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We describe algorithms for computing the greatest common divisor (GCD) of two univariate polynomials with inexactlyknown coefficients. Assuming that an estimate for the GCD degree is available (eg, using an SVD-based algorithm), we formulate and solve a nonlinear optimization problem in order to determine the coefficients of the\best" GCD. We discuss various issues related to the implementation of the algorithms and present some preliminary test results.