Graeffe's, Chebyshev-like, and Cardinal's Processes for Splitting a Polynomial into Factors

Abstract

Numerical splitting of a real or complex univariate polynomial into factors is the basic step of the divide-and-conquer algorithms for approximating complex polynomial zeros. Such algorithms are optimal (up to polylogarithmic factors) and are quite promising for practical computations. In this paper, we develop some new techniques, which enable us to improve numerical analysis, performance, and computational cost bounds of the known splitting algorithms. In particular, we study a Chebyshev-like modification of Graeffe's lifting iteration (which is a basic block of the splitting algorithms, as well as of several other known algorithms for approximating polynomial zeros), analyze its numerical performance, compare it with Graeffe's, prove some results on numerical stability of both lifting processes (that is, Graeffe's and Chebyshev-like), study their incorporation into polynomial root-finding algorithms, and propose some improvements of Cardinal's recent effective technique for numerical splitting of a polynomial into factors. Our improvement relies, in particular, on a modification of the matrix sign iteration, based on the analysis of some conformal mappings of the complex plane and of techniques of recursive lifting/recursive descending. The latter analysis reveals some otherwise hidden correlations among Graeffe's, Chebyshev-like, and Cardinal's iterative processes, and we exploit these correlations in order to arrive at our improvement of Cardinal's algorithm. Our work may also be of some independent interest for the study of applications of conformal maps of the complex plane to polynomial root-finding and of numerical properties of the fundamental techniques for polynomial root-finding such as Graeffe's and Chebyshev-like iterations.