Dynamics of Newton-like root finding methods

When exploring the literature, it can be observed that the operator obtained when applying Newton-like root finding algorithms to the quadratic polynomials z(2) & mdash; c has the same form regardless of which algorithm has been used. In this paper, we justify why this expression is obtained. This is done by studying the symmetries of the operators obtained after applying Newton-like algorithms to a family of degree d polynomials p(z) = z(d) & mdash; c. Moreover, we provide an iterative procedure to obtain the expression of new Newton-like algorithms. We also carry out a dynamical study of the given generic operator and provide general conclusions of this type of methods.

Automated summary

In this paper, the author explores the operator obtained when applying Newton-like root finding algorithms to quadratic polynomials and justifies why the same form of the operator is obtained regardless of the algorithm used. The symmetries of the operators obtained after applying Newton-like algorithms to a family of polynomials are studied, and an iterative procedure to obtain the expression of new Newton-like algorithms is provided. The paper also includes a dynamical study of the given generic operator and derives general conclusions of this type of methods.