Factorization of analytic functions by means of Koenig's theorem and Toeplitz computations

By providing a matrix version of Koenig's theorem we reduce the problem of evaluating the coeecients of a monic factor r(z) of degree h of a power series f (z) to that of approximating the rst h entries in the rst column of the inverse of an innnite Toeplitz matrix in block Hessenberg form. This matrix is reduced to a band matrix if f (z) is a polynomial. We devise a suitable algorithm, based on cyclic reduction and on the concept of displacement rank, for generating a sequence of vectors v (2 j) that quadratically converges to the vector v having as components the coeecients of the factor r(z). In the case of a polynomial f (z) of degree N , the cost of computing the entries of v (2 j) given v (2 j?1) is O(N log N + (N)) arithmetic operations, where (N) = O(N log 2 N) is the cost of solving an N N Toeplitz-like system. In the case of analytic functions the cost depends on the numerical degree of the power series involved in the computation. The algorithm is strictly related to the Graeee method for lifting the roots of a polynomial. From the numerical experiments performed with several test polynomials and power series, the algorithm has shown good numerical properties and promises to be a good candidate for implementing polynomial root-nders based on recursive splitting strategies.