Order Iterative Methods Research Articles

A convergence analysis of a family of third order iterative methods in Riemannian manifold

A convergence analysis of a family of third order iterative methods in Riemannian manifold

A three step seventh order iterative method for solution nonlinear equation using Lagrange interpolation technique

This research paper comprehensively presents the derivation of a seventh-order iteration scheme designed to obtain simple roots of nonlinear equations through the utilization of Lagrange interpolation technique. The scheme is characterized by the requirement for three function evaluations and one evaluation of the first derivative in each iteration. A detailed convergence analysis is also carried out to assess the efficacy of the proposed method. Additionally, the paper includes comprehensive numerical experiments aimed at confirming the theoretical results and illustrating the competitive performance of the derived iteration scheme.

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.

Constructing optimal fourth and eighth order iterative methods by using variant of Newton’s method

In this paper, we have presented an optimal fourth order iterative method and an optimal eighth order iterative method without memory using weight functions. In terms of computational point of view, our first method require three evaluations (two function and one first derivatives) per iteration to get fourth order and the second method require four evaluations (three functions and one derivatives) per iteration to get eighth order. Hence, these methods have high efficiency indices 1.587 and 1.682 respectively. Some numerical examples are tested to know the performance of the new methods which verifies the theoretical results.

Optimal fourth order iterative method for solving nonlinear equation

The principal objective of this work is to propose a fourth order scheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourth order method uses two evaluation of the function and one evaluation of first derivative. So the method has satisfied the Kung-Traub optimality conjecture. In addition, the theoretical convergence properties of our schemes are fully explored with the help of the main theorem that demonstrates the convergence order. The performance and effectiveness of our optimal iteration functions have compared with the existing competitors on some standard academic problems.

Extraction of deflating subspaces using disk function of a matrix pencil via matrix sign function with application in generalized eigenvalue problem

The objective of this study is to extract two pairs of left and right deflating subspaces of a regular matrix pencil A−λB corresponding to the eigenvalues lying inside and outside the unit disk via computation of left and right disk functions diskL(A−λB) and diskR(A−λB) which are obtained from matrix sign evaluations of two suitable matrices. We propose a novel fifth order iterative method for evaluating matrix sign function which can be implemented to compute disk functions. Convergence analysis of the proposed method has been investigated globally along with the illustration of attraction basins. Asymptotic stability of the method is also presented. Then an application of deflating subspaces in solving generalized eigenvalue problem has been explored. Numerical test problems have been performed by considering matrices of various sizes to justify the efficiency and superiority of the proposed method.

Modifying Kurchatov's method to find multiple roots of nonlinear equations

We present a modification of Kurchatov's iterative method in order to solve a nonlinear equation with multiple roots, that is, for approximating solutions with multiplicity greater than one. One of its principal advantages is that you do not have to know a priori the multiplicity of the root, since it does not appear in the iterative expression. In order to examine the behaviour of the proposed method, we perform a dynamical analysis. Furthermore, we carry out some numerical experiments in order to confirm the theoretical results and compare the proposed method with other known methods for dealing with multiple roots.

Employing a Modified Sumudu with a Modified Iteration Method to Solve the System of Nonlinear Partial Differential Equations

The Sumudu transform is presented in this paper in a modified form which is aimed at improving its performance and employing it along with a modified iteration method in order to determine the solution to a system of nonlinear partial differential equations. This includes a theoretical analysis of the associated modified Sumudu transform. It also includes an explanation of the mathematical method for utilizing the transform in conjunction with the modified iteration technique. The iteration method is employed to determine the nonlinear terms of the equations. The research is valuable in the sense that it allows approximate and exact solution configurations to be determined by combining the modified Sumudu transform with a modified iteration method. As another benefit, the modified Sumudu transform can be developed and enhanced to be applicable to a wide range of equations, making it an effective solution tool. By combining techniques, a final advantage is that the solutions can be derived quickly and easily as a result of the combined approach. Finally, an old transformation which has been modified from the Sumudu transform is combined with the modified iteration method to examine its capability of yielding convergent solutions by incorporating the modified iteration method into it.

A parallel fractional explicit group modified AOR iterative method for solving fractional Poisson equation with multi-core architecture

This research studies the Multi-core Architecture for the Modified Accelerated Overrelaxation (MAOR) scheme for solving the fractional Poisson equation. The equation is discretized using the fractional explicit group (FEG) finite difference method. This research also presents the parallel implementation of the fractional order iterative methods with the chessboard (CB) ordering strategies. The experimental results of the test problems were compared with other well known relaxation schemes to test their feasibility on parallel environment. The parameters that is measured include number of iteration, error, execution time and computational cost for various number of processors.

New efficient bi-parametric families of iterative methods with engineering applications and their basins of attraction

We present in this paper two new families of bi-parametric multipoint higher order iterative methods of optimal order for determining simple roots of the nonlinear equation Ω(s)=0. The proposed families of methods are derivative-free with the optimal three-point fourth order methods requiring only three function evaluations per iteration and the optimal four-point eighth order methods consuming four function evaluations per iteration. Taylor’s series expansion and divided difference techniques are employed for the formulation of the methods. Their theoretical convergence properties are thoroughly analysed through the main theorems. Numerical experiments on nonlinear functions with some engineering applications are presented and are compared with some existing methods to demonstrate the effectiveness, applicability and validity of the proposed families of methods. Finally, graphical comparison is made through the basins of attraction which further provide useful information about their dynamical behaviour in the complex plane.

New higher order iterative methods for solving nonlinear equations and their basins of attraction

In this work, we have developed a family of eighth order Newton-type methods for solving nonlinear equations with four functional evaluations. Further, we have improved a new member of eighth order method to sixteenth order iterative method with five function evaluations. These two methods are satisfying Kung-Traub Conjecture and their efficiency indices are 1.682 and 1.741. Numerical results are carried out to validate the new methods and some existing compared methods. We consider a concrete variety of real life problems such as projectile motion, Planck’s radiation law, the trajectory of electron in air gap two parallel plates between, Van der Waal’s equation which explains the behavior of a real gas by introducing in the ideal gas equations, in order to check the applicability and effectiveness of our proposed methods. Finally, the basins of attraction are also given to demonstrate their dynamical behavior in the complex plane.

Application of Jacobi and Gauss–Seidel Numerical Iterative Solution Methods for the Stationary Distribution of Markov Chain

The Physical or Mathematical behaviour of this model may be represented by describing all the different states it may occupy and by indicating how it moves among these states. In this study, the stationary distribution of Markov chains was solved using iterative methods that begin with an initial estimate of the solution vector and then modified it in a way that brings it closer and closer to the real solution with each step or iteration. These methods also involved matrix operations like multiplication with one or more vectors, which preserves the transition matrices while speeding up the process. We computed the solutions using Jacobi iterative method and Gauss-Seidel iterative method in order to shed more light on the solutions of stationary distribution in Markov chain. This was done with the aid of several already-existing laws, theorems, and formulas of Markov chain and the application of normalization principle and matrix operations such as lower, upper, and diagonal matrices. The stationary distribution vector’s 𝜋𝑖,𝑖=1,2,…,4 are obtained for the illustrative example one as 𝜋(3) = (0.078125,0.109375,0.21875,0.59375) as well as the four eigenvalues of the matrix as 𝜆1=1.0, 𝜆2=−0.7718, 𝜆3,4=−0.1141±0.5576𝑖 using Jacobi iterative technique, and for illustrative example two using Gauss-Siedel method as 𝝅(𝟑) = (0.090909, 0.181818, 0.363636, 0.363636). The research shown that Gauss Siedel method converged faster than Jacobi method.

Extended Convergence of Two Multi-Step Iterative Methods

Iterative methods which have high convergence order are crucial in computational mathematics since the iterates produce sequences converging to the root of a non-linear equation. A plethora of applications in chemistry and physics require the solution of non-linear equations in abstract spaces iteratively. The derivation of the order of the iterative methods requires expansions using Taylor series formula and higher-order derivatives not present in the method. Thus, these results cannot prove the convergence of the iterative method in these cases when such higher-order derivatives are non-existent. However, these methods may still converge. Our motivation originates from the need to handle these problems. No error estimates are given that are controlled by constants. The process introduced in this paper discusses both the local and the semi-local convergence analysis of two step fifth and multi-step 5+3r order iterative methods obtained using only information from the operators on these methods. Finally, the novelty of our process relates to the fact that the convergence conditions depend only on the functions and operators which are present in the methods. Thus, the applicability is extended to these methods. Numerical applications complement the theory.

A New Third-Order Family of Multiple Root-Findings Based on Exponential Fitted Curve

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.

Extended Convergence for Two Sixth Order Methods under the Same Weak Conditions

High-convergence order iterative methods play a major role in scientific, computational and engineering mathematics, as they produce sequences that converge and thereby provide solutions to nonlinear equations. The convergence order is calculated using Taylor Series extensions, which require the existence and computation of high-order derivatives that do not occur in the methodology. These results cannot, therefore, ensure that the method converges in cases where there are no such high-order derivatives. However, the method could converge. In this paper, we are developing a process in which both the local and semi-local convergence analyses of two related methods of the sixth order are obtained exclusively from information provided by the operators in the method. Numeric applications supplement the theory.

Modified Iterative Method for Solving Sine - Gordon Equations

The basic goal of this research is to utilize an analytical method which is called the Modified Iterative Method in order to gain an approximate analytic solution to the Sine-Gordon equation. The suggested method is the amalgamation of the iterative method and a well-known technique, namely the Adomian decomposition method. A method minimizes the computational size, averts round-off errors, transformation and linearization, or takes some restrictive assumptions. Several examples are chosen to show the importance and effectiveness of the proposed method. In addition, a modified iterative method gives faster and easier solutions than other methods. These solutions are accurate and in agreement with the series of solutions that are provided by analytical results. To evaluate the outcomes in the modified iterative process, we have used the Matlab symbolic manipulator.

Semilocal convergence analysis of an eighth order iterative method for solving nonlinear systems

<abstract><p>In this paper, the semilocal convergence of the eighth order iterative method is proved in Banach space by using the recursive relation, and the proof process does not need high order derivative. By selecting the appropriate initial point and applying the Lipschitz condition to the first order Fréchet derivative in the whole region, the existence and uniqueness domain are obtained. In addition, the theoretical results of semilocal convergence are applied to two nonlinear systems, and satisfactory results are obtained.</p></abstract>

Extended Local Convergence for the Chebyshev Method under the Majorant Condition

In this article, we present the study on local convergence behaviour of Chebyshev's method, which is a third order iterative method used to solve a non-linear system in Banach space locale. In contrast to the earlier works, we establish the convergence using restricted-majorant conditions. As a result, we get betterconvergence radius and more tighter error estimates in comparison to the previous researches. Suitable numerical examples complement the theory.

A Unified Local-Semilocal Convergence Analysis of Efficient Higher Order Iterative Methods in Banach Spaces

To deal with the estimation of the locally unique solutions of nonlinear systems in Banach spaces, the local as well as semilocal convergence analysis is established for two higher order iterative methods. The given methods do not involve the computation of derivatives of an order higher than one. However, the convergence analysis was carried out in earlier studies by using the assumptions on the higher order derivatives as well. Such types of assumptions limit the applicability of techniques. In this regard, the convergence analysis is developed in the present study by imposing the conditions on first order derivatives only. The central idea for the local analysis is to estimate the bounds on convergence domain as well as the error approximations of the iterates along with the formulation of sufficient conditions for the uniqueness of the solution. Based on the choice of initial estimate in the given domain, the semilocal analysis is established, which ensures the convergence of iterates to a unique solution in that domain. Further, some applied problems are tested to certify the theoretical deductions.

New techniques to develop higher order iterative methods for systems of nonlinear equations

New techniques to develop higher order iterative methods for systems of nonlinear equations