On some methods for the simultaneous determination of polynomial zeros (Q1922231)

The authors consider some iteration methods for the simultaneous approximation of all zeros of a polynomial \(P(z)\). The methods are related to that of Weierstrass (and other authors independently) which uses the iteration formula \(\widehat z_i=z_i-P(z_i)/\prod^n_{\substack{ k=1\\ k\neq i}} (z_i-z_k)\), where the \(z_i\), \(i=1,2,\dots,n\), are approximations to the zeros of \(P(z)\). A unified convergence analysis is given for some known methods having cubic convergence rate. It is also shown that an SOR-like acceleration of the Durand-Kerner method converges for a real acceleration parameter in the range \((0,2)\). Numerical examples illustrating the theoretical results are given and include examples of solution trajectories.