Optimal inequality factor for Durand-Kerner's and Tanabe's methods

Abstract

The convergence condition for the simultaneous inclusion methods is \(w^{(0)}<c(a,b,n)d^{(0)}\), where \(w^{(0)}\) is the maximum Weierstrass factor \(W^{0}_k\), \(k\in I_n\), and \(d^{0}\) is the minimum distance between \(z^{(0)}_1\), \(z^{(0)}_2\), \(\ldots\), \(z^{(0)}_n\), the distinct approximations of the simple roots of the polynomial \(\zeta_1\), \(\zeta_2\),\(\,\ldots\), \(\zeta_n\). The i-factor (inequality-factor) is the positive real function \(c(a,b,n)=\tfrac{1}{an+b}\). The article presents the optimum i-factor for the simultaneous inclusion methods Durand-Kerner and Tanabe.