Improved schemes on Durand-Kerner method and Aberth method (Q2706068)

A new derivation of the Durand-Kerner method by homotopy is first presented. It gives a geometric interpretation of the method. Then a two-step iterative scheme is proposed which is equivalent to the Durand-Kerner method but no longer needs the polynomial evaluation for each iteration after the first one. For the Aberth method an equivalent form is proposed which no longer needs the evaluation of the first derivative of the polynomial for each iteration. Then two \(r\)-step Aberth methods are presented, and their convergence rates are \(2r+1\).