Tangent Graeffe Iteration
Gregorio Malajovich and Jorge P. Zubelli
Graeffe iteration was the choice algorithm for solving univariate polynomials 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 modern digital computers. We prove that under a certain generic assumption the proposed algorithm converges. We also estimate the error after $N$ iterations and the running cost.
The main ideas from which this algorithm is built are: classical Graeffe iteration and Newton Diagrams, changes of scale (renormalization), and replacement of a difference technique by a differentiation one.
The algorithm was implemented successfully and a number of numerical experiments are displayed.