A new multi-step method for solving nonlinear systems with high efficiency indices

Solving nonlinear problems stands as a pivotal domain in scientific exploration. This study introduces a novel method comprising basic and multi-step components. The proposed iterative method has a convergence order of \(2m+1\) $2m+1$ , where m is the step number for \(m\ge 2\) $m\ge 2$ . Since our proposed method has only one Fréchet derivative evaluation and its inversion, the method has a higher efficiency index than previous methods. To comprehensively evaluate the method’s performance in efficiency, accuracy, and attraction basin behavior, numerical tests are presented. Furthermore, we applied the proposed method to solve renowned equations like Hammerstein’s integral equation and Berger’s equation after transforming them into nonlinear systems.