577 research outputs found
Numerical and Asymptotic Aspects of Parabolic Cylinder Functions
Several uniform asymptotics expansions of the Weber parabolic cylinder
functions are considered, one group in terms of elementary functions, another
group in terms of Airy functions. Starting point for the discussion are
asymptotic expansions given earlier by F.W.J. Olver. Some of his results are
modified to improve the asymptotic properties and to enlarge the intervals for
using the expansions in numerical algorithms. Olver's results are obtained from
the differential equation of the parabolic cylinder functions; we mention how
modified expansions can be obtained from integral representations. Numerical
tests are given for three expansions in terms of elementary functions. In this
paper only real values of the parameters will be considered.Comment: 16 pages, 1 figur
Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments
We describe a variety of methods to compute the functions ,
and their derivatives for real and positive . These
functions are numerically satisfactory independent solutions of the
differential equation . 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
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 the
asymptotic methods are enough for a double precision accuracy computation
(- digits) of the nodes and weights of the Gauss--Hermite and
Gauss--Laguerre quadratures.Comment: Submitted to Studies in Applied Mathematic
Integral Representations for Computing Real Parabolic Cylinder Functions
Integral representations are derived for the parabolic cylinder functions
, and 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 , and 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.Comment: 31 pages, 3 figures. To appear in Numer. Mat
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
Computation of the modified Bessel function of the third kind of imaginary orders: uniform Airy-type asymptotic expansion
AbstractThe use of a uniform Airy-type asymptotic expansion for the computation of the modified Bessel functions of the third kind of imaginary orders (Kia(x)) near the transition point x=a, is discussed. In A. Gil et al., Evaluation of the modified Bessel functions of the third kind of imaginary orders, J. Comput. Phys. 17 (2002) 398–411, an algorithm for the evaluation of Kia(x) was presented, which made use of series, a continued fraction method and nonoscillating integral representations. The range of validity of the algorithm was limited by the singularity of the steepest descent paths near the transition point. We show how uniform Airy-type asymptotic expansions fill the gap left by the steepest descent method
Numerical and asymptotic aspects of parabolic cylinder functions
Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver's results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered
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