An algorithm for adaptive quadrature is presented which attempts to ensure that the quadrature used is valid for the behaviour of the integrand. This is achieved by using another algorithm to examine the finite differences of the integrand. The order of the quadrature and the interval subdivision strategy used form an integral part of this algorithm. The resulting adaptive quadrature algorithm is very reliable while its efficiency appears to be of the order of other adaptive quadrature algorithms in the literature.
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