On the convergent condition of Durand-Kerner method in parallel circular iteration of multi-step

In this paper, a new theorem for the convergence of Durand-Kerner method in parallel circular iteration are given. The uniform convergent condition of multi-step method's circular iteration is obtained accurately.The constant concerned with the condition is [ln ϕ]−1, which emerged from the convergent conditions of single-step and double-step method, where ϕ=(5+1)/2≈1.61803399.