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.Comment: 12 pages, 5 figure

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.Comment: 14 pages, 1 figure. To appear in ACM T. Math. Sof

A numerical method for oscillatory integrals with coalescing saddle points

The value of a highly oscillatory integral is typically determined asymptotically by the behaviour of the integrand near a small number of critical points. These include the endpoints of the integration domain and the so-called stationary points or saddle points -- roots of the derivative of the phase of the integrand -- where the integrand is locally non-oscillatory. Modern methods for highly oscillatory quadrature exhibit numerical issues when two such saddle points coalesce. On the other hand, integrals with coalescing saddle points are a classical topic in asymptotic analysis, where they give rise to uniform asymptotic expansions in terms of the Airy function. In this paper we construct Gaussian quadrature rules that remain uniformly accurate when two saddle points coalesce. These rules are based on orthogonal polynomials in the complex plane. We analyze these polynomials, prove their existence for even degrees, and describe an accurate and efficient numerical scheme for the evaluation of oscillatory integrals with coalescing saddle points

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

Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals

High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of convergence is equal, numerical stability strongly depends on r. We give a comprehensive study of this effect; in particular we show that there is a unique radius that minimizes the loss of accuracy caused by round-off errors. For large classes of functions, though not for all, this radius actually gives about full accuracy; a remarkable fact that we explain by the theory of Hardy spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and by the saddle-point method of asymptotic analysis. Many examples and non-trivial applications are discussed in detail.Comment: Version 4 has some references and a discussion of other quadrature rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Mat

Computing special functions by using quadrature rules

The usual tools for computing special functions are power series, asymptotic expansions, continued fractions, differential equations, recursions, and so on. Rather seldom are methods based on quadrature of integrals. Selecting suitable integral representations of special functions, using principles from asymptotic analysis, we develop reliable algorithms which are valid for large domains of real or complex parameters. Our present investigations include Airy functions, Bessel functions and parabolic cylinder functions. In the case of Airy functions we have improvements in both accuracy and speed for some parts of Amos's code for Bessel functions

Basic Methods for Computing Special Functions

This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website