On a global descent method for polynomials
For a given complex polynomial p of degree n the author derives a function \(f=f_ p\) with the property \(| p(f(z))| \leq \kappa (z)| p(z)|,\) \(\kappa (z)<1\) in \({\mathbb{C}}\), and then uses the descent algorithm \(z_{k+1}=f(z_ k)\) with an arbitrary starting point \(z_ 0\) for the construction of a zero of p. A key role in the underlying theory is played by the ''Cauchy radius of p at z'' which is defined as the non-negative solution of the equation \[ \sum^{n}_{k=1}| p^{(k)}(z)| x^ k/k!=| p(z)|. \] There is a detailed discussion of numerical aspects of the associated root-finding algorithm (which is based on a combination of the global descent method with the local Newton method) as well as a report on results of some numerical experiments.
scientific article; zbMATH DE number 3881777
scientific article