Review of some iterative root-finding methods from a dynamical point of view. (English) Zbl 1137.37316

Summary: From a dynamical point of view applied to complex polynomials, we study a number of root-finding iterative methods. We consider Newton’s method, Newton’s method for multiple roots, Jarratt’s method, the super-Halley method, the convex as well as the double convex acceleration of Whittaker’s method, the methods of Chebyshev, Stirling, and Steffensen, among others. Since all of the iterative root-finding methods we study satisfy the Scaling Theorem, except for Stirling’s method and that of Steffensen, we obtain their conjugacy classes.


37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)