Gauss type quadrature rules for Cauchy principal value integrals
Authors:
David Elliott and D. F. Paget
Journal:
Math. Comp. 33 (1979), 301309
MSC:
Primary 65D32; Secondary 41A55
MathSciNet review:
514825
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Abstract: Two quadrature rules for the approximate evaluation of Cauchy principal value integrals, with nodes at the zeros of appropriate orthogonal polynomials, are discussed. An expression for the truncation error, in terms of higher order derivatives, is given for each rule. In addition, two theorems, containing sufficient conditions for the convergence of the sequence of quadrature rules to the integral, are proved.
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 W. BARRETT, "An asymptotic formula relating to orthogonal polynomials," J. London Math. Soc. (2), v. 6, 1973, pp. 701704. MR 48 #593. MR 0322231 (48:593)
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 M. M. CHAWLA & T. R. RAMAKRISHNAN, "Modified GaussJacobi quadrature formulas for the numerical evaluation of Cauchy type singular integrals," BIT, v. 14, 1974, pp. 1421. MR 51 #9437. MR 0331729 (48:10061)
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 D. ELLIOTT & D. F. PAGET, "On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals," Numer. Math., v. 23, 1975, pp. 311319; v. 25, 1976, pp. 287289. MR 52 #1115; 53 #14870. MR 0380215 (52:1115)
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 D. B. HUNTER, "Some Gausstype formulae for the evaluation of Cauchy principal values of integrals," Numer. Math., v. 19, 1972, pp. 419424. MR 47 #7899. MR 0319355 (47:7899)
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 R. PIESSENS, "Numerical evaluation of Cauchy principal values of integrals," BIT, v. 10, 1970, pp. 476480. MR 43 #7066. MR 0281348 (43:7066)
 [16]
 R. PIESSENS, M. VAN ROYBRANDERS & I. MERTENS, "The automatic evaluation of Cauchy principal value integrals," Angewandte Informatik, v. 1, 1976, pp. 3135.
 [17]
 J. F. PRICE, Discussion of Quadrature Formulas for Use on Digital Computers, Rep. D1820052, Boeing Sci. Res. Labs., 1960.
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 G. N. PYKHTEEV, "On the evaluation of certain singular integrals with a kernel of the Cauchy type," J. Appl. Math. Mech., v. 23, 1959, pp. 15361548. MR 22 #12698. MR 0121971 (22:12698)
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 V.J.E. STARK, "A generalized quadrature formula for Cauchy integrals," AIAA J., v. 9, 1971, pp. 18541855. MR 45 #4639. MR 0295573 (45:4639)
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 G. SZEGÖ, Orthogonal Polynomials, 3rd ed., Amer. Math. Soc. Colloq. Publ., v. 23, Amer. Math. Soc., Providence, R. I., 1967. MR 1, 14.
 [21]
 P. S. THEOCARIS & N. I. IOAKIMIDIS, "Numerical solution of Cauchy type singular integral equations," Trans. Acad. Athens, v. 40, 1977, pp. 139. MR 0659483 (58:31969)
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 G. J. TSAMASPHYROS & P. S. THEOCARIS, "On the convergence of a Gauss quadrature rule for evaluation of Cauchy type singular integrals," BIT, v. 17, 1977, pp. 458464. MR 0468120 (57:7959)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197905148252
PII:
S 00255718(1979)05148252
Article copyright:
© Copyright 1979
American Mathematical Society
