Numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures

Computers and Mathematics with Applications

Volumn 36, Issue 8, 1998, Pages 19-39

  Milovanovic G V ^a  

^a UNIVERSITY OF NI   (Serbia)

Author keywords

Bessel functions; Error function; Gaussian quadratures; Legendre functions; Numerical integration; Orthogonal polynomials; Oscillatory kernel; Product rules; Singular kernel

Indexed keywords

ERROR ANALYSIS; FUNCTIONS; INTEGRATION; POLYNOMIALS; PROBLEM SOLVING;

BESSEL FUNCTIONS; GAUSSIAN QUADRATURES; LEGENDRE FUNCTIONS; PRODUCT INTEGRATION RULES;

NUMERICAL METHODS;

EID: 0032182391     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0898-1221(98)00180-1     Document Type: Article

References (58)

1. G.V. Milovanović, Orthogonal polynomial systems and some applications, In Inner Product Spaces, (Edited by Th.M. Rassias), pp. 115-182, Addison Wesley Longman, Harlow, (1997).
2. G.V. Milovanović, Numerical Analysis, Part I, (in Serbian), Naučna knjiga, Belgrade, (1991).
3. R. Wong, Quadrature formulas for oscillatory integral transforms, Numer. Math. 39, 351-360, (1982).
4. B. Gabutti, On high precision methods for computing integrals involving Besse functions, Math. Comp. 147, 1049-1057, (1979).
5. L.N.G. Filon, On a quadrature formula for trigonometric integrals, Proc. Roy. Soc. Edinburgh 49, 38-47, (1928).
6. T. Håvie, Remarks on an expansion for integrals of rapidly oscillating functions, BIT 13, 16-29, (1973).
7. U.T. Ehrenmark, On the error term of the Filon quadrature formulae, BIT 27, 85-97, (1987).
8. E.A. Flinn, A modification of Filon's method of numerical integration, J. Assoc. Comput. Mach. 7, 181-184, (1960).
9. Y.L. Luke, On the computation of oscillatory integrals, Proc. Camb. Phil. Soc. 50, 269-277, (1954).
10. L. Buyst and L. Schotsmans, A method of Gaussian type for the numerical integration of oscillating functions, ICC Bull. 3, 210-214, (1964).
11. E.O. Tuck, A simple Filon-trapezoidal rule, Math. Comp. 21, 239-241, (1967).
12. B. Einarsson, Numerical calculation of Fourier integrals with cubic splines, BIT 8, 279-286, (1968).
13. A.I. van de Vooren and H. J. van Linde, Numerical calculation of integrals with strongly oscillating integrand, Math. Comp. 20, 232-245, (1966).
14. H.J. Stetter, Numerical approximation of Fourier transforms, Numer. Math. 8, 235-249, (1966).
15. J. Miklosko, Numerical integration with weight functions cos kcursive chi, sin kcursive chi on [0, 2π/t], t = 1,2,..., Apl. Mat. 14, 179-194, (1969).
16. W. Gautschi, Tables of Gaussian quadrature rules for the calculation of Fourier coefficients, Math. Comp. 24, (1970).
17. R. Piessens, Gaussian quadrature formulas for the integration of oscillating functions, Z. Angew. Math. Mech. 50, 698-700, (1970).
18. R. Piessens, Gaussian quadrature formulas for the integration of oscillating functions, Math. Comp. 24, (1970).
19. R. Piessens, Gaussian quadrature formulas for the evaluation of Fourier-cosine coefficients, Z. Angew. Math. Mech. 52, 56-58, (1972).
20. V.I. Krylov and L.G. Kruglikova, A Handbook on Numerical Harmonic Analysis (in Russian), Izdat. Nauka i Tehnika, Minsk, (1968).
21. P.J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, (1975).
22. I.M. Longmann, A method for the numerical evaluation of finite integrals of oscillatory functions, Math. Comp. 14, 53-59, (1960).
23. R. Piessens and M. Branders, Tables of Gaussian Quadrature Formulas, Appl. Math. Progr. Div., University of Leuven, Leuven, (1975).
24. R. Piessens and A. Haegemans, Numerical calculation of Fourier transform integrals, Electron. Lett. 9, 108-109, (1973)).
25. J.F. Price, Discussion of quadrature formulas for use on digital computers, Boeing Scientific Research Labs., Report D1-82-0052, (1960).
26. I.M. Longmann, Note on a method for computing infinite integrals of oscillatory functions, Proc. Camb. Phil. Soc. 52, 764-768, (1956).
27. H. Hurwitz, Jr. and P.F. Zweifel, Numerical quadrature of Fourier transform integrals, MTAC 10, 140-149, (1956).
28. G.V. Milovanović and S. Wrigge, Least squares approximation with constraints, Math. Comp. 46, 551-565, (1986).
29. N.S. Bakhvalov and L.G. Vasil'eva, Evaluation of the integrals of oscillating functions by interpolation at nodes of Gaussian quadratures, U.S.S.R. Comput. Math. and Math. Phys. 8, 241-249, (1968).
30. T.N.L. Patterson, On high precision methods for the evaluation of Fourier integrals with finite and infinite limits, Numer. Math. 27, 41-52, (1976).
31. R. Piessens and F. Poleunis, A numerical method for the integration of oscillatory functions, BIT 11, 317-327, (1971).
32. C.L. Frenzen and R. Wong, A note on asymptotic evaluation of some Hankel transforms, Math. Comp. 45, 537-548, (1985).
33. B. Gabutti, An asymptotic approximation for a class of oscillatory infinite integrals, SIAM J. Numer. Anal. 22, 1191-1199, (1985).
34. B. Gabutti and P. Lepora, A novel approach for the determination of asymptotic expansions of certain oscillatory integrals, J. Comput. Appl. Math. 19, 189-206, (1987).
35. D. Elliott and D.F. Paget, Product-integration rules and their convergence, BIT 16, 32-40, (1976).
36. D. Elliott and D.F. Paget, The convergence of product integration rules, BIT 18, 137-141, (1978).
37. I.H. Sloan and W.E. Smith, Product integration with the Clenshaw-Curtis and related points. Convergence properties, Numer. Math. 30, 415-428, (1978).
38. I.H. Sloan and W.E. Smith, Product integration with the Clenshaw-Curtis: Implementation and error estimates, Numer. Math. 34, 387-401, (1980).
39. I.H. Sloan and W.E. Smith, Properties of interpolatory product integration rules, SIAM J. Numer. Anal. 19, 427-442, (1982).
40. W.E. Smith and I.H. Sloan, Product integration rules based on the zeros of Jacobi polynomials, SIAM J. Numer. Anal. 17, 1-13, (1980).
41. P. Nevai, Mean convergence of Lagrange interpolation. I, J. Approx. Theory 18, 363-377, (1976).
42. P. Nevai, Mean convergence of Lagrange interpolation. II, J, Approx. Theory 30, 263-276, (1980).
43. P. Nevai, Mean convergence of Lagrange interpolation. III, Trans. Amer. Math. Soc. 282, 669-698, (1984).
44. G. Mastroianni and P. Vértesi, Error estimates of product quadrature formulae, In Numerical Integration IV, (Edited by H. Brass and G. Hämmerlin), ISNM, Volume 112, pp. 241-252, Birkhäuser Verlag, Basel, (1993).
45. G. Mastroianni and P. Vértesi, Mean convergence of Lagrange interpolation on arbitrary system of nodes, Acta Sci. Math. (Szeged) 57, 429-441, (1993).
46. G. Mastroianni and G. Monegato, Polynomial approximations of functions with endpoint singularities and product integration formulas, Math. Comp. 62, 725-738, (1994).
47. G. Mastroianni, Boundedness of Lagrange operator in some functional spaces. A survey, In Approximation Theory and Functional Series, (Budapest, 1995), pp. 117-139, Bolyai Society Math. Studies, Volume 5, Budapest, (1996).
48. G.V. Milovanović and M.Ć. Stefanović, A contribution in numerical integration of the error function, (in Serbian), In Proceedings of the Second Meeting "Numerical Methods in Technics", (Stubičke Toplice, 1980), pp. 395-400, Faculty of Electrical Engineering, Zagreb, (1980).
49. H. Bateman and A. Erdélyi, Higher Transcendental Functions, Volume 2, McGraw-Hill, New York, (1953).
50. th Conference ETAN, pp. 567-573, Yugoslav Committee for ETAN, Mostar, (1981).
51. rd Internat. Symp. "Computer at the University", (Cavtat, 1981), The University Computing Center, Zagreb, (1981).
52. th Conference ETAN, pp. 559-566, Yugoslav Committee for ETAN, Mostar, (1981).
53. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, (1970).
54. R.V. Lascenov, On a class of orthogonal polynomials, (in Russian), Učen. Zap. Leningrad. Gos. Ped. Inst. 89, 191-206, (1953).
55. T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, (1978).
56. th Conference on Applied Mathematics, (Edited by Z. Bohte), pp. 81-88, University of Ljubljana, Ljubljana, (1986).
57. G.V. Milovanović and M.A. Kovačević, Least squares approximation with constraint: Generalized Gegenbauer case, Facta Univ. Ser. Math. Inform. 1, 73-81, (1986).
58. A.D. Bandrauk, Problem 97-7: A family of integrals occurring in quantum mechanics, SIAM Rev. 39, 317-318, (1997).