Numerical integration in the presence of an interior singularity. (English) Zbl 0627.41022
This paper is one of a series of investigations by the author on numerical integration when the integrand has a singularity in the interval. He first summarizes the previous results. His method is the Gauss-Jordan rule based on the zeros of the function \((1-x^ 2)P_{n- 2m}^{(\alpha,\beta)}(x),\) \(m=0,1\) where \(\alpha\) and \(\beta\) satisfy \(- \leq \alpha,\beta \leq\) or \(-1<\alpha =\beta\). He then discusses interpolatory integration rules and finally applies his methods to the study of convergence of the methods of D. B. Hunter [Numer. Math. 19, 419-424 (1972; Zbl 0231.65028)] for Cauchy principal value integrals.
Reviewer: S.Hitotumatu
MSC:
| 41A55 | Approximate quadratures |
| 65D32 | Numerical quadrature and cubature formulas |
| 30E20 | Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane |
Keywords:
Hunter’s method; Diophantine approximation; interior singularity; Gauss- Jordan rule; interpolatory integration rules; convergence; Cauchy principal value integralsCitations:
Zbl 0231.65028References:
| [1] | Cassels, J. W.S., An Introduction to Diophantine Approximation (1957), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0077.04801 |
| [2] | Cassels, J. W.S., Some metrical theorems in diophantine approximation. I, Proc. Cambridge Philos. Soc., 46, 209-218 (1950) · Zbl 0035.31902 |
| [3] | Davis, P. J.; Rabinowitz, P., Ignoring the singularity in approximate integration, SIAM J. Numer. Anal., 2, 367-383 (1965) · Zbl 0141.13901 |
| [4] | Eggan, L. C., On diophantine approximation, Trans. Amer. Math. Soc., 99, 102-117 (1961) · Zbl 0100.03904 |
| [5] | Elliott, D.; Paget, D. F., Gauss type quadrature rules for Cauchy principal value integrals, Math. Comp., 33, 301-309 (1979) · Zbl 0415.65019 |
| [6] | Gatteschi, L., On the construction of some Gaussian quadrature rules, (Hämmerlin, G., Numerische Integration (1979), Birkhäuser: Birkhäuser Basel), 138-146 · Zbl 0411.65018 |
| [7] | Gatteschi, L.; Pittaluga, G.; Rassias, J. M., An asymptotic expansion for the zeros of Jacobi polynomials, Mathematical Analysis, 79, 70-86 (1985), Teubner-Texte zur Math. · Zbl 0598.33009 |
| [8] | Hunter, D. B., Some Gauss-type formulae for the evaluation of Cauchy principal values of integrals, Numer. Math., 19, 419-424 (1972) · Zbl 0231.65028 |
| [9] | Koksma, J. F., Sur l’approximation des nombres irrationels sous une condition supplementaire, Simon Stevin, 28, 199-202 (1951) · Zbl 0044.04303 |
| [10] | Kütz, M., Fehlerschranken und Fehlerasymptotik für eine Klasse von Interpolation Quadraturen, (Ph.D. Thesis (1982), Technischen Universität Carolo-Wilhelmina: Technischen Universität Carolo-Wilhelmina Braunschweig) · Zbl 0506.41033 |
| [11] | Lubinsky, D. S.; Rabinowitz, P., Rates of convergence of Gaussian quadrature for singular integrands, Math. Comp., 43, 219-242 (1984) · Zbl 0574.41028 |
| [12] | D.S. Lubinsky and A. Sidi, Convergence of product integration rules for functions with interior and endpoints singularities over bounded and unbounded intervals, Tech. Report No. 215, Computer Science Dept., Technion, Haifa, Israel.; D.S. Lubinsky and A. Sidi, Convergence of product integration rules for functions with interior and endpoints singularities over bounded and unbounded intervals, Tech. Report No. 215, Computer Science Dept., Technion, Haifa, Israel. · Zbl 0602.41032 |
| [13] | Miller, R. K., On ignoring the singularity in numerical quadrature, Math. Comp., 25, 521-532 (1971) · Zbl 0221.65045 |
| [14] | Rabinowitz, P., Gaussian integration in the presence of a singularity, SIAM J. Numer. Anal., 4, 191-201 (1967) · Zbl 0183.18102 |
| [15] | Rabinowitz, P., Ignoring the singularity in numerical integration, (Miller, J. J.H., Topics in Numerical Analysis III (1977), Academic Press: Academic Press London), 361-368 · Zbl 0435.65018 |
| [16] | Rabinowitz, P., Generalized composite integration rules in the presence of a singularity, Calcolo, 20, 231-238 (1983) · Zbl 0543.41031 |
| [17] | Rabinowitz, P., On the convergence and divergence of Hunter’s method for Cauchy principal value integrals, (Gerasoulis, A.; Vichnevetsky, R., Numerical Solution of Singular Integral Equations (1984), IMACS: IMACS New Brunswick, NJ), 86-88 |
| [18] | Rabinowitz, P., On the convergence of closed interpolatory integration rules based on the zeros of Gegenbauer polynomials, J. Comput. Appl. Math., 17, 43-46 (1987), (this issue) · Zbl 0625.41020 |
| [19] | Rabinowitz, P.; Sloan, I. H., Product integration in the presence of a singularity, SIAM J. Numer. Anal., 21, 149-166 (1984) · Zbl 0547.65011 |
| [20] | Smith, W. E.; Sloan, I. H., Product-integration rules based on the zeros of Jacobi polynomials, SIAM J. Numer. Anal., 17, 1-13 (1980) · Zbl 0429.41023 |
| [21] | Szegö, G., Ortogonal Polynomials (1959), Amer. Math. Soc: Amer. Math. Soc New York · Zbl 0089.27501 |
| [22] | Tsamasphyros, G. J.; Theocaris, P. S., On the convergence of a Gauss quadrature rule for evaluation of Gauss type singular integrals, BIT, 17, 458-464 (1977) · Zbl 0391.65004 |
| [23] | Tsamasphyros, G. J.; Theocaris, P. S., On the convergence of some quadrature rules for Cauchy principal-value and finite-part integrals, Computing, 31, 105-114 (1983) · Zbl 0504.65011 |
| [24] | Rabinowitz, P., Rates of convergence of Gauss, Radau and Lobatto integration for singular integrands, Math. Comp., 47 (1986) · Zbl 0619.41022 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
