functions where integral representations are computed ($-1<x<0$).

math.CA↗
arXiv2004

The ABC of Hyper Recursions

Each family of Gauss hypergeometric functions $$ f_n={}_2F_1(a+\epsilon_1n, b+\epsilon_2n ;c+\epsilon_3n; z), $$ for fixed $\epsilon_j=0,\pm1$ (not all $\epsilon_j$ 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 $\epsilon_j$ 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.

math.CA↗
arXiv2004

Algorithm xxx: Modified Bessel functions of imaginary order and positive argument

Fortran 77 programs for the computation of modified Bessel functions of purely imaginary order are presented. The codes compute the functions $K_{ia}(x)$, $L_{ia}(x)$ and their derivatives for real $a$ and positive $x$; these functions are independent solutions of the differential equation $x^2 w'' +x w' +(a^2 -x^2)w=0$. The code also computes exponentially scaled functions. The range of computation is $(x,a)\in (0,1500]\times [-1500,1500]$ when scaled functions are considered and it is larger than $(0,500]\times [-400,400]$ for standard IEEE double precision arithmetic. The relative accuracy is better than $10^{-13}$ in the range $(0,200]\times [-200,200]$ and close to $10^{-12}$ in $(0,1500]\times [-1500,1500]$.

cs.MS↗
arXiv2004

Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments

We describe a variety of methods to compute the functions $K_{ia}(x)$, $L_{ia}(x)$ and their derivatives for real $a$ and positive $x$. These functions are numerically satisfactory independent solutions of the differential equation $x^2 w'' +x w' +(a^2 -x^2)w=0$. In an accompanying paper (Algorithm xxx: Modified Bessel functions of imaginary order and positive argument) we describe the implementation of these methods in Fortran 77 codes.

math.NA↗
arXiv2004

Integral Representations for Computing Real Parabolic Cylinder Functions

Integral representations are derived for the parabolic cylinder functions $U(a,x)$, $V(a,x)$ and $W(a,x)$ and their derivatives. The new integrals will be used in numerical algorithms based on quadrature. They follow from contour integrals in the complex plane, by using methods from asymptotic analysis (saddle point and steepest descent methods), and are stable starting points for evaluating the functions $U(a,x)$, $V(a,x)$ and $W(a,x)$ and their derivatives by quadrature rules. In particular, the new representations can be used for large parameter cases. Relations of the integral representations with uniform asymptotic expansions are also given. The algorithms will be given in a future paper.

math.NA↗
arXiv2004

Numerical algorithms for the real zeros of hypergeometric functions

Algorithms for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. The algorithms are based on global fixed point iterations which apply to families of functions satisfying first order linear difference differential equations with continuous coefficients. In order to compute the zeros of arbitrary solutions of the hypergeometric equations, we have at our disposal several different sets of difference differential equations (DDE). We analyze the behavior of these different sets regarding the rate of convergence of the associated fixed point iteration. It is shown how combinations of different sets of DDEs, depending on the range of parameters and the dependent variable, is able to produce efficient methods for the computation of zeros with a fairly uniform convergence rate for each zero.

math.NA↗
arXiv2001

On Non-Oscillating Integrals for Computing Inhomogeneous Airy Functions

Integral representations are considered of solutions of the inhomogeneous Airy differential equation $w''-z w=\pm1/\pi$. The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane. By using steepest descent methods from asymptotics, the standard integral representations of the Scorer functions are modified in order to obtain non-oscillating integrals for complex values of $z$. In this way stable representations for numerical evaluations of the functions are obtained. The methods are illustrated with numerical results.

math.NA↗