Accelerated Approximation of the Complex Roots and Factors of a Univariate Polynomial

To appearInternational audienceThe known algorithms approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time. They are, however, quite involved and require a high precision of computing when the degree of the input polynomial is large, which causes numerical stability problems. We observe that these difficulties do not appear at the initial stages of the algorithms, and in our present paper we extend one of these stages, analyze it, and avoid the cited problems, still achieving the solution within a nearly optimal complexity estimates, provided that some mild initial isolation of the roots of the input polynomial has been ensured. The resulting algorithms promise to be of some practical value for root-finding and can be extended to the problem of polynomial factorization, which is of interest on its own right. We conclude with outlining such an extension, which enables us to cover the cases of isolated multiple roots and root clusters

Root-Refining for a Polynomial Equation

Abstract. Polynomial root-finders usually consist of two stages. At first a crude approximation to a root is slowly computed; then it is much faster refined by means of the same or distinct iteration. The efficiency of com-puting an initial approximation resists formal study, and the users rely on empirical data. In contrast, the efficiency of refinement is formally measured by the classical concept q1/d where q denotes the convergence order, whereas d denotes the number of function evaluations per iter-ation. In our case of a polynomial of a degree n we use 2n arithmetic operations per its evaluation of at a point. Noting this we extend the def-inition to cover iterations that are not reduced to function evaluations alone, including iterations that simultaneously refine n approximations to all n roots of a degree n polynomial. By employing two approaches to the latter task, both based on recursive polynomial factorization, we yield refinement with the efficiency 2d, d = cn / log2 n for a positive constant c. For large n this is a dramatic increase versus the record efficiency 2 of refining an approximation to a single root of a polynomial. The advance could motivate practical use of the proposed root-refiners