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The numerical evaluation of one-dimensional Cauchy principal value integrals

Die numerische Bestimmung der eindimensionalen Cauchyschen Hauptwertintegralen

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Abstract

In this paper we examine the numerical integration (in the Cauchy principal value sense) of functions having (several) first order real poles. We give a survey of results concerning some quadrature formulas of interpolatory type proposed by Delves, Hunter, Elliott and Paget, and several other authors; along with the description we present some minor generalizations and make comments on the computational aspects. Finally, we propose an alternative algorithm for the numerical evaluation of integrals of the form

Zusammenfassung

In dieser Arbeit untersuchen wir die numerische Integration (d. h. die Bestimmung des Hauptwertes im Sinne von Cauchy) von Funktionen mit mehreren reellen Polen erster Ordnung. Wir beschreiben Quadraturformeln vom interpolatorischen Typus, die von Delves, Hunter, Elliott-Paget und anderen Autoren gegeben sind: einige einfache Verallgemeinerungen werden vorgeschlagen und berechnungstechnische Fragen werden diskutiert. Endlich geben wir einen alternativen Algorithmus zum Ausrechnen von Integralen der Form

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References

  1. Abramowitz, M., Stegun, I. A.: Handbook of mathematical functions. Washington, D. C.: Nat. Bureau of Standards, A.M.S. 55. 1967.

    Google Scholar 

  2. Acharya, B. P., Das, R. N.: Numerical determination of Cauchy principal value integrals. Computing27, 373–378 (1981).

    Google Scholar 

  3. Askey, R.: Positivity of the Cotes number for some Jacobi abscissas. Numer. Math.19, 46–48 (1972).

    Google Scholar 

  4. Chawla, M. M., Kumar, S.: Convergence of quadratures for Cauchy principal value integrals. Computing23, 67–72 (1979).

    Google Scholar 

  5. Chawla, M. M., Ramakrishnan, T. R.: Modified Gauss-Jacobi quadrature formulas for the numerical evaluation of Cauchy type singular integrals. BIT14, 14–21 (1974).

    Google Scholar 

  6. Davis, P. J., Rabinowitz, P.: Methods of numerical integration. New York: Academic Press 1975.

    Google Scholar 

  7. Delves, L. M.: The numerical evaluation of principal value integrals. Comput. J.10, 389–391 (1968).

    Google Scholar 

  8. Elliott, D.: On the convergence of Hunter's quadrature rule for Cauchy principal value integrals. BIT19, 457–462 (1979).

    Google Scholar 

  9. Elliott, D., Paget, D. F.: Gauss type quadrature rules for Cauchy principal value integrals. Math. Comp.33, 301–309 (1979).

    Google Scholar 

  10. Elliott, D., Paget, D. F.: On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals. Numer. Math.23, 311–319 (1975); Addendum: Numer. Math.25, 287–289 (1976).

    Google Scholar 

  11. Ergodan, F., Gupta, G. D.: On the numerical solution of singular integral equations. Quart. Appl. Math.29, 525–534 (1972).

    Google Scholar 

  12. Gautschi, W.: A survey of Gauss-Christoffel quadrature formulas, in: E. B. Christoffel; the influence of his work in Mathematics and Physical Sciences. International Christoffel Symposium. A collection of articles in honour of Christoffel on the 150th anniversary of his birth (Butzer, P. L., Fehér, F., eds.). Basel: Birkhäuser 1981.

    Google Scholar 

  13. Hunter, D. B.: Some Gauss-type formulae for the evaluation of Cauchy principal values of integrals. Numer. Math.19, 419–424 (1972).

    Google Scholar 

  14. Ioakimidis, N. I., Theocaris, P. S.: A comparison between the direct and the classical numerical methods for the solution of Cauchy type singular integral equations. SIAM J. Numer. Anal.17, 115–118 (1980).

    Google Scholar 

  15. Ioakimidis, N. I.: On the numerical evaluation of derivatives of Cauchy principal value integrals. Computing27, 81–88 (1981).

    Google Scholar 

  16. Korneičuk, A. A.: Quadrature formulae for singular integrals. Ž. Vyčisl. Mat. i Mat. Fiz.4, n. 4, suppl., 64–74 (1964). (In Russian.)

    Google Scholar 

  17. Krenk, S.: On quadrature formulas for singular integral equations of the first and the second kind. Quart. Appl. Math.33, 225–232 (1975).

    Google Scholar 

  18. Krenk, S.: Quadrature formulae of closed type for solution of singular integral equations. J. Inst. Maths. Applics.22, 99–107 (1978).

    Google Scholar 

  19. Kumar, S.: A note on quadrature formulae for Cauchy principal value integrals. J. Inst. Maths. Applics.26, 447–451 (1980).

    Google Scholar 

  20. Mastjanica, V. S.: Application of parabolic splines to the approximate computation of a singular integral. Vesci. Akad. Navuk. BSSR Ser. Fiz.-Mat. Navuk1979, n. 2, 124–126. MR. 80f:65028.

  21. Noble, B., Beighton, S.: Error estimates for three methods of evaluating Cauchy principal value integrals. J. Inst. Maths. Applics.26, 431–446 (1980).

    Google Scholar 

  22. Paget, D. F., Elliott, D.: An algorithm for the numerical evaluation of certain Cauchy principal value integrals. Numer. Math.19, 373–385 (1972).

    Google Scholar 

  23. Piessens, R.: Numerical evaluation of Cauchy principal values of integrals. BIT10, 476–480 (1970).

    Google Scholar 

  24. Piessens, R., Van Roy-Branders, M., Mertens, I.: The automatic evaluation of Cauchy principal value integrals. Angew. Informatik1, 31–35 (1976).

    Google Scholar 

  25. Pykhteev, G. N., Shokamolov, I.: Interpolated quadrature formulae containing derivatives for some Cauchy type integrals and for their principal values. Zh. Vychisl. Mat. i Mat. Fiz.10, 2, 438–444 (1970).

    Google Scholar 

  26. Sanikidze, D. G.: The convergence of a quadrature process for certain singular integrals. Ž. Vyčisl. Mat. i Mat. Fiz.10, 1, 189–196 (1970).

    Google Scholar 

  27. Sloan, I. H.: The numerical evaluation of principal-value integrals. J. Comp. Phys.3, 332–333 (1968).

    Google Scholar 

  28. Stark, V. J. E.: A generalized quadrature formula for Cauchy integrals. AIAA J.9, 1854–1855 (1971).

    Google Scholar 

  29. Szegö, G.: Orthogonal Polynomials. Amer. Math. Soc. Colloquium Publications23. Providence, R. I.: 1975.

  30. Takahasi, H., Mori, M.: Estimation of errors in the numerical quadrature of analytic functions. Appl. Anal.1, 201–229 (1971).

    Google Scholar 

  31. Theocaris, P. S., Ioakimidis, N. I.: Numerical integration methods for the solution of singular integral equations. Quart. Appl. Math.35, 173–187 (1977).

    Google Scholar 

  32. Tsamasphyros, G., Theocaris, P. S.: Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations. Computing27, 71–80 (1981).

    Google Scholar 

  33. Van der Sluis, A., Zweerus, J. R.: An appraisal of some methods for computing Cauchy principal values of integrals, in: Numerische Integration (Hämmerlin, G., ed.), pp. 264–287. Basel: Birkhäuser 1979.

    Google Scholar 

  34. Kalandiya, A. I.: On a direct method of solution of an equation in wing theory and its application to the theory of elasticity. Mat. sb.42, 249–272 (1957). (In Russian.)

    Google Scholar 

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  1. Politecnico di Torino, Istituto Matematico, Corso Duca degli Abruzzi, 24, I-10129, Torino, Italy

    G. Monegato

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Work sponsored by the Italian Research Council under contract n. 80.02188.01.

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Monegato, G. The numerical evaluation of one-dimensional Cauchy principal value integrals. Computing 29, 337–354 (1982). https://doi.org/10.1007/BF02246760

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