Root-finding and Root-refining for a Polynomial Equation

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 computing 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 q where q denotes the convergence order, whereas d denotes the number of function evaluations per iteration. 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 definition 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 2, d = cn/ log 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.