A new two step iterative method based on Newton and fixed point for solving absolute value equations

The absolute value equation (AVE) is particularly important in numerical mathematics, mathematical programming, and optimization. Due to its close connection to linear complementarity problems, significant research efforts have been devoted to developing efficient numerical solvers for AVEs. In this study, we presented a novel iterative methodology that combines Newton's method with fixed-point iteration, known as the Newton technique fixed-point iteration (NTFPI) method, to solve AVEs. The proposed solution integrated the optimal elements of Newton methods and fixed-point iterations to enhance the convergence speed and stabilize calculations. A theoretical examination was performed to determine the convergence characteristics of the approach. In addition, numerical assessments were performed to determine the efficacy of the NTFPI method in relation to other recognized Newton and fixed-point iterative techniques. The findings indicated that the proposed method enhanced accuracy and reduced repetitions, particularly for complex and large-scale tasks. These findings improved the numerical approaches for AVEs and demonstrate their potential applications in engineering, computational economics, and numerical optimization.