Product integration over infinite intervals. I. Rules based on the zeros of Hermite polynomials

Authors:
William E. Smith, Ian H. Sloan and Alex H. Opie

Journal:
Math. Comp. **40** (1983), 519-535

MSC:
Primary 65D32

DOI:
https://doi.org/10.1090/S0025-5718-1983-0689468-1

MathSciNet review:
689468

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The paper discusses both theoretical properties and practical implementation of product integration rules of the form \[ \int _{ - \infty }^\infty {k(x)f(x) dx \approx \sum \limits _{i = 1}^n {{w_{ni}}f({x_{ni}}),} } \] where *f* is continuous, *k* is absolutely integrable, the nodes $\{ {x_{ni}}\}$ are roots of the Hermite polynomials ${H_n}(x)$, and the weights $\{ {w_{ni}}\}$ are chosen so that the rule is exact if *f* is any polynomial of degree $< n$. Convergence of the rule to the exact integral as $n \to \infty$ is proved for a wide class of functions *f* and *k* (including singular or oscillatory functions *k*), and rates of convergence are estimated. The rules are shown to have the property of asymptotic positivity, and as a consequence exhibit good numerical stability. Numerical calculations for some practical cases are presented, which show the method to be computationally effective for integrands (including highly oscillatory ones) that decay suitably at infinity. Applications of the method to integration over $[0,\infty )$ are also discussed.

- Milton Abramowitz and Irene A. Stegun,
*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642**
N. S. Bakhvalov & L. G. Vasil’eva, "Evaluation of the integrals of oscillating functions by interpolation at nodes of Gaussian quadratures," - M. Blakemore, G. A. Evans, and John Hyslop,
*Comparison of some methods for evaluating infinite range oscillatory integrals*, J. Comput. Phys.**22**(1976), no. 3, 352–376. MR**455300**, DOI https://doi.org/10.1016/0021-9991%2876%2990054-1 - C. W. Clenshaw,
*A note on the summation of Chebyshev series*, Math. Tables Aids Comput.**9**(1955), 118–120. MR**71856**, DOI https://doi.org/10.1090/S0025-5718-1955-0071856-0 - David Elliott and D. F. Paget,
*Product-integration rules and their convergence*, Nordisk Tidskr. Informationsbehandling (BIT)**16**(1976), no. 1, 32–40. MR**405809**, DOI https://doi.org/10.1007/bf01940775 - David Elliott and D. F. Paget,
*The convergence of product integration rules*, BIT**18**(1978), no. 2, 137–141. MR**483319**, DOI https://doi.org/10.1007/BF01931690 - Géza Freud,
*A contribution to the problem of weighted polynomial approximation*, Linear operators and approximation (Proc. Conf., Oberwolfach, 1971), Birkhäuser, Basel, 1972, pp. 431–447. Internat. Ser. Numer. Math., Vol. 20. MR**0402358**
G. Freud, - Bruno Gabutti,
*On high precision methods for computing integrals involving Bessel functions*, Math. Comp.**33**(1979), no. 147, 1049–1057. MR**528057**, DOI https://doi.org/10.1090/S0025-5718-1979-0528057-5
I. S. Gradshteyn & I. M. Ryzhik, - Gene H. Golub and John H. Welsch,
*Calculation of Gauss quadrature rules*, Math. Comp. 23 (1969), 221-230; addendum, ibid.**23**(1969), no. 106, loose microfiche suppl, A1–A10. MR**0245201**, DOI https://doi.org/10.1090/S0025-5718-69-99647-1 - D. R. Lehman, William C. Parke, and L. C. Maximon,
*Numerical evaluation of integrals containing a spherical Bessel function by product integration*, J. Math. Phys.**22**(1981), no. 7, 1399–1413. MR**626130**, DOI https://doi.org/10.1063/1.525061 - Benjamin Muckenhoupt,
*Mean convergence of Hermite and Laguerre series. I, II*, Trans. Amer. Math. Soc. 147 (1970), 419-431; ibid.**147**(1970), 433–460. MR**0256051**, DOI https://doi.org/10.1090/S0002-9947-1970-99933-9 - Paul G. Nevai,
*Mean convergence of Lagrange interpolation. II*, J. Approx. Theory**30**(1980), no. 4, 263–276. MR**616953**, DOI https://doi.org/10.1016/0021-9045%2880%2990030-1
D. F. Paget, - T. N. L. Patterson,
*On high precision methods for the evaluation of Fourier integrals with finite and infinite limits*, Numer. Math.**27**(1976/77), no. 1, 41–52. MR**433932**, DOI https://doi.org/10.1007/BF01399083 - Ian H. Sloan,
*On the numerical evaluation of singular integrals*, BIT**18**(1978), no. 1, 91–102. MR**501799**, DOI https://doi.org/10.1007/BF01947747 - Ian H. Sloan,
*On choosing the points in product integration*, J. Math. Phys.**21**(1980), no. 5, 1032–1039. MR**574876**, DOI https://doi.org/10.1063/1.524552 - Ian H. Sloan and W. E. Smith,
*Product-integration with the Clenshaw-Curtis and related points. Convergence properties*, Numer. Math.**30**(1978), no. 4, 415–428. MR**494863**, DOI https://doi.org/10.1007/BF01398509 - Ian H. Sloan and William E. Smith,
*Product integration with the Clenshaw-Curtis points: implementation and error estimates*, Numer. Math.**34**(1980), no. 4, 387–401. MR**577405**, DOI https://doi.org/10.1007/BF01403676 - Ian H. Sloan and William E. Smith,
*Properties of interpolatory product integration rules*, SIAM J. Numer. Anal.**19**(1982), no. 2, 427–442. MR**650061**, DOI https://doi.org/10.1137/0719027 - William E. Smith and Ian H. Sloan,
*Product-integration rules based on the zeros of Jacobi polynomials*, SIAM J. Numer. Anal.**17**(1980), no. 1, 1–13. MR**559455**, DOI https://doi.org/10.1137/0717001
G. Szegö, - J. V. Uspensky,
*On the convergence of quadrature formulas related to an infinite interval*, Trans. Amer. Math. Soc.**30**(1928), no. 3, 542–559. MR**1501444**, DOI https://doi.org/10.1090/S0002-9947-1928-1501444-8 - Andrew Young,
*Approximate product-integration*, Proc. Roy. Soc. London Ser. A**224**(1954), 552–561. MR**63778**, DOI https://doi.org/10.1098/rspa.1954.0179

*U.S.S.R. Comput. Math. and Math. Phys.*, v. 8, 1968, pp. 241-249.

*Orthogonal Polynomials*, Pergamon Press, Oxford, 1971, Problem 1, pp. 130-131.

*Table of Integrals Series and Products*, Academic Press, New York, 1965.

*Generalised Product Integration*, Ph.D. thesis, University of Tasmania, 1976.

*Orthogonal Polynomials*, Amer. Math. Soc. Colloq. Publ., Vol. 23, Amer. Math. Soc., Providence, R. I., 1939.

Retrieve articles in *Mathematics of Computation*
with MSC:
65D32

Retrieve articles in all journals with MSC: 65D32

Additional Information

Keywords:
Numerical integration,
infinite interval,
product integration,
interpolation,
Hermite polynomials

Article copyright:
© Copyright 1983
American Mathematical Society