Istituto di Scienza e Tecnologie dell'Informazione     
Di M., Favati P., Lotti G., Romani F. Asymptotic behaviour of automatic quadrature. Internal note IEI-B4-35, 1992.
The complexity of automatic quadrature programs is investigated under the hypothesis of exactness or asymptotical consistence of local error estimates. The complexity measure used is the number N of functional evaluations in real exact arithmetic versus the number E of exact decimal digits in the result. The methods of integration analysed are m-panel rules, Clenshaw-Curtis quadrature, Romberg method, global adaptive quadrature, and double adaptive quadrature. For m-panel and global adaptive quadrature, based on a local rule of degree r-1 the constants hidden by the "Oh" notation are determined in terms of the r-th derivative of the integrand and the numerical properties of the chosen local rule. The complexity of global adaptive quadrature results to be of order Θ(10ΔE/r), regardless of the regularity of the integrand. The double adaptive quadrature achieves O(E) complexity for regular integrands and O(EΔ2) for singular ones.

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