Numerical evaluation of Cauchy principal value integrals with singular integrands
HTML articles powered by AMS MathViewer
- by Philip Rabinowitz PDF
- Math. Comp. 55 (1990), 265-275 Request permission
Abstract:
Convergence results are proved for sequences of interpolatory integration rules for Cauchy principal value integrals of the form \[ \oint k(x)(f(x)/(x - \lambda )) dx,\quad - 1 < \lambda < 1,\] when $f(x)$ is singular at a point $\xi \ne \lambda$ and the singularity is ignored or avoided.References
- Giuliana Criscuolo and Giuseppe Mastroianni, On the convergence of an interpolatory product rule for evaluating Cauchy principal value integrals, Math. Comp. 48 (1987), no. 178, 725–735. MR 878702, DOI 10.1090/S0025-5718-1987-0878702-4 G. Criscuolo, G. Mastroianni, and P. Nevai, Associated generalized Jacobi functions and polynomials (submitted).
- G. Monegato, The numerical evaluation of one-dimensional Cauchy principal value integrals, Computing 29 (1982), no. 4, 337–354 (English, with German summary). MR 684742, DOI 10.1007/BF02246760
- Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, DOI 10.1090/memo/0213
- Paul Nevai and Péter Vértesi, Mean convergence of Hermite-Fejér interpolation, J. Math. Anal. Appl. 105 (1985), no. 1, 26–58. MR 773571, DOI 10.1016/0022-247X(85)90095-2 P. Rabinowitz, Some practical aspects in the numerical evaluation of Cauchy principal value integrals, Internat. J. Comput. Math. 20 (1986), 283-298.
- Philip Rabinowitz, Rates of convergence of Gauss, Lobatto, and Radau integration rules for singular integrands, Math. Comp. 47 (1986), no. 176, 625–638. MR 856707, DOI 10.1090/S0025-5718-1986-0856707-6
- Philip Rabinowitz, Numerical integration in the presence of an interior singularity, J. Comput. Appl. Math. 17 (1987), no. 1-2, 31–41. MR 884259, DOI 10.1016/0377-0427(87)90036-7
- Philip Rabinowitz, On an interpolatory product rule for evaluating Cauchy principal value integrals, BIT 29 (1989), no. 2, 347–355. MR 997540, DOI 10.1007/BF01952688
- Philip Rabinowitz and Ian H. Sloan, Product integration in the presence of a singularity, SIAM J. Numer. Anal. 21 (1984), no. 1, 149–166. MR 731219, DOI 10.1137/0721010
- Philip Rabinowitz and William E. Smith, Interpolatory product integration for Riemann-integrable functions, J. Austral. Math. Soc. Ser. B 29 (1987), no. 2, 195–202. MR 905804, DOI 10.1017/S0334270000005713
- Ian H. Sloan and William E. Smith, Properties of interpolatory product integration rules, SIAM J. Numer. Anal. 19 (1982), no. 2, 427–442. MR 650061, DOI 10.1137/0719027
- William E. Smith and Ian H. Sloan, Product-integration rules based on the zeros of Jacobi polynomials, SIAM J. Numer. Anal. 17 (1980), no. 1, 1–13. MR 559455, DOI 10.1137/0717001 G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1959.
- P. Vértesi, Remarks on convergence of Gaussian quadrature for singular integrals, Acta Math. Hungar. 53 (1989), no. 3-4, 399–405. MR 1014923, DOI 10.1007/BF01953377
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Math. Comp. 55 (1990), 265-275
- MSC: Primary 65D30
- DOI: https://doi.org/10.1090/S0025-5718-1990-1023768-4
- MathSciNet review: 1023768