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 K(ωt)f(t) and ∫ ∞ a L(ωt)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
W. L. Anderson, Fast Hankel transform using related and lagged convolutions, ACM Trans. Math. Software 8 (1982) 344–368.
M. Branders and R. Piessens, An extension of Clenshaw-Curtis quadrature, J. Comput. Appl. Math. 1 (1975) 55–65.
J. R. Cash, A note on the numerical solution of linear recurrence relations, Numer. Math. 34 (1980) 371–386.
C. W. Clenshaw and A. R. Curtis, A method for numerical integration on an automatic computer, Numer. Math. 2 (1960) 197–205.
B. Davies,Integral Transforms and Their Applications (Springer, New York, 2nd. ed., 1985).
P. J. Davis and P. Rabinowitz,Methods of Numerical Integration (Academic Press, Orlando, FL, 2nd ed., 1984).
D. Elliott, Truncation errors in two Chebyshev series approximations, Math. Comp. 19 (1965) 234–248.
T. O. Espelid and K. J. Overholt, DQAINF: an algorithm for automatic integration of infinite oscillating tails, Numer. Algorithms 8 (1994) 83–101.
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.
W. Gautschi, Computational aspect of three-term recurrence relations, SIAM Rev. 9 (1967) 24–82.
W. Gautschi and R. S. Varga, Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal. 20 (1983) 1170–1186.
W. M. Gentleman, Implementing Clenshaw-Curtis quadrature II. Computing the cosine transformation. Comm. ACM 15 (1972) 343–346.
I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, and Products, translated by A. Jeffrey (Academic Press, New York, 1980).
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.
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).
T. Hasegawa and T. Torii, Indefinite integration of oscillatory functions by the Chebyshev series expansion, J. Comput. Appl. Math. 17 (1987) 21–29.
T. Hasegawa amd T. Torii, An automatic quadrature for Cauchy principal value integrals, Math. Comp. 56 (1991) 741–754.
T. Hasegawa and T. Torii, Application of a modified FFT to product type integration, J. Comput. Appl. Math. 38 (1991) 157–168.
T. Hasegawa and T. Torii, An algorithm for nondominant solutions of linear second-order inhomogeneous difference equations, Math. Comp. 64 (1995) 1199–1214.
T. Hasegawa, T. Torii and I. Ninomiya, Generalized Chebyshev interpolation and its application to automatic quadrature, Math. Comp. 41 (1983) 537–553.
T. Hasegawa, T. Torii and H. Sugiura, An algorithm based on the FFT for a generalized Chebyshev interpolation, Math. Comp. 54 (1990) 195–210.
D. K. Kahaner, C. Moler and S. Nash,Numerical Methods and Software (Prentice-Hall, Englewood Cliffs, NJ, 1989).
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.
P. Linz, A method for computing Bessel function integrals, Math. Comp. 26 (1972) 509–513.
I. M. Longman, Tables for rapid and accurate numerical evaluation of certain infinite integrals involving Bessel functions, MTAC 11 (1957) 166–180.
I. M. Longman, A method for the numerical evaluation of finite integrals of oscillatory functions, Math. Comp., 14 (1960) 53–59.
D. W. Lozier, Numerical solution of linear difference equations, Report NBSIR 80-1976, NBS, Washington (1980).
Y. L. Luke,Mathematical Functions and Their Approximations (Academic Press, New York, 1975).
J. Lund, Bessel transforms and rational extrapolation, Numer. Math. 47 (1985) 1–14.
J. Lyness and G. Hines, Algorithm 639, to integrate some infinite oscillating tails, ACM Trans. Math. Software 12 (March 1986) 24–25.
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.
R. Piessens and M. Branders, Modified Clenshaw-Curtis method for the computation of Bessel function integrals, BIT 23 (1983) 370–381.
R. Piessens, E. deDoncker-Kapenga, C. W. Überhuber and D. K. Kahaner,QUADPACK, A Subroutine Package for Automatic Integration (Springer, Berlin, 1983).
R. Piessens and A. Haegemans, Algorithm 22, algorithm for the automatic integration of highly oscillatory functions, Computing 13 (1974) 183–193.
A. Sidi, Extrapolation methods for oscillatory infinite integrals, J. Inst. Math. Appl. 26 (1980) 1–20.
A. Sidi, The numerical evaluation of very oscillatory infinite integrals by extrapolation, Math. Comp. 38 (1982) 517–529.
A. Sidi, An algorithm for a special case of a generalization of the Richardson extrapolation process, Numer. Math. 38 (1982) 299–307.
A. Sidi, A user-friendly extrapolation method for oscillatory infinite integrals, Math. Comp. 51 (1988) 249–266.
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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
Keywords
- Automatic integration
- infinite oscillatory integral
- Bessel function
- Hankel transform
- extrapolation
- modified W-transformation
- acceleration
- Chebyshev expansion
- FFT