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We present an error analysis for two related quadrature methods (the Delves–Lyness method and its modification by Kravanja, Sakurai and Van Barel) for computing all the zeros of an analytic function that lie inside the unit circle. We consider the forward as well as the backward approximation error in case the integrals are computed via the trapezoidal rule on the unit circle. Contrary to the Delves–Lyness method, the quadrature error that arises from the zeros located inside the unit circle does not affect the results of the approach of Kravanja et al. Numerical experiments illustrate our main results.