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The classical Levinson algorithm for computing the predictor polynomial relative to a real positive definite Toeplitz matrix is shown to be redundant in complexity. It can be broken down into two simpler algorithms, either of which needs only to be processed. This result can be interpreted in the framework of the theory of orthogonal polynomials on the real line as follows: the symmetric and antisymmetric parts of the predictors relative to the sequence of Toeplitz matrices constitute two families of polynomials orthogonal on the interval [- 1,1] with respect to some even weight functions. It turns out that the recurrence relations for these orthogonal polynomials can be used efficiently to compute the desired predictor. The resulting "split Levinson algorithm" requires roughly one-half the number of multiplications and the same number of additions as the classical Levinson algorithm. A simple derivation of Cybenko's method for computing the Pisarenko frequencies is obtained from the recurrence relations underlying the split Levinson algorithm.