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
Abramowitz, M., Stegun, I. A.: Handbook of mathematical functions. Washington, D. C.: Nat. Bureau of Standards, A.M.S. 55. 1967.
Acharya, B. P., Das, R. N.: Numerical determination of Cauchy principal value integrals. Computing27, 373–378 (1981).
Askey, R.: Positivity of the Cotes number for some Jacobi abscissas. Numer. Math.19, 46–48 (1972).
Chawla, M. M., Kumar, S.: Convergence of quadratures for Cauchy principal value integrals. Computing23, 67–72 (1979).
Chawla, M. M., Ramakrishnan, T. R.: Modified Gauss-Jacobi quadrature formulas for the numerical evaluation of Cauchy type singular integrals. BIT14, 14–21 (1974).
Davis, P. J., Rabinowitz, P.: Methods of numerical integration. New York: Academic Press 1975.
Delves, L. M.: The numerical evaluation of principal value integrals. Comput. J.10, 389–391 (1968).
Elliott, D.: On the convergence of Hunter's quadrature rule for Cauchy principal value integrals. BIT19, 457–462 (1979).
Elliott, D., Paget, D. F.: Gauss type quadrature rules for Cauchy principal value integrals. Math. Comp.33, 301–309 (1979).
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).
Ergodan, F., Gupta, G. D.: On the numerical solution of singular integral equations. Quart. Appl. Math.29, 525–534 (1972).
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.
Hunter, D. B.: Some Gauss-type formulae for the evaluation of Cauchy principal values of integrals. Numer. Math.19, 419–424 (1972).
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).
Ioakimidis, N. I.: On the numerical evaluation of derivatives of Cauchy principal value integrals. Computing27, 81–88 (1981).
Korneičuk, A. A.: Quadrature formulae for singular integrals. Ž. Vyčisl. Mat. i Mat. Fiz.4, n. 4, suppl., 64–74 (1964). (In Russian.)
Krenk, S.: On quadrature formulas for singular integral equations of the first and the second kind. Quart. Appl. Math.33, 225–232 (1975).
Krenk, S.: Quadrature formulae of closed type for solution of singular integral equations. J. Inst. Maths. Applics.22, 99–107 (1978).
Kumar, S.: A note on quadrature formulae for Cauchy principal value integrals. J. Inst. Maths. Applics.26, 447–451 (1980).
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.
Noble, B., Beighton, S.: Error estimates for three methods of evaluating Cauchy principal value integrals. J. Inst. Maths. Applics.26, 431–446 (1980).
Paget, D. F., Elliott, D.: An algorithm for the numerical evaluation of certain Cauchy principal value integrals. Numer. Math.19, 373–385 (1972).
Piessens, R.: Numerical evaluation of Cauchy principal values of integrals. BIT10, 476–480 (1970).
Piessens, R., Van Roy-Branders, M., Mertens, I.: The automatic evaluation of Cauchy principal value integrals. Angew. Informatik1, 31–35 (1976).
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).
Sanikidze, D. G.: The convergence of a quadrature process for certain singular integrals. Ž. Vyčisl. Mat. i Mat. Fiz.10, 1, 189–196 (1970).
Sloan, I. H.: The numerical evaluation of principal-value integrals. J. Comp. Phys.3, 332–333 (1968).
Stark, V. J. E.: A generalized quadrature formula for Cauchy integrals. AIAA J.9, 1854–1855 (1971).
Szegö, G.: Orthogonal Polynomials. Amer. Math. Soc. Colloquium Publications23. Providence, R. I.: 1975.
Takahasi, H., Mori, M.: Estimation of errors in the numerical quadrature of analytic functions. Appl. Anal.1, 201–229 (1971).
Theocaris, P. S., Ioakimidis, N. I.: Numerical integration methods for the solution of singular integral equations. Quart. Appl. Math.35, 173–187 (1977).
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).
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.
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.)
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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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DOI: https://doi.org/10.1007/BF02246760