Numerical Algorithms 34: 217–227, 2003.  2003 Kluwer Academic Publishers. Printed in the Netherlands. Effective fast algorithms for polynomial spectral factorization D.A. Bini, G. Fiorentino, L. Gemignani and B. Meini Dipartimento di Matematica, Università di Pisa, via Buonarroti 2, 56127 Pisa, Italy E-mail: {bini;fiorent;gemignan;meini}@dm.unipi.it Received 3 December 2001; accepted 7 January 2003 Let p(z) be a polynomial of degree n having zeros |ξ 1 |  ···  |ξ m | < 1 < |ξ m+1 |  ···  |ξ n |. This paper is concerned with the problem of efficiently computing the coeffi- cients of the factors u(z) =  m i =1 (z - ξ i ) and l(z) =  n i =m+1 (z - ξ 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 ,...x n-1 of the Laurent series x(z) = ∑ +∞ i =-∞ x i z 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 is quadratically convergent. Differently, the second al- gorithm directly employs evaluation/interpolation techniques at the roots of 1 and it is linearly convergent only. Algorithmic issues and numerical experiments are discussed. Keywords: spectral factorization, polynomial factorization, Graeffe method, Laurent polyno- mial inversion AMS subject classification: 65H05, 12Y05, 12D05 1. Introduction Let p(z) be a polynomial of degree n with complex coefficients, p(z) = n  i =0 p i z i = p n n  i =1 (z - ξ i ), |ξ 1 |  ···  |ξ m | < 1 < |ξ m+1 |  ···  |ξ n | (1) and define its factors u(z) = m  i =1 (z - ξ i ) = m  i =0 u m-i z i , l(z) = n  i =m+1 (z - ξ i ) = n-m  i =0 l i z i (2)