A multi-step Ulm-Chebyshev-like method for solving nonlinear operator equations

In this paper, based on the Ulm-Chebyshev iterative procedure, we present a multi-step Ulm-Chebyshev-like method to solve systems of nonlinear equations $ F(x) = 0 $,$ \begin{equation*} \left\{\begin{array}{l} \quad {\bf{y}}_{n} = {\bf{x}}_{n}-B_{n}F( {\bf{x}}_{n}),\\ \quad {\bf z}_{n} = {\bf{y}}_{n}-B_{n}F( {\bf{y}}_{n}),\\ {\bf{x}}_{n+1} = {\bf z}_{n}-B_{n}F( {\bf z}_{n}),\\ \quad \bar{B}_{n} = 2B_{n}-B_{n}A_{n+1}B_{n},\\ B_{n+1} = \bar{B}_{n}+\bar{B}_{n}(2I-A_{n+1}\bar{B}_{n})(I-A_{n+1}\bar{B}_{n}),\quad n = 0,1,2,\ldots, \end{array}\right. \end{equation*} $where $ A_{n+1} $ is an approximation of the derivative $ F'({\bf{x}}_{n+1}) $. This method does not contain inverse operators in its expression, and does not require computing Jacobian matrices for solving Jacobian equations. We have proved that the multi-step Ulm-Chebyshev-like method converges locally to the solution with $ R $-convergence rate 4 under appropriate conditions. Some applications are given, compared with other existing methods, where the most important features of the method are shown.