DQAINF: An algorithm for automatic integration of infinite oscillating tails
type: Item
instance of: scholarly article
title: DQAINF: An algorithm for automatic integration of infinite oscillating tails
zbMATH Open document ID: 0811.65017
DOI: 10.1007/BF02145697
author: K. J. Overholt
author: Terje O. Espelid
published in: Numerical Algorithms
publication date: 1995-05-02T00:00:00Z
review text: The paper describes a quadrature routine designed to integrate a (scalar or vector) function with a certain type of infinite oscillating, decaying tails over an infinite interval. The algorithm is based on the assumption that the oscillating behavior is due to the superposition of periodic functions which change sign when evaluated at points of distance half a period. Hence, following the basic ideas of \textit{J. N. Lyness} [J. Comp. Appl. Math. 12/13, 109-117 (1985; Zbl 0574.65013)], partitioning the original integral into an infinite series of integrals all of same interval length yields close relations to alternating series, and the Euler transformation (together with some modifications) implies very stable schemes for accelerated approximations. A FORTRAN subroutine for the algorithm is described in detail. Six examples show the efficiency of the method. Reviewer's remark: The explicit (exact) integration is possible for more examples than for the two indicated by the authors, in some cases even through the explicit representation of primitives with the help of higher transcendental functions. For instance: The integral \(I= \int^ \infty_ 1 p(x) dx/\sqrt x\) with \(p(x)= (-1)^ k\) for \(k- 1\leq x< k\) (Example 1) is related to the Riemann \(\zeta\)-function; we find \(I= 2- (2\sqrt 2- 1)\cdot \zeta(3/2)/\pi= 0.479580749561263933\). The integral of Example 2 may be represented by the Fresnel integrals, the special integral appearing here is \(\sqrt{2\pi} f(\sqrt{2/\pi})= 0.80952548174708844\) with a customary ``auxiliary function'' \(f(z)\), studied thoroughly by \textit{J. Boersma} [Math. Comput. 14, 380 (1960; Zbl 0091.293)] and tabulated in the handbook of \textit{M. Abramowitz} and \textit{I. A. Stegun} [Handbook of mathematical functions (1964; Zbl 0171.385)]. Similarly, the integral \(\int^ \infty_ 1 \sin(x+ 1/x)dx/\sqrt x\) of Example 3 is \(\sqrt{\pi/2}[\cos 2-S_ 1(2)\cos 2+ C_ 1(2)\sin 2]= 0.232948197094002526\), again with the Fresnel integrals. These…
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65D20
zbMATH DE Number: 698814
zbMATH Keywords: quadrature formulas
zbMATH Keywords: Riemann zeta function
zbMATH Keywords: infinite oscillating, decaying tails
zbMATH Keywords: infinite interval
zbMATH Keywords: algorithm
zbMATH Keywords: superposition of periodic functions
zbMATH Keywords: Euler transformation
zbMATH Keywords: FORTRAN subroutine
zbMATH Keywords: Fresnel integrals
reviewed by: E. Lanckau
describes a project that uses: HURRY
describes a project that uses: Algorithm 639
describes a project that uses: QUADPACK
describes a project that uses: DQAINT
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 698814