Multi-step Iterative Methods Research Articles

Multistep Iterative Methods for Solving Equations in Banach Space

The novelty of this article lies in the fact that we extend the use of a multistep method for developing a sequence whose limit solves a Banach space-valued equation. We suggest the error estimates, local convergence, and semi-local convergence, a radius of convergence, the uniqueness of the required solution that can be computed under ω-continuity, and conditions on the first derivative, which is on the method. But, earlier studies used high-order derivatives, even though those derivatives do not appear in the body structure of the proposed method. In addition to this, they did not propose computable estimates and semi-local convergence. We checked the applicability of our study to three real-life problems for semi-local convergence and two problems chosen for local convergence. Based on the obtained results, we conclude that our approach improves its applicability and makes it suitable for challenges in applied science.

Achieving Optimal Order in a Novel Family of Numerical Methods: Insights from Convergence and Dynamical Analysis Results

In this manuscript, we introduce a novel parametric family of multistep iterative methods designed to solve nonlinear equations. This family is derived from a damped Newton’s scheme but includes an additional Newton step with a weight function and a “frozen” derivative, that is, the same derivative than in the previous step. Initially, we develop a quad-parametric class with a first-order convergence rate. Subsequently, by restricting one of its parameters, we accelerate the convergence to achieve a third-order uni-parametric family. We thoroughly investigate the convergence properties of this final class of iterative methods, assess its stability through dynamical tools, and evaluate its performance on a set of test problems. We conclude that there exists one optimal fourth-order member of this class, in the sense of Kung–Traub’s conjecture. Our analysis includes stability surfaces and dynamical planes, revealing the intricate nature of this family. Notably, our exploration of stability surfaces enables the identification of specific family members suitable for scalar functions with a challenging convergence behavior, as they may exhibit periodical orbits and fixed points with attracting behavior in their corresponding dynamical planes. Furthermore, our dynamical study finds members of the family of iterative methods with exceptional stability. This property allows us to converge to the solution of practical problem-solving applications even from initial estimations very far from the solution. We confirm our findings with various numerical tests, demonstrating the efficiency and reliability of the presented family of iterative methods.

A novel families of higher‐order multistep iterative methods for solving nonlinear systems

In this paper, we propose the first time fifth‐ and sixth‐order two‐step schemes. Based on the proposed two‐step methods, we develop three‐step iterative methods with the ninth‐ and tenth‐order of convergence. We also suggest two parametric families of these schemes. The obtained results are extended to ‐step iterations. The main advantages of the proposed methods are that they are higher order of convergence and utilize only one matrix inversion per iteration which makes these methods computationally more efficient. Numerical experiments are performed to confirm the theoretical results concerning the order of convergence and computational efficiency.

New Family of Multi-Step Iterative Methods Based on Homotopy Perturbation Technique for Solving Nonlinear Equations

This research aims to propose a new family of one-parameter multi-step iterative methods that combine the homotopy perturbation method with a quadrature formula for solving nonlinear equations. The proposed methods are based on a higher-order convergence scheme that allows for faster and more efficient convergence compared to existing methods. It aims also to demonstrate that the efficiency index of the proposed iterative methods can reach up to 43≈1.587 and 84≈1.681, respectively, indicating a high degree of accuracy and efficiency in solving nonlinear equations. To evaluate the effectiveness of the suggested methods, several numerical examples including their performance are provided and compared with existing methods.

Reasons for stability in the construction of derivative‐free multistep iterative methods

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.

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.

Constructing a Class of Frozen Jacobian Multi-Step Iterative Solvers for Systems of Nonlinear Equations

In this paper, in order to solve systems of nonlinear equations, a new class of frozen Jacobian multi-step iterative methods is presented. Our proposed algorithms are characterized by a highly convergent order and an excellent efficiency index. The theoretical analysis is presented in detail. Finally, numerical experiments are presented for showing the performance of the proposed methods, when compared with known algorithms taken from the literature.

New Multi-Step Iterative Methods for Solving Systems of Nonlinear Equations and Their Application on GNSS Pseudorange Equations.

A two-step fifth and a multi-step order iterative method are derived, for finding the solution of system of nonlinear equations. The new two-step fifth order method requires two functions, two first order derivatives, and the multi-step methods needs a additional function per step. The performance of this method has been tested with finding solutions to several test problems then applied to solving pseudorange nonlinear equations on Global Navigation Satellite Signal (GNSS). To solve the problem, at least four satellite’s measurements are needed to locate the user position and receiver time offset. In this work, a number of satellites from 4 to 8 are considered such that the number of equations is more than the number of unknown variables to calculate the user position. Moreover, the Geometrical Dilution of Precision (GDOP) values are computed based on the satellite selection algorithm (fuzzy logic method) which could be able to bring the best suitable combination of satellites. We have restricted the number of satellites to 4 to 6 for solving the pseudorange equations to get better GDOP value even after increasing the number of satellites beyond six also yields a 0.4075 GDOP value. Actually, the conventional methods utilized in the position calculation module of the GNSS receiver typically converge with six iterations for finding the user position whereas the proposed method takes only three iterations which really decreases the computation time which provide quicker position calculation. A practical study was done to evaluate the computation efficiency index (CE) and efficiency index (IE) of the new model. From the simulation outcomes, it has been noted that the new method is more efficient and converges 33% faster than the conventional iterative methods with good accuracy of 92%.

Monodisperse Macromolecules by Self-Interrupted Living Polymerization.

Biological macromolecules such as proteins and nucleic acids are monodisperse just as low-molar-mass organic compounds are. However, synthetic macromolecules contain mixtures of different chain lengths, the most uniform being generated by living polymerizations, which exhibit a maximum of 1-3% of chains with the desired length. Monodisperse natural and synthetic oligomers can be obtained in low quantities by tedious, multistep iterative methods. Here we report a methodology to synthesize monodisperse synthetic macromolecules by self-interrupted living polymerization. This methodology relies on a concept that combines supramolecular and macromolecular chemistry and differs from the conventional reactivity principles employed in the synthesis of polymers for over 100 years.

On the local and semilocal convergence of a parameterized multi-step Newton method

This paper is devoted to a family of Newton-like methods with frozen derivatives used to approximate a locally unique solution of an equation. We perform a convergence study and an analysis of the efficiency. This analysis gives us the opportunity to select the most efficient method in the family without the necessity of their implementation. The method can be applied to many type of problems, including the discretization of ordinary differential equations, integral equations, integro-differential equations or partial differential equations. Moreover, multi-step iterative methods are computationally attractive.

Single-step calibration method for nano indentation testing machines

Nano indentation is an effective method for materials mechanical characterisation at grain scale. Literature underlined relevance of testing machine calibration to measurement uncertainty of the mechanical characterisation. ISO 14577-2 defines multi-step iterative methods for calibrating frame compliance and indenter area function that do not require high-resolution microscopes. Previous research demonstrated that standard's recommendations are unsatisfactory and result in high calibration uncertainty. This work defines an improved calibration method based on a single-step procedure that achieves, as proved by experimental tests, definite advantages in terms of implementation and measurement uncertainty of both calibration and mechanical characterisation with respect to current procedures.

Modified Potra-Pták Multi-step Schemes with Accelerated Order of Convergence for Solving Systems of Nonlinear Equations

In this study, an iterative scheme of sixth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of third order Potra-Pták method and last is weighted-Newton step. Furthermore, we generalize our work to derive a family of multi-step iterative methods with order of convergence 3 r + 6 , r = 0 , 1 , 2 , … . The sixth order method is the special case of this multi-step scheme for r = 0 . The family gives a four-step ninth order method for r = 1 . As much higher order methods are not used in practice, so we study sixth and ninth order methods in detail. Numerical examples are included to confirm theoretical results and to compare the methods with some existing ones. Different numerical tests, containing academical functions and systems resulting from the discretization of boundary problems, are introduced to show the efficiency and reliability of the proposed methods.

Some multi-step iterative methods for solving nonlinear equations

Some multi-step iterative methods for solving nonlinear equations

Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs

In this paper, we present and illustrate a frozen Jacobian multistep iterative method to solve systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). We have used Jacobi-Gauss-Lobatto collocation (J-GL-C) methods to discretize the IVPs and BVPs. Frozen Jacobian multistep iterative methods are computationally very efficient. They require only one inversion of the Jacobian in the form of LU-factorization. The LU factors can then be used repeatedly in the multistep part to solve other linear systems. The convergence order of the proposed iterative method is 5m-11, where m is the number of steps. The validity, accuracy, and efficiency of our proposed frozen Jacobian multistep iterative method is illustrated by solving fifteen IVPs and BVPs. It has been observed that, in all the test problems, with one exception in this paper, a single application of the proposed method is enough to obtain highly accurate numerical solutions. In addition, we present a comprehensive comparison of J-GL-C methods on a collection of test problems.

Higher order multi-step interval iterative methods for solving nonlinear equations in $$R^n$$ R n

In this paper, higher order multi-step interval iterative methods are proposed for solving nonlinear equations in \(R^n\). Each method leads to an an interval vector enclosing the approximate solution along with the rigorous error bounds automatically. These methods require solving linear interval systems of equations. Interval extension of Gaussian elimination algorithm is described and used for solving them. The convergence analysis of both the methods is established to show their third and fourth order of convergence. A number of numerical examples are worked out and the performance in terms of iterations count and diameters of resulting interval vectors are measured.

Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs

A general class of multi-step iterative methods for finding approximate real or complex solutions of nonlinear systems is presented. The well-known technique of undetermined coefficients is used to construct the first method of the class while the higher order schemes will be attained by a frozen Jacobian. The point of attraction theory will be taken into account to prove the convergence behavior of the main proposed iterative method. Then, it will be observed that an m-step method converges with 2m-order. A discussion of the computational efficiency index alongside numerical comparisons with the existing methods will be given. Finally, we illustrate the application of the new schemes in solving nonlinear partial differential equations.

Multi-step implicit iterative methods with regularization for minimization problems and fixed point problems

Abstract In this paper we introduce a multi-step implicit iterative scheme with regularization for finding a common solution of the minimization problem (MP) for a convex and continuously Fréchet differentiable functional and the common fixed point problem of an infinite family of nonexpansive mappings in the setting of Hilbert spaces. The multi-step implicit iterative method with regularization is based on three well-known methods: the extragradient method, approximate proximal method and gradient projection algorithm with regularization. We derive a weak convergence theorem for the sequences generated by the proposed scheme. On the other hand, we also establish a strong convergence result via an implicit hybrid method with regularization for solving these two problems. This implicit hybrid method with regularization is based on the CQ method, extragradient method and gradient projection algorithm with regularization. MSC:49J30, 47H09, 47J20.

Increasing the order of convergence of iterative schemes for solving nonlinear systems

A set of multistep iterative methods with increasing order of convergence is presented, for solving systems of nonlinear equations. One of the main advantages of these schemes is to achieve high order of convergence with few Jacobian and functional evaluations, joint with the use of the same matrix of coefficients in the most of the linear systems involved in the process. Indeed, the application of the pseudocomposition technique on these proposed schemes allows us to increase their order of convergence, obtaining new high-order, efficient methods. Finally, some numerical tests are performed in order to check their practical behavior.

Strong convergence theorems and rate of convergence of multi-step iterative methods for continuous mappings on an arbitrary interval

In this article, by using the concept of W-mapping introduced by Atsushiba and Takahashi and K-mapping introduced by Kangtunyakarn and Suantai, we define W(T,N)-iteration and K(T,N)-iteration for finding a fixed point of continuous mappings on an arbitrary interval. Then, a necessary and sufficient condition for the strong convergence of the proposed iterative methods for continuous mappings on an arbitrary interval is given. We also compare the rate of convergence of those iterations. It is proved that the W(T,N)-iteration and K(T,N)-iteration are equivalent and the K(T,N)-iteration converges faster than the W(T,N)-iteration. Moreover, we also present numerical examples for comparing the rate of convergence between W(T,N)-iteration and K(T,N)-iteration.MSC: 26A18; 47H10; 54C05.

Some multi-step iterative methods for solving nonlinear equations

In this paper, we develop some new iterative methods for solving nonlinear equations by using the techniques introduced in Golbabai and Javidi (2007) [1] and Rafiq and Rafiullah (2008) [20] . We establish the convergence analysis of the proposed methods and then demonstrate their efficiency by taking some test problems.