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Asymptotically optimal error bounds for quadrature rules of given degree


Author: H. Brass
Journal: Math. Comp. 61 (1993), 785-798
MSC: Primary 41A55; Secondary 65D32
DOI: https://doi.org/10.1090/S0025-5718-1993-1192968-8
MathSciNet review: 1192968
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Abstract: If the quadrature rule Q is applied to the function f, then the error can in many situations be bounded by $ {\rho _s}(Q){\left\Vert {{f^{(s)}}} \right\Vert _\infty }$, where $ {\rho _s}(Q)$ is independent of f. We obtain the asymptotics of these numbers for the Gaussian method $ Q_n^{\text{G}}\;(n = 1,2, \ldots )$ with very general weight functions and show that $ {\rho _s}(Q_n^{\text{G}})$ is (asymptotically) an upper bound for $ {\rho _s}(Q)$, if Q is any quadrature rule with the same degree as $ Q_n^{\text{G}}$.


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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1993-1192968-8
Article copyright: © Copyright 1993 American Mathematical Society