Tangent Graeffe iteration

Graeffe iteration was the choice algorithm for solving univariate polynomia ls in the XIX-th and early XX-th century. In this paper, a new variation of Graeffe iteration is given, suitable to IEEE floating-point arithmetics of modem digital computers. We prove that under a certain generic assumption the proposed algorithm converges. We also estimate the error after N iterat ions and the running cost. The main ideas from which this algorithm is buil t are: classical Graeffe iteration and Newton Diagrams, changes of scale (r enormalization), and replacement of a difference technique by a differentia tion one. The algorithm was implemented successfully and a number of numeri cal experiments are displayed.