Abstract. Let $p(z)$ be a polynomial of degree $n$ having zeros $ |\xi_1|\leq\ldots\leq |\xi_m|<1<|\xi_{m+1}|\leq\ldots\leq|\xi_n|.$ This paper is concerned with the problem of efficiently computing the coefficients of the %spectral factors $u(z)=\prod_{i=1}^m(z-\xi_i)$ and $l(z)=\prod_{i=m+1}^n(z-\xi_i)$ of $p(z)$ such that $a(z)=z^{-m}p(z)=(z^{-m}u(z)) l(z)$ is the spectral factorization of $a(z)$. To perform this task the following two-stage approach is considered: First we approximate the central coefficients $x_{-n+1},\ldots x_{n-1}$ of the Laurent series $x(z)=\sum_{i=-\infty}^{+\infty}x_iz^i$ satisfying $x(z)a(z)=1$; then we determine the entries in the first column and in the first row of the inverse of the Toeplitz matrix $T=(x_{i-j})_{i,j=-n+1,n-1}$ which provide the sought coefficients of $u(z)$ and $l(z)$. Two different algorithms are analyzed for the reciprocation of Laurent polynomials. One algorithm makes use of Graeffe's iteration which %reduces to perform polynomial multiplications and is quadratically convergent. %, the convergence rate depending on the %isolation ratio $\rho=\max\{|\xi_m|, |1/\xi_{m+1}|\}$. Differently, the second algorithm directly employs evaluation/interpolation techniques at the roots of 1 and it is linearly convergent only. Algorithmic issues and numerical experiments are discussed.

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