Iterative Methods For Nonlinear Equations Research Articles

Modification of Adomian decomposition technique in multiplicative calculus and application for nonlinear equations

Multiplicative calculus is a mathematical system that offers an alternative to traditional calculus. Instead of using addition and subtraction to measure change, as in traditional calculus, it uses multiplication and division. The framework of nonlinear equations is an incredibly powerful tool that has proven invaluable in advancing our understanding of various phenomena across a wide range of applied sciences. This framework has enabled researchers to gain deeper insights into a vast array of scientific problems. The physical interpretation of iterative methods for nonlinear equations using multiplicative calculus offers a unique perspective on solving such equations and opens up potential applications across various scientific disciplines. Multiplicative calculus naturally aligns with processes characterized by exponential growth or decay. In many physical, biological, and economic systems, quantities change in a manner proportional to their current state. Multiplicative calculus models these processes more accurately than traditional additive approaches. For example, population growth, radioactive decay, and compound interest are all better described multiplicatively. The primary objective of this work is to modify and implement the Adomian decomposition method within the multiplicative calculus framework and to develop an effective class of multiplicative numerical algorithms for obtaining the best approximation of the solution of nonlinear equations. We build up the convergence criteria of the multiplicative iterative methods. To demonstrate the application and effectiveness of these new recurrence relations, we consider some numerical examples. Comparison of the multiplicative iterative methods with the similar ordinary existing methods is presented. Graphical comparison is also provided by plotting log of residuals. The purpose in constructing new algorithms is to show the implementation and effectiveness of multiplicative calculus.

Extended Higher Order Iterative Method for Nonlinear Equations and its Convergence Analysis in Banach Spaces

In this article, a novel higher order iterative method for solving nonlinear equations is developed. The new iterative method obtained from fifth order Newton-Özban method attains eighth order of convergence by adding a single step with only one additional function evaluation. The method is extended to Banach spaces and its local as well as semi-local convergence analysis is done under generalized continuity conditions. The existence and uniqueness results of solution are also provided along with radii of convergence balls. From the numerical experiments, it can be inferred that the proposed method is more accurate and effective in high precision computations than existing eighth order methods. The computation of error analysis and norm of functions demonstrate that proposed method takes a lead over the considered methods.

Extending the Applicability of Highly Efficient Iterative Methods for Nonlinear Equations and Their Applications

Numerous three-step methods of high convergence order have been developed to produce sequences approximating solutions of equations usually defined on the Euclidean space with a finite dimension. The local convergence order is determined by Taylor expansions requiring the existence of derivatives that are not present on the methods. The more interesting semi-local convergence analysis for these methods has not been considered before. The semi-local is also provided based on generalized ω-continuity conditions on the derivative of the operator involved and the majorizing sequences, thus limiting their usage to only solving equations with operators that are many times differentiable. However, these methods may convergence to a solution of the equation even if these high-order derivatives do not exist. That is why a methodology is utilized on two sixth convergence order methods and in the more general setting of a Banach space. This time, the convergence depends only on the operators and the first derivative on the method. Therefore, by this methodology the applicability of the methods is in the extended area. Although this methodology is demonstrated on two competing and efficient methods, it can also be utilized for the same reasons on other methods involving inverses of operators that are linear. This is the motivation and novelty of the paper. The numerical applications further validate the theoretical results both in the local as well as the semi-local convergence case.

Optimal fourth and eighth-order iterative methods for non-linear equations

Optimal fourth and eighth-order iterative methods for non-linear equations

An optimal fourth-order second derivative free iterative method for nonlinear scientific equations

In the present paper, we develop an efficient second derivative free two-step optimal fourth-order iterative method for nonlinear equations. We explore the convergence criteria of the proposed method and also exhibit its validity and efficiency by considering some test problems. We present both numerical as well as graphical comparisons. Further, the dynamical behavior of the proposed method is explored.

A new family of optimal fourth-order iterative methods for nonlinear equations

A new two-parameter family of fourth-order iterative methods for the numerical solution of nonlinear equations of the form f(x)=0 has been introduced and their convergence analysis have been performed. The new methods in the family are optimal in the sense of Kung–Traub conjecture. Numerical tests which are made to support the theoretical results also show that our new family of optimal fourth order methods perform well and in many cases some members of the family are superior to other well-known and recent fourth order optimal methods in the literature.

Extended local and semilocal convergence for interpolatory iterative methods for nonlinear equations

Extended local and semilocal convergence for interpolatory iterative methods for nonlinear equations

Dynamical Comparison of Several Third-Order Iterative Methods for Nonlinear Equations

Dynamical Comparison of Several Third-Order Iterative Methods for Nonlinear Equations

Axis optimisation of arch-shaped pylons for high-speed railway cable-stayed bridges

In order to obtain reasonable arch-shaped pylon axes for high-speed railway cable-stayed bridges, an iterative optimisation method based on force-balance analysis, a fixed point iterative method for nonlinear equations, and finite element analysis was proposed. The developed method was used to optimise the axes of the arch-shaped pylons of two high-speed railway arch-shaped pylon cable-stayed bridges. The results showed that the arch-shaped pylon axis of a cable-stayed bridge with main spans of 2 × 80 m converged after seven iterations. The absolute maximum bending moments under the service load of the optimised arch-shaped pylons were reduced by 53.15–83.53%, and the maximum and minimum stresses under the service load were reduced by 79.81–106.80% and 52.63–66.74%, respectively. Under a double-line ZK-live load, the maximum upward and minimum downward deflections of the optimised arch-shaped pylons were reduced by 21.30–25.0% and 12.4–21.6%, respectively, which resulted in increased stiffness of the bridge. The optimised axes obtained after inputting different initial axes to the proposed method were identical, and the obtained axis was adopted as the final arch-shaped pylon axis design for the actual bridge. For the second arch-shaped pylon cable-stayed bridge, which had main spans of 2 × 90 m, the absolute maximum bending moments of the optimised arch-shaped pylons were reduced by 77.05–92.08%, and the maximum and minimum stresses under the service load decreased by 117.82–173.91% and 39.00–41.38%, respectively. The results obtained illustrate the suitability of the proposed arch-shaped pylon axis optimisation method and demonstrate its reliability.

Ehrlich-type Methods with King's Correction for the Simultaneous Approximation of Polynomial Complex Zeros

There are many simultaneous iterative methods for approximating complex polynomial zeros, from more traditional numerical algorithms, such as the well-known third order Ehrlich–Aberth method, to the more recent ones. In this paper, we present a new family of combined iterative methods for the simultaneous determination of simple complex zeros of a polynomial, which uses the Ehrlich iteration and a correction based on King's family of iterative methods for nonlinear equations. The use of King's correction allows increasing the convergence order of the basic method from three to six. Some numerical examples are given to illustrate the convergence behaviour and effectiveness of the proposed sixth order Ehrlich-like family of combined iterative methods for the simultaneous approximation of simple complex polynomial zeros.

A general class of four parametric with- and without memory iterative methods for nonlinear equations

In this paper, we have constructed a derivative-free weighted eighth-order iterative class of methods with and without-memory for solving nonlinear equations. These methods are optimal as they satisfy Kung–Traub’s conjecture. We have used four accelerating parameters, univariate and multivariate weight functions at the second and third step of the method respectively. This family of schemes is converted into with-memory one by approximating the parameters using Newton’s interpolating polynomials of appropriate degree to increase the order of convergence to 15.51560 and the efficiency index is nearly two. Numerical and dynamical comparison of our methods is done with some recent methods of the same order applying them on some applied problems from chemical engineering, such as fractional conversion in a chemical reactor. The stability of the proposed schemes and their comparison with existing ones is made by using dynamical planes of the different methods, showing the wideness of the sets of converging initial estimations for all the test functions. The proposed schemes show to have good stability properties, as in their eighth-order version as well as in the case of methods with memory.

Optimal fourth- and eighth-order of convergence derivative-free modifications of King’s method

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.

Eighth order family of iterative methods for nonlinear equations and their basins of attraction

We present a new family of eighth order methods for solving nonlinear equations. The order of convergence of the considered methods is proved and corresponding asymptotic error constants are expressed in terms of four parameters. Numerical examples, obtained using Mathematica with high precision arithmetic, demonstrate convergence and efficacy of our family of methods. For some combinations of parameter values, the new eighth order methods produce very good results on tested examples compared to the results produced by some of the eighth order methods existing in the related literature. An exploration of the relevant dynamics of the proposed methods is presented along with illustrative basins of attraction for various polynomials.

Ball convergence of some iterative methods for nonlinear equations in Banach space under weak conditions

The aim of this paper is to expand the applicability of a fast iterative method in a Banach space setting. Moreover, we provide computable radius of convergence, error bounds on the distances involved and a uniqueness of the solution result based on Lipschitz-type functions not given before. Furthermore, we avoid hypotheses on high order derivatives which limit the applicability of the method. Instead, we only use hypotheses on the first derivative. The convegence order is determined using the computational order of convergence or the approximate order of convergence. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study.

On a robust Aitken–Newton method based on the Hermite polynomial

We introduce an Aitken–Newton iterative method for nonlinear equations, which is obtained by using the Hermite inverse interpolation polynomial of degree 2, with two nodes given by the Newton method.The local convergence of these iterates is shown to be 8, and the efficiency index is 85≈1.51, which is not optimal in the sense of Kung and Traub. However, we show that under supplementary conditions (sometimes easy to verify) the inner and outer iterates converge monotonically to the solution. This aspect allows an improved control of the iteration stopping (avoiding divisions by zero) and offer an alternative way to the estimation of radius of attraction balls in ensuring the convergence of the iterates. Numerical examples show that this method may become competitive and in certain circumstances even more robust than certain optimal methods of same convergence order.

On an efficient [formula omitted]-step iterative method for nonlinear equations

This paper is devoted to the construction and analysis of an efficient k-step iterative method for nonlinear equations. The main advantage of this method is that it does not need to evaluate any high order Fréchet derivative. Moreover, all the k-step have the same matrix, in particular only one LU decomposition is required in each iteration. We study the convergence order, the efficiency and the dynamics in order to motivate the proposed family. We prove, using some recurrence relations, a semilocal convergence result in Banach spaces. Finally, a numerical application related to nonlinear conservative systems is presented.

Local convergence of a family of iterative methods for Hammerstein equations

In this paper we give a local convergence result for a uniparametric family of iterative methods for nonlinear equations in Banach spaces. We assume boundedness conditions involving only the first Frechet derivative, instead of using boundedness conditions for high order derivatives as it is usual in studies of semilocal convergence, which is a drawback for solving some practical problems. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained. We apply this theory to different examples, including a nonlinear Hammerstein equation that have many applications in chemistry and appears in problems of electro-magnetic fluid dynamics or in the kinetic theory of gases. With these examples we illustrate the advantages of these results. The global convergence of the method is addressed by analysing the behaviour of the methods on complex polynomials of second degree.

New iterative methods for nonlinear equations in R

A parameter based on two-step iterative method combining two known third order methods is proposed for solving nonlinear equations in R. The convergence analysis of the method is established to show its fourth order of convergence. In order to enhance the order of convergence from four to seven, its three-step extension is also developed. The increase in the number of function evaluations is reduced by approximating the involved derivative in the third step by the second order divided differences. A number of numerical examples are worked out to demonstrate the efficacy of both the methods. The performance measures used are the number of iterations and the total number of function evaluations required to get an approximation to the root correct up to fifteen decimal places. The informational index and efficiency index of the proposed methods are computed. On comparison with the existing methods, it is observed that our methods give improved performance in terms of computational speed and efficiency.

Ball convergence theorems and the convergence planes of an iterative method for nonlinear equations

We study the local convergence of a method presented by Cordero et al. of convergence order at least five to approximate a locally unique solution of a nonlinear equation. These studies show the convergence under hypotheses on the third derivative or even higher. The convergence in this study is shown under hypotheses on the first derivative. Hence, the applicability of the method is expanded. The dynamical analysis of this method is also studied. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.

Applied Technology with a Modified Sixth-Order Convergent Iterative Method for Solving Nonlinear Equations

With the rapid development and wide applications of information science and applied technology, nonlinear problems become an important direction of research in the field of numerical analysis. In this paper, we mainly study the iterative method for nonlinear equations. We propose and analyze a modified Newton-type method with order of convergence six for solving nonlinear equations. The method is free from second derivatives. The efficiency index of the presented method is 1.565, which is better than that of the classical Newton’s method 1.414. Some numerical experiments illustrate the efficiency and performance of the proposed method.