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Z. Kovarik proposed in 1970 a method for approximate orthogonalization of a finite set of linearly independent vectors from a Hilbert space. This method uses at each iteration a symmetric and positive definite matrix inversion. In this paper we describe an algorithm in which the above matrix inversion step is replaced by an arbitrary odd degree polynomial matrix expression. We prove that this new algorithm converges to the same orthonormal set of vectors as the original Kovarik's method. Some numerical experiments presented in the last section of the paper show us that, even for small degree polynomial expressions the convergence properties of the new algorithm are comparable with those of the original one.