Abstract
Ehrenmark (this journal, 1988) and Vanden Berghe et al. (this journal, 1990) introduced quadrature formulas of a compound type, which are obtained by interpolation with the null space of the linear differential operator D r f + λ 2 i D r-2 f, where λ i may be chosen differently for each subinterval. In this paper, the asymptotic behaviour of the error of this type of compound quadrature rule is determined for smooth functions. It is shown that an improvement of the rate of convergence by a factor H 2 can be obtained by choosing λ i = λ opt for all i and N. This optimal λ can be determined explicitly (if the derivatives of f are available) and approximations to λ opt can be obtained numerically.
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