Showing papers by "Alicia Cordero published in 2023"
TL;DR: In this paper , a conformable fractional Newton-type method was proposed for solving nonlinear equations, which involves a lower computational cost compared to other fractional iterative methods.
Abstract: Abstract In a recent paper, a conformable fractional Newton-type method was proposed for solving nonlinear equations. This method involves a lower computational cost compared to other fractional iterative methods. Indeed, the theoretical order of convergence is held in practice, and it presents a better numerical behaviour than fractional Newton-type methods formerly proposed, even compared to classical Newton-Raphson method. In this work, we design a generalization of this method for solving nonlinear systems by using a new conformable fractional Jacobian matrix, and a suitable conformable Taylor power series; and it is compared with classical Newton’s scheme. The necessary concepts and results are stated in order to design this method. Convergence analysis is made and a quadratic order of convergence is obtained, as in classical Newton’s method. Numerical tests are made, and the Approximated Computational Order of Convergence (ACOC) supports the theory. Also, the proposed scheme shows good stability properties observed by means of convergence planes.
2 citations
TL;DR: In this paper , the authors presented an eight-order optimal derivative-free method to find multiple zeros of the nonlinear equation by weight function approach in a three-step method including the first step as a Traub-Steffensen iteration and the next two as Traub Steffensen-like iterations.
Abstract: The problem of solving a nonlinear equation is considered to be one of the significant domain. Motivated by the requirement to achieve more optimal derivative-free schemes, we present an eighth-order optimal derivative-free method to find multiple zeros of the nonlinear equation by weight function approach in this paper. This family of methods requires four functional evaluations. The technique is based on a three-step method including the first step as a Traub-Steffensen iteration and the next two as Traub-Steffensen-like iterations. Our proposed scheme is optimal in the sense of Kung-Traub conjecture. The applicability of the proposed schemes is shown by using different nonlinear functions that verify the robust convergence behavior. Convergence of the presented family of methods is demonstrated through the graphical regions by drawing basins of attraction.
1 citations
TL;DR: In this paper , a new third-order family of iterative point-to-point methods was proposed to compute the multiple roots of nonlinear equations when the multiplicity (m ≥ 1) is known in advance.
Abstract: In this paper, we present a new third-order family of iterative methods in order to compute the multiple roots of nonlinear equations when the multiplicity (m≥1) is known in advance. There is a plethora of third-order point-to-point methods, available in the literature; but our methods are based on geometric derivation and converge to the required zero even though derivative becomes zero or close to zero in vicinity of the required zero. We use the exponential fitted curve and tangency conditions for the development of our schemes. Well-known Chebyshev, Halley, super-Halley and Chebyshev–Halley are the special members of our schemes for m=1. Complex dynamics techniques allows us to see the relation between the element of the family of iterative schemes and the wideness of the basins of attraction of the simple and multiple roots, on quadratic polynomials. Several applied problems are considered in order to demonstrate the performance of our methods and for comparison with the existing ones. Based on the numerical outcomes, we deduce that our methods illustrate better performance over the earlier methods even though in the case of multiple roots of high multiplicity.
1 citations
TL;DR: In this paper , a family of iterative schemes to estimate the solutions of nonlinear systems is presented, based on the Ermakov-Kalitkin procedure, which widens the set of converging initial estimations.
Abstract: A novel family of iterative schemes to estimate the solutions of nonlinear systems is presented. It is based on the Ermakov-Kalitkin procedure, which widens the set of converging initial estimations. This class is designed by means of a weight function technique, obtaining 6th-order convergence. The qualitative properties of the proposed class are analyzed by means of vectorial real dynamics. Using these tools, the most stable members of the family are selected, and also the chaotical elements are avoided. Some test vectorial functions are used in order to illustrate the performance and efficiency of the designed schemes.
TL;DR: In this paper , a parametric family of derivative-free three-step iterative methods with a weight function for solving nonlinear equations is presented, and various ways of introducing memory to this family are discussed.
Abstract: In this manuscript, we present a parametric family of derivative-free three-step 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 from different points of view.
TL;DR: In this article , a deep dynamical analysis is made, by using tools from multidimensional real discrete dynamics, of some derivative-free iterative methods with memory, showing which is the best in terms of wideness of basins of convergence or the existence of free critical points that would yield to convergence toward different elements from the desired zeros of the nonlinear function.
Abstract: In this paper, a deep dynamical analysis is made, by using tools from multidimensional real discrete dynamics, of some derivative-free iterative methods with memory. All of them have good qualitative properties, but one of them (due to Traub) shows to have 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 wideness of basins of convergence or the existence of free critical points that would yield to convergence toward 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 in their simplicity. These results have been checked with numerical and graphical comparison with many other known methods with and without memory, with different order of convergence, with excellent performance.
TL;DR: In this paper , a Kurchatov-type scheme and Steffensen's method with memory were developed for estimating the inverse of a nonsingular square complex matrix and the Moore-Penrose inverse of the singular square matrix or an arbitrary m×n complex matrix.
Abstract: Some iterative schemes with memory were designed for approximating the inverse of a nonsingular square complex matrix and the Moore–Penrose inverse of a singular square matrix or an arbitrary m×n complex matrix. A Kurchatov-type scheme and Steffensen’s method with memory were developed for estimating these types of inverses, improving, in the second case, the order of convergence of the Newton–Schulz scheme. The convergence and its order were studied in the four cases, and their stability was checked as discrete dynamical systems. With large matrices, some numerical examples are presented to confirm the theoretical results and to compare the results obtained with the proposed methods with those provided by other known ones.
TL;DR: In this paper , the authors proposed an iterative step that, combined with any other method, allows to obtain a polynomial approximation scheme for approximating the simple roots of polynomials simultaneously.
TL;DR: In this paper , the authors design point-to-point higher-order conformable Newton-type and multipoint procedures for solving nonlinear equations and propose a general technique to deduce the conformable version of any classical iterative method with integer derivatives.
Abstract: In recent years, some Newton-type schemes with noninteger derivatives have been proposed for solving nonlinear transcendental equations by using fractional derivatives (Caputo and Riemann–Liouville) and conformable derivatives. It has also been shown that the methods with conformable derivatives improve the performance of classical schemes. In this manuscript, we design point-to-point higher-order conformable Newton-type and multipoint procedures for solving nonlinear equations and propose a general technique to deduce the conformable version of any classical iterative method with integer derivatives. A convergence analysis is given and the expected orders of convergence are obtained. As far as we know, these are the first optimal conformable schemes, beyond the conformable Newton procedure, that have been developed. The numerical results support the theory and show that the new schemes improve the performance of the original methods in some aspects. Additionally, the dependence on initial guesses is analyzed, and these schemes show good stability properties.
TL;DR: In this paper , a parametric family of iterative methods is obtained as a generalization of Traub, which is also a member of it, and a dynamical analysis is performed after applying the family for solving a system cubic polynomials by means of multidimensional real dynamics.
Abstract: In this work, we modify the iterative structure of Traub's method to include a real parameter α $$ \alpha $$ . A parametric family of iterative methods is obtained as a generalization of Traub, which is also a member of it. The cubic order of convergence is proved for any value of α $$ \alpha $$ . Then, a dynamical analysis is performed after applying the family for solving a system cubic polynomials by means of multidimensional real dynamics. This analysis allows to select the best members of the family in terms of stability as a preliminary study to be generalized to any nonlinear function. Finally, some iterative schemes of the family are used to check numerically the previous developments when they are used to approximate the solutions of academic nonlinear problems and a chemical diffusion reaction problem.
TL;DR: In this article , a new parametric class of multi-step iterative procedures to find the solutions of systems of nonlinear equations is presented, where the convergence order of the family can be accelerated to six by setting two parameters, resulting in a uniparametric family.
Abstract: This manuscript is focused on a new parametric class of multi-step iterative procedures to find the solutions of systems of nonlinear equations. Starting from Ostrowski’s scheme, the class is constructed by adding a Newton step with a Jacobian matrix taken from the previous step and employing a divided difference operator, resulting in a triparametric scheme with a convergence order of four. The convergence order of the family can be accelerated to six by setting two parameters, resulting in a uniparametric family. We performed dynamic and numerical development to analyze the stability of the sixth-order family. Previous studies for scalar functions allow us to isolate those elements of the family with stable performance for solving practical problems. In this regard, we present dynamical planes showing the complexity of the family. In addition, the numerical properties of the class are analyzed with several test problems.
TL;DR: In this article , a generalization of a known family of multipoint scalar iterative processes for approximating the solutions of nonlinear systems is presented, and the convergence analysis of the proposed class under various smooth conditions is provided.
Abstract: In this manuscript, we carry out a study on the generalization of a known family of multipoint scalar iterative processes for approximating the solutions of nonlinear systems. The convergence analysis of the proposed class under various smooth conditions is provided. We also study the stability of this family, analyzing the fixed and critical points of the rational operator resulting from applying the family on low-degree polynomials, as well as the basins of attraction and the orbits (periodic or not) that these points produce. This dynamical study also allows us to observe which members of the family are more stable and which have chaotic behavior. Graphical analyses of dynamical planes, parameter line and bifurcation planes are also studied. Numerical tests are performed on different nonlinear systems for checking the theoretical results and to compare the proposed schemes with other known ones.
TL;DR: In this article , a derivative-free multi-step iterative scheme based on Steffensen's method is proposed to avoid excessively increasing the number of functional evaluations and, at the same time, to increase the order of convergence.
TL;DR: In this paper , an iterative step that can be added to any numerical process for solving systems of nonlinear equations is designed, by means of this addition, the resulting iterative scheme obtains, simultaneously, all the solutions to the vectorial problem.
Abstract: Abstract In this manuscript, we design an iterative step that can be added to any numerical process for solving systems of nonlinear equations. By means of this addition, the resulting iterative scheme obtains, simultaneously, all the solutions to the vectorial problem. Moreover, the order of this new iterative procedure duplicates that of their original partner. We apply this step to some known methods and analyse the behaviour of these new algorithms, obtaining simultaneously the roots of several nonlinear systems.
TL;DR: The fixed point theory is a fascinating subject that has a wide range of applications in many areas of mathematics and has been used extensively in many applications in the past few decades as discussed by the authors .
Abstract: Fixed point theory is a fascinating subject that has a wide range of applications in many areas of mathematics [...]
TL;DR: Ermakov and Kalitkin this paper proposed Ermakov's Hyperfamily, which is a family of two-parameter fourth-order optimal methods with good stability properties.
Abstract: In this paper, we generalize the scheme proposed by Ermakov and Kalitkin and present a class of two-parameter fourth-order optimal methods, which we call Ermakov’s Hyperfamily. It is a substantial improvement of the classical Newton’s method because it optimizes one that extends the regions of convergence and is very stable. Another novelty is that it is a class containing as particular cases some classical methods, such as King’s family. From this class, we generate a new uniparametric family, which we call the KLAM, containing the classical Ostrowski and Chun, whose efficiency, stability, and optimality has been proven but also new methods that in many cases outperform these mentioned, as we prove. We demonstrate that it is of a fourth order of convergence, as well as being computationally efficienct. A dynamical study is performed allowing us to choose methods with good stability properties and to avoid chaotic behavior, implicit in the fractal structure defined by the Julia set in the related dynamic planes. Some numerical tests are presented to confirm the theoretical results and to compare the proposed methods with other known methods.
TL;DR: In this article , the stability of the family of iterative methods designed by Jarratt using complex dynamics tools is analyzed. And the authors deduce that classical Jarratter's scheme is not the only stable element of the group. And they also obtain information about the members of the class with chaotical behavior.
Abstract: In this paper, we analyze the stability of the family of iterative methods designed by Jarratt using complex dynamics tools. This allows us to conclude whether the scheme known as Jarratt’s method is the most stable among all the elements of the family. We deduce that classical Jarratt’s scheme is not the only stable element of the family. We also obtain information about the members of the class with chaotical behavior. Some numerical results are presented for confirming the convergence and stability results.
TL;DR: In this article , a derivative-free multi-step iterative scheme based on Steffensen's method is proposed, where a weight function on already evaluated operators is used to avoid excessively increasing the number of functional evaluations and, at the same time, to increase the order of convergence.
TL;DR: In this paper , two parametric classes of iterative methods without memory were designed to solve nonlinear systems, whose convergence order is 4 and 7, respectively, and memory is introduced in these families of different forms.
Abstract: In this paper, we design two parametric classes of iterative methods without memory to solve nonlinear systems, whose convergence order is 4 and 7, 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 4 to 6 the convergence order in the first family and from 7 to 11 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.