Computation of integrals with oscillatory and singular integrands
Authors:
Bing Yuan Ting and Yudell L. Luke
Journal:
Math. Comp. 37 (1981), 169183
MSC:
Primary 65D30; Secondary 41A60
MathSciNet review:
616369
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Abstract: This paper is concerned with evaluation of integrals whose integrands are oscillatory and contain singularities at the endpoints of the interval of integration. A typical form is , where a and b can be finite or infinite, is a parameter which is usually large, is analytic in the range of integration, and the singularities are encompassed in the weight function . We suppose that can be expanded in series of polynomials which are orthogonal over the interval of integration with respect to . There are two such expansions for . One is an infinite series which follows from the usual orthogonality property. The other is a polynomial approximation plus a remainder. The relations between the coefficients in these representations are detailed and methods for the evaluation of these are analyzed. Error analyses are provided. A numerical example is given to illustrate the effectiveness of the schemes developed.
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 [1]
 L. N. G. Filon, "On a quadrature formula for trigonometric integrals," Proc. Roy. Soc. Edinburgh, v. 49, 1929, pp. 3847.
 [2]
 Y. L. Luke, "On the computation of oscillatory integrals," Part 2, Proc. Cambridge Philos. Soc., v. 50, 1954, pp. 269277. MR 0062518 (15:992b)
 [3]
 J. N. Lyness, "The calculation of Fourier coefficients by Möbius inversion of the Poisson summation formula. Part I, Functions whose early derivatives are continuous," Math. Comp., v. 24, 1970, pp. 101135. MR 0260230 (41:4858)
 [4]
 J. N. Lyness, "The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. Part II, Piecewise continuous functions and functions with poles near [0, 1]," Math. Comp., v. 25, 1971, pp. 5978. MR 0293846 (45:2922)
 [5]
 J. N. Lyness, "The calculation of Fourier coefficients by the Möbius inversion of the Poisson summation formula. Part III, Functions having algebraic singularities," Math. Comp., v. 25, 1971, pp. 483493. MR 0297162 (45:6220)
 [6]
 Y. L. Luke, "On the error in a certain interpolation formula and in the Gaussian integration formula," J. Austral. Math. Soc. Ser. A (2), v. 19, 1975, pp. 196209. MR 0386232 (52:7090)
 [7]
 Y. L. Luke, B. Y. Ting & M. J. Kemp, "On generalized Gaussian quadrature," Math. Comp., v. 29, 1975, pp. 10831093. MR 0388740 (52:9574)
 [8]
 N. S. Bakhvalov & L. G. Vasiléva, "The calculation of integrals of oscillatory functions by interpolation of the Gaussian quadrature nodes," Ž. Vyčisl. Mat. i Mat. Fiz., v. 8, 1968, pp. 175181. MR 0226851 (37:2437)
 [9]
 R. Piessens & F. Poleunis, "A numerical method for the integration of oscillatory functions," BIT, v. 11, 1971, pp. 317327. MR 0288959 (44:6154)
 [10]
 T. N. L. Patterson, "On higher precision methods for the evaluation of Fourier integrals with finite and infinite limits," Numer. Math., v. 27, 1976, pp. 4152. MR 0433932 (55:6902)
 [11]
 R. K. Littlewood & V. Zakian, "Numerical evaluation of Fourier integrals," J. Inst. Math. Appl., v. 18, 1976, pp. 331339. MR 0448822 (56:7127)
 [12]
 P. J. Davis & P. Rabinowitz, Numerical Integration, Blaisdell, Boston, Mass., 1967. MR 0211604 (35:2482)
 [13]
 J. D. Donaldson & D. Elliott, "A uniform approach to quadrature rules with asymptotic estimates of their remainders," SIAM J. Numer. Anal., v. 9, 1972, pp. 573602. MR 0317522 (47:6069)
 [14]
 C. W. Clenshaw & A. R. Curtis, "A method for numerical integration on an automatic computer," Numer. Math., v. 2, 1960, pp. 197205. MR 0117885 (22:8659)
 [15]
 G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford and New York, 1971.
 [16]
 A. C. Aitken, "On Bernoulli's numerical solution of algebraic equations," Proc. Roy. Soc. Edinburgh, v. 46, 1926, pp. 289305.
 [17]
 I. P. Natanson, Constructive Function Theory, Vols. 1, 2 and 3, Ungar, New York, 1964.
 [18]
 J. Prasad, "On an approximation of a function and its derivatives," Publ. Inst. Math., v. 14 (28), 1972, pp. 129132. MR 0407524 (53:11297)
 [19]
 A. Erdélyi et al., Higher Transcendental Functions, Vol. 2, McGrawHill, New York, 1954.
 [20]
 Y. L. Luke, The Special Functions and Their Approximations, Vols. 1 and 2, Academic Press, New York and London, 1969.
 [21]
 W. Gautschi, "Computational aspects of threeterm recurrence relations," SIAM Rev., v. 9, 1967, pp. 2482. MR 0213062 (35:3927)
 [22]
 J. Wimp, On recursive computation, Report ARL 690186, Aerospace Res. Labs., WrightPatterson Air Force Base, Ohio, 1969. MR 0253554 (40:6768)
 [23]
 J. Wimp, "Recent development in recursive computation," SIAM Studies in Appl. Math., Vol. VI, Philadelphia, Pa., 1970, pp. 101123.
 [24]
 Y. L. Luke, "On the error in the Padé approximants for a form of the incomplete gamma function including the exponential function," SIAM J. Math. Anal., v. 6, 1975, pp. 829839. MR 0385413 (52:6275)
 [25]
 B. Y. Ting, Evaluation of Integrals Whose Integrands Are Oscillatory and Singular, Ph.D. Thesis, University of Missouri, Kansas City, Mo., 1979.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198106163695
PII:
S 00255718(1981)06163695
Keywords:
Fourier coefficients,
numerical quadrature,
integration of oscillatory and singular integrands,
interpolation
Article copyright:
© Copyright 1981
American Mathematical Society
