Abstract
The problem considered is that of evaluating numerically an integral of the form
where the integrand has one or more simple poles in the interval (O,p). Modified forms of the trapezoidal and mid-ordinate rules, taking account of the singularities, are obtained; it is then shown that the resulting approximations can be extrapolated by Romberg's method. Further modifications to deal with the case when the integrand has an integrable branch singularity at one or both ends of the interval of integration are also briefly discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I. A.: Handbook of mathematical functions. New York: Dover Publications Inc. 1965
Bulirsch, R.: Bemerkungen zur Romberg-Integration. Numer. Math.6, 6–16 (1964)
Davis, P. J., Rabinowitz, P.: Numerical integration. Blaisdell 1967
Fox, L.: Romberg integration for a class of singular integrands. Computer J.10, 87–93 (1967)
Gragg, W. B.: On extrapolation algorithms for ordinary initial value problems. J. SIAM (Num. Anal.)2, 384–403 (1965)
Hunter, D. B.: Some Gauss-type formulae for the evaluation of Cauchy principal values of integrals. Numer. Math.19, 419–424 (1972)
Kress, R.: Ein ableitungsfreies Restglied für die trigonometrische Interpolation periodischer analytischer Funktionen. Numer. Math.16, 389–396 (1971)
Longman, I. M.: On the numerical evaluation of Cauchy principal values of integrals. MTAC12, 205–207 (1958)
Luke, Y. L.: Simple formulas for the evaluation of some higher transcendental functions. J. Math. and Phys.34, 298–307 (1956)
Lyness, J. N., Ninham, B. W.: Numerical quadrature and asymptotic expansions. Math. Comp.21, 162–178 (1967)
Navot, I.: An extension of the Euler-Maclaurin summation formula to functions with a branch singularity. J. Math. and Phys.40, 271–276 (1961)
Navot, I.: A further extension of the Euler-Maclaurin summation formula. J. Math. and Phys.41, 155–163 (1962)
Ninham, B. W., Lyness, J. N.: Further asymptotic expansions for the error functional. Math. Comp.23, 71–83 (1969)
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. Nordisk Tidskr. Informationsbehandling (BIT)10, 476–480 (1970)
Romberg, W.: Vereinfachte numerische Integration, Det. Kong. Norske Videnskabers Forhandlinger28, 30–36 (1955)
Rutishauser, H.: Ausdehnung des Rombergschen Prinzips. Numer. Math.5, 48–54 (1963)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hunter, D.B. The numerical evaluation of cauchy principal values of integrals by Romberg integration. Numer. Math. 21, 185–192 (1973). https://doi.org/10.1007/BF01436622
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01436622