186 research outputs found
Algorithms and codes for the Macdonald function: recent progress and comparisons
AbstractThe modified Bessel function Kiν(x), also known as the Macdonald function, finds application in the Kontorovich–Lebedev integral transform when x and ν are real and positive. In this paper, a comparison of three codes for computing this function is made. These codes differ in algorithmic approach, timing, and regions of validity. One of them can be tested independent of the other two through Wronskian checks, and therefore is used as a standard against which the others are compared
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
Software needs in special functions
AbstractCurrently available software for special functions exhibits gaps and defects in comparison to the needs of moderm high-performance scientific computing and also, surprisingly, in comparison to what could be constructed from current algorithms. In this paper we expose some of these deficiencies and identify the related need for user-oriented testing software
Resurgent large genus asymptotics of intersection numbers
In this paper, we present a novel approach for computing the large genus
asymptotics of intersection numbers. Our strategy is based on a resurgent
analysis of the -point functions of such intersection numbers, which are
computed via determinantal formulae, and relies on the presence of a quantum
curve. With this approach, we are able to extend the recent results of Aggarwal
for Witten-Kontsevich intersection numbers with the computation of all
subleading corrections, proving a conjecture of Guo-Yang, and to obtain new
results on -spin and Theta-class intersection numbers.Comment: 47 pages, 7 figure
Analytic traveling-wave solutions of the Kardar-Parisi-Zhang interface growing equation with different kind of noise terms
The one-dimensional Kardar-Parisi-Zhang dynamic interface growth equation
with the traveling-wave Ansatz is analyzed. As a new feature additional
analytic terms are added. From the mathematical point of view, these can be
considered as various noise distribution functions. Six different cases were
investigated among others Gaussian, Lorentzian, white or even pink noise.
Analytic solutions are evaluated and analyzed for all cases. All results are
expressible with various special functions Mathieu, Bessel, Airy or Whittaker
functions showing a very rich mathematical structure with some common general
characteristics. This study is the continuation of our former work, where the
same physical phenomena was investigated with the self-similar Ansatz. The
differences and similarities among the various solutions are enlightened.Comment: 14 pages,14 figures. arXiv admin note: text overlap with
arXiv:1904.0183
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
The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series
Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41670/1/10440_2004_Article_193995.pd
Multiple zeta values and the WKB method
The multiple zeta values ζ(d1, . . . , dr ) are natural generalizations
of the values ζ(d) of the Riemann zeta functions at integers d. They have many applications, e.g. in knot theory and in quantum physics. It turns out that some generating functions for the multiple zeta values, like fd(x) = 1 − ζ(d)xd + ζ(d, d)x2d − . . . , are related with hypergeometric equations. More precisely, fd(x) is the value at t = 1 of some hypergeometric series dFd−1(t) = 1 − x t + . . ., a solution to a hypergeometric equation of degree d with parameter x. Our idea is to represent fd(x) as some connection coeffi- cient between certain standard bases of solutions near t = 0 and near t = 1. Moreover, we assume that |x| is large. For large complex x the above basic solutions are represented in terms of so-called WKB solutions. The series which define the WKB solutions are divergent and are subject to so-called Stokes phenomenon. Anyway it is possible to treat them rigorously. In the paper we review our results about application of the WKB method to the generating functions
f
x), focusing on the cases d = 2 and d = 3
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