11,328 research outputs found
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
On the computation of moments of the partial non-central chi-squared distribution function
Properties satisfied by the moments of the partial non-central chi-square
distribution function, also known as Nuttall Q-functions, and methods for
computing these moments are discussed in this paper. The Nuttall Q-function is
involved in the study of a variety of problems in different fields, as for
example digital communications.Comment: 6 page
Computation of the Marcum Q-function
Methods and an algorithm for computing the generalized Marcum function
() and the complementary function () are described.
These functions appear in problems of different technical and scientific areas
such as, for example, radar detection and communications, statistics and
probability theory, where they are called the non-central chi-square or the non
central gamma cumulative distribution functions.
The algorithm for computing the Marcum functions combines different methods
of evaluation in different regions: series expansions, integral
representations, asymptotic expansions, and use of three-term homogeneous
recurrence relations. A relative accuracy close to can be obtained
in the parameter region ,
, while for larger parameters the accuracy decreases (close to
for and close to for ).Comment: Accepted for publication in ACM Trans. Math. Soft
On Non-Oscillating Integrals for Computing Inhomogeneous Airy Functions
Integral representations are considered of solutions of the inhomogeneous
Airy differential equation . 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
. 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
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
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