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Derivative-free error bounds in numerical integration of analytic functions. (Ableitungsfreie Fehlerabschätzungen bei numerischer Integration holomorpher Funktionen.) (German) Zbl 0809.41029
Karlsruhe: Univ. Karlsruhe, Fak. f. Mathematik, 123 S. (1994).
The thesis deals with remainder terms of quadratures rules with respect to nonnegative weight functions $$w$$ over the interval $$[-1,1]$$. For integrands $$f$$ that can be extended analytically in a domain $$G(\supset [-1,1])$$ it is well-known that the remainder term can be expressed as a contour integral with a kernel $$K_ n$$. This leads to derivative-free error bounds. The knowledge of the maximum modulus of the kernel $$K_ n$$ on the contour is essential to obtain sharp error bounds. These maxima were first studied for Gauss quadrature rules by W. Gautschi and R. S. Varga [SIAM J. Numer. Anal. 20, 1170-1186 (1983; Zbl 0545.41040)]. They determined the maxima on circular contours $${\mathcal C}_ r$$ for a large class of weight functions and considered elliptical contours $${\mathcal E}_ \rho$$ (with foci $$\pm 1$$ and semiaxis $${1 \over 2} (\rho \pm \rho^{-1}), \rho > 1)$$ for the four Chebyshev weights. Some of these results were later enlarged to Gauss-Lobatto and Gauss-Radau quadrature rules (see e.g. W. Gautschi’s survey [NATO Adv. Res. Workshop, Bergen/Norw. 1991, NATO ASI Ser., C 357, 133-145 (1992; Zbl 0754.41026)]).
In this thesis the discussion of the kernel $$K_ n$$ is extended for quadrature rules with respect to more general weight functions that have not yet been considered in the literature. For the Gauss-Lobatto quadrature rule with respect to $$w{(1/2)} (x) : = (1 - x^ 2)^{1/2}$$ a conjecture of W. Gautschi [Rocky Mt. J. Math. 21, 209-226 (1991; Zbl 0749.41026)] concerning the maxima of $$K_ n$$ on $${\mathcal E}_ \rho$$ for $$\rho \geq \rho_{n,L}$$ is proved. For Gauss quadrature rules with respect to symmetric weight functions $$w$$ a new series representation of the kernel $$K_ n$$ is developed. This is essential for the determination of the maximum of $$K_ n$$ on ellipses $${\mathcal E}_ \rho$$ with $$\rho \geq \rho_{n,G}$$ for a large class of symmetric weight functions, including the Gegenbauer weights $$w^{(\alpha)} (x) : = (1 - x^ 2)^ \alpha$$ for $$| \alpha | \geq {1 \over 2}$$. The remaining cases $$| \alpha | < {1 \over 2}$$ cannot be handled in this way, but using a well-known expansion of the kernel $$K_ n$$ similar results are obtained. For the quadrature rules of Pólya-type (i.e. Clenshaw-Curtis, Pólya and Filippi rule) special expansions of the corresponding kernel $$K_ n$$ are used in order to determine the maximum of the kernel $$K_ n$$ on ellipses $${\mathcal E}_ \rho$$, $$\rho \geq \rho_{n,X}$$. All the parameters $$\rho_{n,L}$$, $$\rho_{n,G}$$ and $$\rho_{n,X}$$ are explicitly given. The knowledge of the maxima of $$K_ n$$ leads to practical derivative- free error bounds for the mentioned quadrature rules. Additionally, another type of useful derivative-free error bounds is presented, based on the estimation of the kernel $$K_ n$$ with the BOGS-method [see e.g. W. Haußmann, E. Luik, and K. Zeller [Multivariate approximation theory II, Proc. Conf., Oberwolfach 1982 ISNM 61, 191-200 (1982; Zbl 0508.41027)]. Several numerical examples show that both methods lead to similar error bounds. Finally, the bivariate case is treated. Considering several assumptions on the region of analyticity of the bivariate integrand $$f$$ derivative-free error bounds for tensor product and blending cubature formulae are derived.
Reviewer: Thomas Schira

MSC:
 41A55 Approximate quadratures 65D30 Numerical integration 65D32 Numerical quadrature and cubature formulas