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An automatic integration procedure for infinite range integrals involving oscillatory kernels

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Abstract

Let the real functionsK(x) andL(x) be such thatM(x)=K(x)+iL(x)=eix g(x), whereg(x) is infinitely differentiable for all largex and is non-oscillatory at infinity. We develop an efficient automatic quadrature procedure for numerically computing the integrals ∫ a Kt)f(t) and ∫ a Lt)f(t)dt, where the functionf(t) is smooth and nonoscillatory at infinity. One such example for which we also provide numerical results is that for whichK(x)=J ν(x) andL(x)=Y ν(x), whereJ ν(x) andY ν(x) are the Bessel functions of order ν. The procedure involves the use of an automatic scheme for Fourier integrals and the modified W-transformation which is used for computing oscillatory infinite integrals.

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References

  1. W. L. Anderson, Fast Hankel transform using related and lagged convolutions, ACM Trans. Math. Software 8 (1982) 344–368.

    Google Scholar 

  2. M. Branders and R. Piessens, An extension of Clenshaw-Curtis quadrature, J. Comput. Appl. Math. 1 (1975) 55–65.

    Google Scholar 

  3. J. R. Cash, A note on the numerical solution of linear recurrence relations, Numer. Math. 34 (1980) 371–386.

    Google Scholar 

  4. C. W. Clenshaw and A. R. Curtis, A method for numerical integration on an automatic computer, Numer. Math. 2 (1960) 197–205.

    Google Scholar 

  5. B. Davies,Integral Transforms and Their Applications (Springer, New York, 2nd. ed., 1985).

    Google Scholar 

  6. P. J. Davis and P. Rabinowitz,Methods of Numerical Integration (Academic Press, Orlando, FL, 2nd ed., 1984).

    Google Scholar 

  7. D. Elliott, Truncation errors in two Chebyshev series approximations, Math. Comp. 19 (1965) 234–248.

    Google Scholar 

  8. T. O. Espelid and K. J. Overholt, DQAINF: an algorithm for automatic integration of infinite oscillating tails, Numer. Algorithms 8 (1994) 83–101.

    Google Scholar 

  9. F. N. Fritsch, D. K. Kahaner and J. N. Lyness, Double integration using one-dimensional adaptive quadrature routines: a software interface problem, ACM Trans. Math. Software 7 (1981) 46–75.

    Google Scholar 

  10. W. Gautschi, Computational aspect of three-term recurrence relations, SIAM Rev. 9 (1967) 24–82.

    Google Scholar 

  11. W. Gautschi and R. S. Varga, Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal. 20 (1983) 1170–1186.

    Google Scholar 

  12. W. M. Gentleman, Implementing Clenshaw-Curtis quadrature II. Computing the cosine transformation. Comm. ACM 15 (1972) 343–346.

    Google Scholar 

  13. I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, and Products, translated by A. Jeffrey (Academic Press, New York, 1980).

    Google Scholar 

  14. H. L. Gray and T. A. Atchison, Nonlinear transformations related to the evaluation of improper integrals I, SIAM J. Numer. Anal. 4 (1967) 363–371.

    Google Scholar 

  15. T. Hasegawa and T. Torii, Stable algorithm for the minimal solution of second order linear difference equations, J. Inform. Process. 23 (1982) 583–590 (in Japanese).

    Google Scholar 

  16. T. Hasegawa and T. Torii, Indefinite integration of oscillatory functions by the Chebyshev series expansion, J. Comput. Appl. Math. 17 (1987) 21–29.

    Google Scholar 

  17. T. Hasegawa amd T. Torii, An automatic quadrature for Cauchy principal value integrals, Math. Comp. 56 (1991) 741–754.

    Google Scholar 

  18. T. Hasegawa and T. Torii, Application of a modified FFT to product type integration, J. Comput. Appl. Math. 38 (1991) 157–168.

    Google Scholar 

  19. T. Hasegawa and T. Torii, An algorithm for nondominant solutions of linear second-order inhomogeneous difference equations, Math. Comp. 64 (1995) 1199–1214.

    Google Scholar 

  20. T. Hasegawa, T. Torii and I. Ninomiya, Generalized Chebyshev interpolation and its application to automatic quadrature, Math. Comp. 41 (1983) 537–553.

    Google Scholar 

  21. T. Hasegawa, T. Torii and H. Sugiura, An algorithm based on the FFT for a generalized Chebyshev interpolation, Math. Comp. 54 (1990) 195–210.

    Google Scholar 

  22. D. K. Kahaner, C. Moler and S. Nash,Numerical Methods and Software (Prentice-Hall, Englewood Cliffs, NJ, 1989).

    Google Scholar 

  23. D. Levin and A. Sidi, Two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series, Appl. Math. Comput. 9 (1981) 204–215.

    Google Scholar 

  24. P. Linz, A method for computing Bessel function integrals, Math. Comp. 26 (1972) 509–513.

    Google Scholar 

  25. I. M. Longman, Tables for rapid and accurate numerical evaluation of certain infinite integrals involving Bessel functions, MTAC 11 (1957) 166–180.

    Google Scholar 

  26. I. M. Longman, A method for the numerical evaluation of finite integrals of oscillatory functions, Math. Comp., 14 (1960) 53–59.

    Google Scholar 

  27. D. W. Lozier, Numerical solution of linear difference equations, Report NBSIR 80-1976, NBS, Washington (1980).

    Google Scholar 

  28. Y. L. Luke,Mathematical Functions and Their Approximations (Academic Press, New York, 1975).

    Google Scholar 

  29. J. Lund, Bessel transforms and rational extrapolation, Numer. Math. 47 (1985) 1–14.

    Google Scholar 

  30. J. Lyness and G. Hines, Algorithm 639, to integrate some infinite oscillating tails, ACM Trans. Math. Software 12 (March 1986) 24–25.

    Google Scholar 

  31. R. Piessens and M. Branders, Approximation for Bessel functions and their application in the computation of Hankel transforms, Comput. Math. Appl. 8 (1982) 305–311.

    Google Scholar 

  32. R. Piessens and M. Branders, Modified Clenshaw-Curtis method for the computation of Bessel function integrals, BIT 23 (1983) 370–381.

    Google Scholar 

  33. R. Piessens, E. deDoncker-Kapenga, C. W. Überhuber and D. K. Kahaner,QUADPACK, A Subroutine Package for Automatic Integration (Springer, Berlin, 1983).

    Google Scholar 

  34. R. Piessens and A. Haegemans, Algorithm 22, algorithm for the automatic integration of highly oscillatory functions, Computing 13 (1974) 183–193.

    Google Scholar 

  35. A. Sidi, Extrapolation methods for oscillatory infinite integrals, J. Inst. Math. Appl. 26 (1980) 1–20.

    Google Scholar 

  36. A. Sidi, The numerical evaluation of very oscillatory infinite integrals by extrapolation, Math. Comp. 38 (1982) 517–529.

    Google Scholar 

  37. A. Sidi, An algorithm for a special case of a generalization of the Richardson extrapolation process, Numer. Math. 38 (1982) 299–307.

    Google Scholar 

  38. A. Sidi, A user-friendly extrapolation method for oscillatory infinite integrals, Math. Comp. 51 (1988) 249–266.

    Google Scholar 

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Communicated by G. Mühlbach

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Hasegawa, T., Sidi, A. An automatic integration procedure for infinite range integrals involving oscillatory kernels. Numer Algor 13, 1–19 (1996). https://doi.org/10.1007/BF02143123

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  • DOI: https://doi.org/10.1007/BF02143123

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