We present a new scheme for solving nonlinear systems that combines simplicity with high performance, offering minimal evaluation cost for an arbitrary number of steps. Given the specified assumptions, we provide its development and a convergence analysis, guaranteeing that the order is twice the number of linear systems or steps. Based on this, we introduce a new eighth-order method with low computational cost, requiring only four linear systems that share the same Jacobian matrix and scalar weight function. Efficiency studies show that this new, high-order method competes effectively with other efficient schemes in the literature. A brief real-dynamics study confirms the scheme's superior stability, covering large, connected convergence regions. Finally, numerical tests validate the theoretical results. The contribution of this work is to enable the use of very high-order schemes with low computational costs, opening new avenues for their practical application.

In this paper, we propose a new parametric family of iterative schemes to compute the inverse of a complex nonsingular matrix. It is shown that the members of this family have at least a fourth-order of convergence. A particular element of the class is extended to approximate the Moore-Penrose inverse of rectangular complex matrices, keeping the convergence order. A dynamic analysis is performed to obtain a parameter domain in which stability is assured and to detect which members of the proposed family have good stability properties and which have chaotic behavior. Some numerical examples, with matrices of different sizes, are tested to confirm the theoretical and dynamical results.

In this paper, a hybrid of Jarratt and Particle Swarm Optimization (PSO) has been proposed to solve nonlinear system of equations and optimization problems efficiently. Jarratt fourth order method has been combined with PSO to accelerate the process of obtaining the solution. The hybrid approach merges the exploratory and stochastic features of PSO with the exact and predictable Jarratt technique. The proposed technique put in work to optimize the work in power flow analysis of IEEE 5, 6 and 9-bus systems. We aim to simultaneously minimize the losses in the power system by finding the best values of voltage angle and magnitude at each bus in power system analysis. The performance of hybrid method has been compared with the previous schemes. The comparison shows that the proposed method is doing a way better job in minimizing the loses in power system as compared to the previous methods. A comparison has been made which show the numerical results for hybrid approach is better as compared to PSO and Jarratt method.

This manuscript is devoted to a derivative-free parametric iterative step to obtain roots simultaneously for both nonlinear systems and equations. We prove that when it is added to an arbitrary scheme, it doubles the convergence order of the original procedure and defines a new algorithm that obtains several solutions simultaneously. Different numerical experiments are carried out to check the behaviour of the iterative methods by changing the value of the parameter and the initial guesses. Also, it is perform a numerical example where the dynamical planes are carried out to see and compare the basins of attraction.

In this manuscript,we present a parametric family of derivative-free 3-steps iterative methods with a weight function for solving nonlinear equations. We study various ways of introducing memory to this parametric family in order to increase the order of convergence without new functional evaluations. We also performed numerical experiments to compare the iterative methods fromdifferent points of view.

In this paper, a deep dynamical analysis is made using tools from multidimensional real discrete dynamics of some derivative-free iterative methods with memory. They all have good qualitative properties, but one of them (due to Traub) shows the same behavior as Newton’s method on quadratic polynomials. Then, the same techniques are employed to analyze the performance of several multipoint schemes with memory, whose first step is Traub’s method, but their construction was made using different procedures. Therefore, their stability is analyzed, showing which is the best in terms of the wideness of basins of convergence or the existence of free critical points that would yield convergence towards different elements from the desired zeros of the nonlinear function. Therefore, the best stability properties are linked with the best estimations made in the iterative expressions rather than their simplicity. These results have been checked with a numerical and graphical comparison with many other known methods with and without memory, with different orders of convergence, with excellent performance.

In this paper, we design two parametric classes of iterative methods without memory to solve nonlinear systems, whose convergence order is four and seven, respectively. From their error equations and to increase the convergence order without performing new functional evaluations, memory is introduced in these families of different forms. That allows us to increase from four to seven the convergence order in the first family and from seven to eleven in the second one. We perform some numerical experiments with big size systems for confirming the theoretical results and comparing the proposed methods along other known schemes.

In this work, we modify the iterative Kurchatov's method to solve nonlinear equations with multiple roots, that is,for approximating the solutions of multiplicity grater than one. Its main feature is that you do not need to know a priori the multiplicity of the root, which does not appear in the iterative expression. We perform a dynamical analysis to see the behaviour of the proposed method. We also carry out some numerical experiments to confirm the theoretical results and compare the proposed method with other known schemes for multiple roots.

In this paper, we study different ways for introducing memory to a parametric family of optimal two-step iterative methods. We study the convergence and the stability, by means of real dynamics, of the methods obtained by introducing memory in order to compare them. We also perform several numerical experiments to see how the methods behave.

The dynamical behavior of the rational vectorial operator associated with a multidimensional iterative method in polynomial systems, gives us interesting information about the stability of the iterative scheme. The stability of fixed points, dynamic planes, bifurcation diagrams, etc. are known tools that act in this sense. In this manuscript, we introduce a new tool, which we call isonormal surface, to complement the information about the stability of the iterative method provided by the dynamic elements mentioned above. These dynamical instruments are used for analyze the stability of a parametric family of multidimensional iterative schemes in terms of the value of the parameter. Some numerical tests confirm the obtained dynamical results.