Our focus in this study was on examining the convergence problem of a novel method, inspired by the Ulm-Chebyshev-like Cayley transform method, which was designed to solve the inverse eigenvalue problems (IEPs) with multiple eigenvalues. Compared with other existing methods, the proposed method has higher convergence order and/or requires less operations. Under the assumption that the relative generalized Jacobian matrices at a solution are nonsingular, the proposed method was proved to be convergent with cubic convergence. Experimental findings demonstrated the practicality and efficiency of the suggested approaches.

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 $F(x)=0$,

$$\{\begin{array}{l}{\mathbf{y}}_n={\mathbf{x}}_n-B_{n}F({\mathbf{x}}_n),\\ {\mathbf{z}}_n={\mathbf{y}}_n-B_{n}F({\mathbf{y}}_n),\\ {\mathbf{x}}_{n+1}={\mathbf{z}}_n-B_{n}F({\mathbf{z}}_n),\\ {\overline{B}}_n=2B_n-B_n{A}_{n+1}{B}_n,\\ B_{n+1}={\overline{B}}_n+{\overline{B}}_n(2I-A_{n+1}{\overline{B}}_n)(I-A_{n+1}{\overline{B}}_n),n=0,1,2,\dots ,\end{array}$$

where $A_{n+1}$ is an approximation of the derivative $F^{\prime }({\mathbf{x}}_{n+1})$. 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.