Systems of nonlinear equations are known as the basis for many models of engineering and data science, and their accurate solutions are very critical in achieving progress in these fields. However, solving a system with multiple nonlinear equations, usually, is not an easy task. Consequently, finding a robust and accurate solution can be a very challenging problem in complex systems. In this work, a novel hybrid method namely Newton-Harris hawks optimization (NHHO) for solving systems of nonlinear equations is proposed. The proposed NHHO combines Newton's method, with a second-order convergence where the correct digits roughly double in every step, and the Harris hawks optimization (HHO) to enhance the search mechanism, avoid local optima, improve convergence speed, and find more accurate solutions. We tested a group of six well-known benchmark systems of nonlinear equations to evaluate the efficiency of NHHO. Further, comparisons between NHHO and other optimization algorithms, including the original HHO algorithm, Particle Swarm Optimization (PSO), Ant Lion Optimizer (ALO), Butterfly Optimization Algorithm (BOA), and Equilibrium Optimization (EO) were performed. The norm of the equation system was calculated as a fitness function to measure the optimization algorithms' performance. A solution with less fitness value is considered a better solution. Furthermore, the experimental results confirmed the superiority of NHHO over the other optimization algorithms, in the comparisons, in different aspects, including best solution, average fitness value, and convergence speed. Accordingly, the proposed NHHO is powerful and more effective in all benchmark problems in solving systems of nonlinear equations compared to the other optimization algorithms. Finally, NHHO overcomes the limitations of Newton's method, including selecting the initial point and divergence problems.

Several problems have been solved by nonlinear equation systems (NESs), including real-life issues in chemistry and neurophysiology. However, the accuracy of solutions is highly dependent on the efficiency of the algorithm used. In this paper, a Modified Sperm Swarm Optimization Algorithm called MSSO is introduced to solve NESs. MSSO combines Newton’s second-order iterative method with the Sperm Swarm Optimization Algorithm (SSO). Through this combination, MSSO’s search mechanism is improved, its convergence rate is accelerated, local optima are avoided, and more accurate solutions are provided. The method overcomes several drawbacks of Newton’s method, such as the initial points’ selection, falling into the trap of local optima, and divergence. In this study, MSSO was evaluated using eight NES benchmarks that are commonly used in the literature, three of which are from real-life applications. Furthermore, MSSO was compared with several well-known optimization algorithms, including the original SSO, Harris Hawk Optimization (HHO), Butterfly Optimization Algorithm (BOA), Ant Lion Optimizer (ALO), Particle Swarm Optimization (PSO), and Equilibrium Optimization (EO). According to the results, MSSO outperformed the compared algorithms across all selected benchmark systems in four aspects: stability, fitness values, best solutions, and convergence speed.

In this paper, we present two new iterative methods, one of them is second derivative free, for solving nonlinear equations. We derive these methods based on the Taylor series expansion and Halley’s method. The convergence analysis of the two methods is discussed. It is established that the new methods have sixth order of convergence. Several numerical examples given show that the new methods are comparable with the well-known existing methods of the same order.

Starting by King’s method, we propose a modified families of fourth- and eighth-order of convergence iterative methods for nonlinear equations. The fourth-order method requires at each iteration three function evaluations, while the eighth-order methods both need four function evaluations. The proposed methods are derivative-free. Based on the conjecture of Kung and Traub, the new methods attain the optimality with efficiency index $1.587$ for the fourth-order method and $1.68$ for the eighth-order methods. The convergence analyses of the methods are given, and comparisons with some well-known schemes having identical order of convergence demonstrate the efficiency of the present techniques.

In this study, we propose a modified predictor‐corrector Newton‐Halley (MPCNH) method for solving nonlinear equations. The proposed sixteenth‐order MPCNH is free of second derivatives and has a high efficiency index. The convergence analysis of the modified method is discussed. Different problems were tested to demonstrate the applicability of the proposed method. Some are real life problems such as a chemical equilibrium problem (conversion in a chemical reactor), azeotropic point of a binary solution, and volume from van der Waals equation. Several comparisons with other optimal and nonoptimal iterative techniques of equal order are presented to show the efficiency of the modified method and to clarify the question, are the optimal methods always good for solving nonlinear equations?

This study implements a very effective derivative-free approach for root determination in the context of nonlinear equations. The use of Homeier’s third-order technique, which is based on the application of Newton’s theorem with regard to the inverse function as well as the development of a novel category of Newton-type procedures that exhibit cubic convergence, is the basis of this study. A variety of nonlinear problems, such as nonlinear equations and nonlinear systems of equations, have been examined to examine the proposed method’s efficacy in contrast to other widely used derivative-free approaches. The results suggest that the proposed method outperforms the currently examined techniques published. The scheme given in this study demonstrates a reduced number of iterations acquired in attaining the intended result. Exhibiting a convergence order of around 2.5. This convergence order surpasses the convergence order observed in comparable approaches. Moreover, by following the prescribed procedure of Broyden’s technique while employing the recommended approach with regards to solving systems of nonlinear equations, the issue pertaining to the Jacobian may be circumvented. Hence, the technique suggested may be regarded as a prominent approach that achieves rapid convergence in determining the roots of nonlinear equations without the use of derivatives. In practice, we employ the novel approach to address practical challenges, such as the mixed Hammerstein integral equation, discretized Poisson equation, and Weierstrass method. Through comparisons and illustrative examples, it is evident that the proposed method proves to be effective and on par with existing techniques of equivalent order.