A user-friendly extrapolation method for computing infinite range integrals of products of oscillatory functions

wheres is an arbitrary integer,̂ φ(x) and θ̂ (x) are real polynomials inx, φ̂(x) ≡ 0 being possible, exp[φ̂(x)] is bounded at infinity andR(x) andgj (x) aresmooth functions that have asymptotic expansions of the formsR(x) ∼ ∑∞ i =0 bi x −i and gj (x) ∼ ∑∞ i =0 c ± j i x δ j −i as x → ∞, with arbitrary and δ j . We denote the class of such functions by B̃(s). These integrals may converge or diverge and in the case of divergence are defined in some summability sense. The mW(s)-transformationwe propose here is analogous to the mW-transformation ofSidi (1988, A user-friendly extrapolation method for oscillatory infinite integrals. Math. Comput., 51, 249–266), which was originally developed for a class of infinite range oscillatory integrals, whose integrands actually belong to a subfamily of B̃(s). ThemW(s)transformationdetermines a two-dimensional array of approximations A j ) n to I [ f ]. We study some of the convergence properties of the A j ) n . We also provide several numerical examples that illustrate the performance of the method.

[1]  Jonathan M. Borwein,et al.  Hand-to-hand combat with thousand-digit integrals , 2012, J. Comput. Sci..

[2]  Avram Sidi,et al.  An algorithm for a special case of a generalization of the Richardson extrapolation process , 1982 .

[3]  Avram Sidi.  Computation of infinite integrals involving Bessel functions of arbitrary order by the D¯ -transformation , 1997 .

[4]  Avram Sidi.  Further convergence and stability results for the generalized Richardson extrapolation process GREP (1) with an application to the D (1) -transformation for infinite intergals , 1999 .

[5]  David Levin,et al.  Two New Classes of Nonlinear Transformations for Accelerating the Convergence of Infinite Integrals and Series , 1981 .

[6]  A. Sidi.  Extrapolation methods for divergent oscillatory infinite integrals that are defined in the sense of summability , 1987 .

[7]  Avram Sidi,et al.  Extrapolation Methods for Oscillatory Infinite Integrals , 1980 .

[8]  Numerische Mathematik.  An Algorithm for a Special Case of a Generalization of the Richardson Extrapolation Process , 2005 .

[9]  Ronald Cools,et al.  Integrating products of Bessel functions with an additional exponential or rational factor , 2008, Comput. Phys. Commun..

[10]  S. K. Lucas,et al.  Evaluating infinite integrals involving products of Bessel functions of arbitrary order , 1995 .

[11]  Ana Slot,et al.  Extrapolation Methods for Oscillatory Infinite Integrals , 2009 .

[12]  Avram Sidi,et al.  Some Properties of a Generalization of the Richardson Extrapolation Process , 1979 .

[13]  Ronald F. Boisvert,et al.  NIST Handbook of Mathematical Functions , 2010 .

[14]  A. Sim,et al.  Some Properties of a Generalization of the Richardson Extrapolation Process , 2022 .

[15]  Ronald Cools,et al.  Algorithm 858: Computing infinite range integrals of an arbitrary product of Bessel functions , 2006, TOMS.

[16]  Avram Sidi,et al.  Practical Extrapolation Methods - Theory and Applications , 2003, Cambridge monographs on applied and computational mathematics.

[17]  David Levin,et al.  Practical extrapolation methods: theory and applications , 2005, Math. Comput..

[18]  Avram Sidi,et al.  A user-friendly extrapolation method for oscillatory infinite integrals , 1988 .

[19]  Avram Sidi,et al.  The numerical evaluation of very oscillatory infinite integrals by extrapolation , 1982 .