On the error term of symmetric

Gauss-Lobatto quadrature formulae

for analytic functions

Authors:
David Hunter and Geno Nikolov

Journal:
Math. Comp. **69** (2000), 269-282

MSC (1991):
Primary 41A55; Secondary 65D30, 65D35

DOI:
https://doi.org/10.1090/S0025-5718-99-01078-9

Published electronically:
March 4, 1999

MathSciNet review:
1642754

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[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****[7]**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:
https://doi.org/10.1090/S0025-5718-99-01078-9

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

Published electronically:
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

Article copyright:
© Copyright 1999
American Mathematical Society