A modified Newton method in parallel circular iteration of single-step and double-step (Q814105)

The Newton method is a classical method for finding zeros of a monic polynomial, \(f(z)\), of degree \(n>1\) with complex coefficients. The well-known iteration \(z^{k+1}=z^k-f(z^k)/f'(z^k)\) is of order \(2\) if the zero is simple. Another higher-order iteration which can be applied to find simultaneously all zeros of \(f(z)\) is constructed as \(z_i^{k+1}=z_i^k-1/[f'(z_i^k)/f(z_i^k)-\sum_{j\not=i}1/(z_i^k-z_j^k)].\) Furthermore, a circular iteration corresponding to this construction can be formed as \(Z_i^{k+1}=z_i^k-1/[f'(z_i^k)/f(z_i^k)-\sum_{j\not=i}1/(z_i^k-Z_j^k)].\) This construction is called modified Newton method in parallel circular iteration. In this paper, two theorems for the convergence of this method are given. The convergence 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 of the fifth order is obtained accurately as well.