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

Summary.

By providing a matrix version of Koenig's theorem we reduce the problem of evaluating the coefficients of a monic factor r(z) of degree h of a power series f(z) to that of approximating the first h entries in the first column of the inverse of an $n\times n$ Toeplitz matrix in block Hessenberg form for sufficiently large values of n. This matrix is reduced to a band matrix if f(z) is a polynomial. We prove that the factorization problem can be also reduced to solving a matrix equation $\phi(X)=0$ for an $h\times h$ matrix X, where $\phi$ is a matrix power series whose coefficients are Toeplitz matrices. The function $\phi$ is reduced to a matrix polynomial of degree 2 if f(z) is a polynomial of degreeN and $h\geq N/2$. These reductions allow us to devise a suitable algorithm, based on cyclic reduction and on the concept of displacement rank, for generating a sequence of vectors $\vec{v}^{(2^j)}$ that quadratically converges to the vector $\vec{v}$ having as components the coefficients of the factor r(z). In the case of a polynomial f(z) of degree N, the cost of computing the entries of $\vec{v}^{(2^j)}$ given $\vec{v}^{(2^{j-1})}$ is $O(N\log N+\theta(N))$ arithmetic operations, where $\theta(N)=O(N\log^2 N)$ is the cost of solving an $N\times N$ Toeplitz-like system. In the case of analytic functions the cost depends on the numerical degree of the power series involved in the computation. 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-finders based on recursive splitting strategies. Applications to solving spectral factorization problems and Markov chains are also shown.