This study focuses on an improved Newton-type iterative scheme for solving systems of nonlinear equations, which is based on a rational approximation with linear numerator and denominator (RALND) model and proposed in (Saheya et al SpringerPlus 5:1–13, 2016). We prove and verify the third-order convergence of this algorithm through theoretical analysis and numerical computations. Numerical experiments and application instances are conducted to demonstrate its effectiveness. The performance profile compares this method with traditional Newton’s method and another third-order convergent method, which shows that this method outperforms the traditional Newton’s method in terms of the number of iterations, and its computational efficiency is comparable to that of the third-order method. The analysis of the basin of attraction maps compares the INM method with the traditional Newton’s method and another third-order convergent method. The results demonstrate that the INM method offers superior characteristics in terms of the contour of the convergence region and the complexity of the boundary compared to the other methods.

We propose an efficient neural network for solving the second-order cone constrained variational inequality (SOCCVI). The network is constructed using the Karush-Kuhn-Tucker (KKT) conditions of the variational inequality (VI), which is used to recast the SOCCVI as a system of equations by using a smoothing function for the metric projection mapping to deal with the complementarity condition. Aside from standard stability results, we explore second-order sufficient conditions to obtain exponential stability. Especially, we prove the nonsingularity of the Jacobian of the KKT system based on the second-order sufficient condition and constraint nondegeneracy. Finally, we present some numerical experiments, illustrating the efficiency of the neural network in solving SOCCVI problems. Our numerical simulations reveal that, in general, the new neural network is more dominant than all other neural networks in the SOCCVI literature in terms of stability and convergence rates of trajectories to SOCCVI solution.