Abstract

We introduce the CIRQUE algorithm, for approximating definite integrals of continuous, univariate, real-valued functions, using positive-coefficient composite interpolatory quadrature. CIRQUE estimates and controls absolute and/or relative error, without the need for a prior estimate of the magnitude of the integral. The limiting effects of roundoff error are catered for, and CIRQUE is able to provide estimates of error bounds as output. Moreover, if these bounds are deemed too large, it is a simple matter to rerun CIRQUE once to obtain an acceptable bound. We have demonstrated the algorithm using the Trapezium rule, Simpson's rule and four-point Gauss-Legendre quadrature.

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