Gauss type quadrature rules for Cauchy principal value integrals

Authors:
David Elliott and D. F. Paget

Journal:
Math. Comp. **33** (1979), 301-309

MSC:
Primary 65D32; Secondary 41A55

DOI:
https://doi.org/10.1090/S0025-5718-1979-0514825-2

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.

**[1]**W. BARRETT, "An asymptotic formula relating to orthogonal polynomials,"*J. London Math. Soc.*(2), v. 6, 1973, pp. 701-704. MR**48**#593. MR**0322231 (48:593)****[2]**M. M. CHAWLA & N. JAYARAJAN, "Quadrature formulas for Cauchy principal value integrals,"*Computing*, v. 15, 1975, pp. 347-355. MR**54**#4068. MR**0415991 (54:4068)****[3]**M. M. CHAWLA & T. R. RAMAKRISHNAN, "Modified Gauss-Jacobi quadrature formulas for the numerical evaluation of Cauchy type singular integrals,"*BIT*, v. 14, 1974, pp. 14-21. MR**51**#9437. MR**0331729 (48:10061)****[4]**C. W. CLENSHAW, "A note on the summation of Chebyshev series,"*Math. Comp.*, v. 9, 1955, pp. 118-120. MR**17**, 194. MR**0071856 (17:194e)****[5]**P. J. DAVIS & P. RABINOWITZ,*Methods of Numerical Integration*, Academic Press, New York, 1975. MR**0448814 (56:7119)****[6]**L. M. DELVES, "The numerical evaluation of principal value integrals,"*Comput. J.*, v. 10, 1967/1968, pp. 389-391. MR**36**#4802. MR**0221750 (36:4802)****[7]**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. 311-319; v. 25, 1976, pp. 287-289. MR**52**#1115;**53**#14870. MR**0380215 (52:1115)****[8]**F. ERDOGAN & G. D. GUPTA, "On the numerical solution of singular integral equations,"*Quart. Appl. Math.*, v. 29, 1972, pp. 525-534. MR**53**#12042. MR**0408277 (53:12042)****[9]**D. B. HUNTER, "Some Gauss-type formulae for the evaluation of Cauchy principal values of integrals,"*Numer. Math.*, v. 19, 1972, pp. 419-424. MR**47**#7899. MR**0319355 (47:7899)****[10]**N. I. IOAKIMIDIS & P. S. THEOCARIS, "The Gauss-Hermite numerical integration method for the solution of the plane elastic problem of semi-infinite periodic cracks,"*Internat. J. Engrg. Sci.*, v. 15, 1977, pp. 271-280. MR**0495560 (58:14230)****[11]**A. I. KALANDIYA, "On a direct method of solution of an equation in wing theory with an application to the theory of elasticity,"*Mat. Sb.*, v. 42, 1957, pp. 249-272. (Russian) MR**21**#6800. MR**0108079 (21:6800)****[12]**V. I. LEBEDEV & O. V. BABURIN, "Calculation of the principal values, weights and nodes of the Gauss quadrature formulae of integrals,"*U.S.S.R. Comput. Math. and Math. Phys.*, v. 5, 1965, pp. 81-92. MR**32**#6670. MR**0189243 (32:6670)****[13]**D. F. PAGET,*Generalized Product Integration*, Ph. D. thesis, Univ. of Tasmania, 1976.**[14]**D. F. PAGET & D. ELLIOTT, "An algorithm for the numerical evaluation of certain Cauchy principal value integrals,"*Numer. Math.*, v. 19, 1972, pp. 373-385. MR**51**#2256. MR**0366004 (51:2256)****[15]**R. PIESSENS, "Numerical evaluation of Cauchy principal values of integrals,"*BIT*, v. 10, 1970, pp. 476-480. MR**43**#7066. MR**0281348 (43:7066)****[16]**R. PIESSENS, M. VAN ROY-BRANDERS & I. MERTENS, "The automatic evaluation of Cauchy principal value integrals,"*Angewandte Informatik*, v. 1, 1976, pp. 31-35.**[17]**J. F. PRICE,*Discussion of Quadrature Formulas for Use on Digital Computers*, Rep. D1-82-0052, Boeing Sci. Res. Labs., 1960.**[18]**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. 1536-1548. MR**22**#12698. MR**0121971 (22:12698)****[19]**V.J.E. STARK, "A generalized quadrature formula for Cauchy integrals,"*AIAA J.*, v. 9, 1971, pp. 1854-1855. MR**45**#4639. MR**0295573 (45:4639)****[20]**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. 1-39. MR**0659483 (58:31969)****[22]**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. 458-464. MR**0468120 (57:7959)**

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DOI:
https://doi.org/10.1090/S0025-5718-1979-0514825-2

Article copyright:
© Copyright 1979
American Mathematical Society