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We introduce a conjugate gradient method for estimating and tracking the minor eigenvector of a data correlation matrix. This new algorithm is less computationally demanding and converges faster than other methods derived from the conjugate gradient approach. It can also be applied in the context of minor subspace tracking, as a pre-processing step for the YAST algorithm, in order to enhance its performance. Simulations show that the resulting algorithm converges much faster than existing minor subspace trackers.