Quadrature formulas for functions with poles near the interval of integration

Monegato, Giovanni

doi:10.1090/S0025-5718-1986-0842137-X

Quadrature formulas for functions with poles near the interval of integration
HTML articles powered by AMS MathViewer

by Giovanni Monegato PDF

Math. Comp. 47 (1986), 301-312 Request permission

Abstract:

In this paper, we examine the construction of quadrature rules of interpolatory type, using only real function values, for functions with complex conjugate pairs of poles near the interval of integration.

References

Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0448814
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 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 10.1007/BF01931690
Walter Gautschi, A survey of Gauss-Christoffel quadrature formulae, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 72–147. MR 661060

Numerical Integration

Walter Gautschi, Minimal solutions of three-term recurrence relations and orthogonal polynomials, Math. Comp. 36 (1981), no. 154, 547–554. MR 606512, DOI 10.1090/S0025-5718-1981-0606512-6
Walter Gautschi, Discrete approximations to spherically symmetric distributions, Numer. Math. 44 (1984), no. 1, 53–60. MR 745085, DOI 10.1007/BF01389754
Walter Gautschi and Gradimir V. Milovanović, Gaussian quadrature involving Einstein and Fermi functions with an application to summation of series, Math. Comp. 44 (1985), no. 169, 177–190. MR 771039, DOI 10.1090/S0025-5718-1985-0771039-1

Tables of Integrals, Series, and Products

Peter Henrici, Applied and computational complex analysis, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros. MR 0372162
Frank G. Lether, Modified quadrature formulas for functions with nearby poles, J. Comput. Appl. Math. 3 (1977), no. 1, 3–9. MR 440876, DOI 10.1016/0771-050X(77)90017-1
G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR 0213785
G. Monegato, The numerical evaluation of one-dimensional Cauchy principal value integrals, Computing 29 (1982), no. 4, 337–354 (English, with German summary). MR 684742, DOI 10.1007/BF02246760
Ian H. Sloan, On the numerical evaluation of singular integrals, BIT 18 (1978), no. 1, 91–102. MR 501799, DOI 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 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 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 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 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 10.1137/0717001
William E. Smith, Ian H. Sloan, and Alex H. Opie, Product integration over infinite intervals. I. Rules based on the zeros of Hermite polynomials, Math. Comp. 40 (1983), no. 162, 519–535. MR 689468, DOI 10.1090/S0025-5718-1983-0689468-1

Orthogonal Polynomials