Article
Full entry |
PDF
(1.5 MB)
Feedback
PDF
(1.5 MB)
Feedback
Keywords:
rate-of-convergence; quadrature rules; Cauchy type principal value integrals; finite interval; Gauss quadrature; interpolatory quadrature
rate-of-convergence; quadrature rules; Cauchy type principal value integrals; finite interval; Gauss quadrature; interpolatory quadrature
Summary:
New convergence and rate-of-convergence results are established for two well-known quadrature rules for the numerical evaluation of Cauchy type principal value integrals along a finite interval, namely the Gauss quadrature rule and a similar interpolatory quadrature rule where the same nodes as in the Gauss rule are used. The main result concerns the convergence of the interpolatory rule for functions satisfying the Hölder condition with exponent less or equal to $\frac12$. The results obtained here supplement a series of previous results on the convergence of the aforementioned quadrature rules.
References:
[1] E. W. Cheney: Introduction to Approximation Theory. McGraw-Hill, New York, 1966. MR 0222517 | Zbl 0161.25202
[2] D. Elliott: On the Convergence of Hunter's Quadrature Rule for Cauchy Principal Value Integrals. Nordisk Tidskrift for Informationsbehandling (BIT), 1979, Vol. 19, pp. 457-462. MR 0559954 | Zbl 0418.65011
[3] D. Elliott, D. F. Paget: On the Convergence of a Quadrature Rule for Evaluating Certain Cauchy Principal Value Integrals. Numerische Mathematik, 1975, Vol. 23, pp. 311 - 319. DOI 10.1007/BF01438258 | MR 0380215 | Zbl 0313.65019
[4] D. Elliott, D. F. Paget: On the Convergence of a Quadrature Rule for Evaluating Certain Cauchy Principal Value Integrals: An Addendum. Numerische Mathematik, 1976, Vol. 25, pp. 287-289. DOI 10.1007/BF01399417 | MR 0411131 | Zbl 0321.65017
[5] D. Elliott, D. F. Paget: Gauss Type Quadrature Rules for Cauchy Principal Value Integrals. Mathematics of Computation, 1979, Vol. 33, pp. 301 - 309. DOI 10.1090/S0025-5718-1979-0514825-2 | MR 0514825 | Zbl 0415.65019
[6] G. Freud: Orthogonal Polynomials. 1st English ed. Pergamon Press, Oxford, 1971. MR 0294981 | Zbl 0226.33014
[7] D. B. Hunter: Some Gauss-Type Formulae for the Evaluation of Cauchy Principal Values of Integrals. Numerische Mathematik, 1972, Vol. 19, pp. 419-424. DOI 10.1007/BF01404924 | MR 0319355 | Zbl 0231.65028
[8] N. I. Ioakimidis: General Methods for the Solution of Crack Problems in the Theory of Plane Elasticity. Doctoral Thesis at the National Technical University of Athens, Athens, 1976. [Available from University Microfilms, Ann Arbor, Michigan; Order No. 76-21, 056.] MR 0521392 | Zbl 0351.73109
[9] N. I. Ioakimidis, P. S. Theocaris: On the Numerical Evaluation of Cauchy Principal Value Integrals. Revue Roumaine des Sciences Techniques - Série de Mécanique Appliquée, 1977, Vol. 22, pp. 803-818. MR 0483321 | Zbl 0376.65009
[10] A. I. Kalandiya: Mathematical Methods of Two-dimensional Elasticity. 1st English ed. Mir Publishers, Moscow, 1975. MR 0400846
[11] A. A. Korneichuk: Quadrature Formulae for Singular Integrals. In: Numerical Methods for the Solution of Differential and Integral Equations and Quadrature Formulae, Moscow, Nauka, 1964, pp. 64-74 (in Russian).
[12] D. F. Paget, D. Elliott: An Algorithm for the Numerical Evaluation of Certain Cauchy Principal Value Integrals. Numerische Mathematik, 1972, Vol. 19, pp. 373 - 385. DOI 10.1007/BF01404920 | MR 0366004 | Zbl 0229.65091
[13] T. J. Rivlin: An Introduction to the Approximation of Functions. 1st ed. Blaisdell, Waltham, Massachusetts, 1969. MR 0249885 | Zbl 0189.06601
[14] D. G. Sanikidze: On a Uniform Estimation of Approximation of Singular Integrals with Chebyshev's Weight Function by Sums of Interpolating Type. Soobščenija Akademii Nauk Gruzinskoī SSR, 1974, Vol. 75 (No. 1), pp. 53-55 (in Russian). MR 0358242
[15] M. A. Sheshko: On the Convergence of Quadrature Processes for a Singular Integral. Soviet Mathematics (Izvestija Vysših Učebnyh Zavedeniī. Matematika), 1976, Vol. 20 (No. 12), pp. 86-94.
[16] G. Szegö: Orthogonal Polynomials. revised ed. American Mathematical Society, New York, 1959. MR 0106295
[17] P. S. Theocaris, G. J. Tsamasphyros: On the Convergence of a Gauss Quadrature Rule for Evaluation of Cauchy Type Singular Integrals. Nordisk Tidskrift for Informationsbehandling (BIT), 1977, Vol. 17, pp. 458-464. MR 0468120 | Zbl 0391.65004
Search
Partner of
