Effective Fast Algorithms for Polynomial Spectral Factorization

Abstract

Let p(z) be a polynomial of degree n having zeros |ξ₁|≤⋅⋅⋅≤|ξ _m |<1<|ξ _m+1|≤⋅⋅⋅≤|ξ _n |. This paper is concerned with the problem of efficiently computing the coefficients of the factors u(z)=∏ _i=1 ^m(z−ξ _i ) and l(z)=∏ _i=m+1 ⁿ(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 algorithm directly employs evaluation/interpolation techniques at the roots of 1 and it is linearly convergent only. Algorithmic issues and numerical experiments are discussed.