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

Amat, Sergio; Busquier, Sonia; Plaza, Sergio

Sci., Ser. A, Math. Sci. (N.S.) 10, 3-35 (2004).

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.

MSC:

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)