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We present a semilocal convergence result for a Newton-type method applied to a polynomial operator equation of degree 2. The method consists in fact in evaluating the Jacobian at every two steps, and it has the r-convergence order at least 3. We apply the method in order to approximate the eigenpairs of matrices. We perform some numerical examples on some test matrices and compare the method to the Chebyshev method. The norming function we have proposed in a previous paper shows a betterfatcat:mszrieymprbunjstrwr5tjr6ay