Modified iterative double step length model for solving nonlinear systems with application to motion control

The double-step-length procedure presents an inventive approach by incorporating two corrections during each iteration, which improves the robustness of the iterative process. The idea that if one correction fails, the other might still help guide the system in the right direction seems likely to reduce the chance of getting divergence, a common challenge in nonlinear solvers. This article presents a more effective approach to solving systems of nonlinear equations by combining the double-step length iterative method with the Picard-Mann hybrid technique. This combination improves performance. We streamline the process by employing a positive scalar to approximate the Jacobian matrix in Newton's method, thereby eliminating the need to calculate derivatives. An inexact line search technique was used to determine the values of the two-step lengths. The proposed method converges globally under mild conditions. The numerical tests highlight its efficiency, proving that it is more effective than existing double-direction and length methods. Furthermore, we have effectively implemented this technique in motion control applications, showcasing its practicality and efficacy.