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Mathematics of Computation

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Error bounds for Gauss-Kronrod quadrature formulae


Author: Sven Ehrich
Journal: Math. Comp. 62 (1994), 295-304
MSC: Primary 65D32; Secondary 41A55
MathSciNet review: 1208221
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Abstract: The Gauss-Kronrod quadrature formula $ Q_{2n + 1}^{GK}$, is used for a practical estimate of the error $ R_n^G$ of an approximate integration using the Gaussian quadrature formula $ Q_n^G$. Studying an often-used theoretical quality measure, for $ Q_{2n + 1}^{GK}$ we prove best presently known bounds for the error constants

$\displaystyle {c_s}(R_{2n + 1}^{GK}) = \mathop {\sup }\limits_{{{\left\Vert {{f^{(s)}}} \right\Vert}_\infty } \leq 1} \vert R_{2n + 1}^{GK}[f]\vert$

in the case $ s = 3n + 2 + \kappa ,\kappa = \left\lfloor {\frac{{n + 1}}{2}} \right\rfloor - \left\lfloor {\frac{n}{2}} \right\rfloor $. A comparison with the Gaussian quadrature formula $ Q_{2n + 1}^G$ shows that there exist quadrature formulae using the same number of nodes but having considerably better error constants.

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DOI: https://doi.org/10.1090/S0025-5718-1994-1208221-0
Keywords: Gauss-Kronrod quadrature formulae, error constants
Article copyright: © Copyright 1994 American Mathematical Society