Abstract
An automatic quadrature routine was presented by de Boor which was based on the use of the trapezoidal rule with “cautious Romberg extrapolation”. It is the object of this paper to modify the approach of de Boor to cautious Romberg extrapolation by making a comparison between the results obtained from the trapezoidal and Simpson rules. This has the advantage that it gives some guidance regarding the estimation of the dominant terms involved, requires no additional function evaluations, and only minor changes in de Boor's program CADRE are necessary to effect the proposed modifications. Special account is taken of integrands with end-point singularities and the possibilities of logarithmic terms being involved in the extrapolation. Further, indications are given of where automatic Romberg extrapolation can fail.
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