On the convergent conditions of Durand–Kerner method in parallel circular iteration of single-step and double-step

摘要

In this paper, two theorems for the convergence of Durand–Kerner method in parallel circular iteration are given. The convergent condition of single-step method's circular iteration is relaxed compared with the classical theorem in the same field, while the one of the first proposed double-step method's is obtained accurately as well. An unique phenomenon appears that both of the constants concerned with either condition are [lnφ]−1, where φ=(5+1)/2≈1.61803399.