Abstract
We consider Smolyak's construction for the numerical integration over the d‐dimensional unit cube. The underlying class of integrands is a tensor product space consisting of functions that are analytic in the Cartesian product of ellipses. The Kronrod–Patterson quadrature formulae are proposed as the corresponding basic sequence and this choice is compared with Clenshaw–Curtis quadrature formulae. First, error bounds are derived for the one‐dimensional case, which lead by a recursion formula to error bounds for higher dimensional integration. The applicability of these bounds is shown by examples from frequently used test packages. Finally, numerical experiments are reported.
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Petras, K. On the Smolyak cubature error for analytic functions. Advances in Computational Mathematics 12, 71–93 (2000). https://doi.org/10.1023/A:1018904816230
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DOI: https://doi.org/10.1023/A:1018904816230