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On the error term of symmetric Gauss-Lobatto quadrature formulae for analytic functions
Author(s):
David
Hunter;
Geno
Nikolov.
Journal:
Math. Comp.
69
(2000),
269-282.
MSC (1991):
Primary 41A55;
Secondary 65D30, 65D35
Posted:
March 4, 1999
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Abstract:
Gauss-Lobatto quadrature formulae associated with symmetric weight functions are considered. The kernel of the remainder term for classes of analytic functions is investigated on elliptical contours. Sufficient conditions are found ensuring that the kernel attains its maximal absolute value at the intersection point of the contour with either the real or the imaginary axis. The results obtained here are an analogue of some recent results of T. Schira concerning Gaussian quadratures.
References:
- [1]
- W. Gautschi, On the remainder term for analytic functions of Gauss-Lobatto and Gauss-Radau quadratures, Rocky Mountain J. Math. 21 (1991), 209-226. Corrected in W. Gautschi, Rocky Mountain J. Math. 21 (1991), 1143. MR 93a:41071a; MR 93a:41071b
- [2]
- W. Gautschi, Remainder estimates for analytic functions, Numerical Integration (T. O. Espelid, A. Genz, eds.), Kluwer Academic Publishers, 1992, 133-145. MR 94e:41049
- [3]
- W. Gautschi and S. Li, The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules with multiple end points, J. Comp. Appl. Math. 33 (1990), 315-329. MR 92a:65078
- [4]
- W. Gautschi and R. S. Varga, Error bounds for Gaussian quadrature of analytic functions, SIAM J. Numer. Anal. 20 (1983), 1170-1186. MR 86j:65010
- [5]
- W. Gautschi, E. Tychopoulos and R. S. Varga, A note on the contour representation of the remainder term for a Gauss-Chebyshev quadrature rule, SIAM J. Numer. Anal. 27 (1990), 219-224. MR 91d:65044
- [6]
- D. B. Hunter, Some error expansions for Gaussian quadrature, BIT 35 (1995), 64-82. MR 97i:65040
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- F. Peherstorfer, On the remainder of Gaussian quadrature formulas for Bernstein-Szegö weight functions, Math. Comp. 60 (1993), 317-325. MR 93d:65030
- [8]
- T. Schira, Ableitungsfreie Fehlerabschätzungen bei numerischer Integration holomorpher Funktionen, Ph.D. Dissertation, Univ. Karlsruhe, 1994.
- [9]
- T. Schira, The remainder term for analytic functions of Gauss-Lobatto quadratures, J. Comp. Appl. Math. 76 (1996), 171-193. MR 97m:41033
- [10]
- T. Schira, The remainder term for analytic functions of symmetric Gaussian quadratures, Math. Comp. 66 (1997), 297-310. MR 97c:65050
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Additional Information:
David
Hunter
Affiliation:
Department of Computing and Mathematics, University of Bradford, BD7 Bradford, West Yorkshire, United Kingdom
Geno
Nikolov
Affiliation:
Department of Mathematics, University of Sofia, blvd. James Bourchier 5, 1164 Sofia, Bulgaria
Email:
geno@fmi.uni-sofia.bg
DOI:
10.1090/S0025-5718-99-01078-9
PII:
S 0025-5718(99)01078-9
Keywords:
Gauss-Lobatto quadrature,
remainder term for analytic functions,
contour integral representation,
kernel function
Received by editor(s):
October 14, 1997
Received by editor(s) in revised form:
March 26, 1998
Posted:
March 4, 1999
Additional Notes:
The second author did this work while he was on leave from the Department of Mathematics, University of Sofia, blvd J. Bourchier 5, 1164 Sofia, Bulgaria. He was supported by a grant from the Royal Society, and by the Bulgarian Ministry of Science, Education and Technologies under Grant MM-513/95
Copyright of article:
Copyright
1999,
American Mathematical Society
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