Majorizing Sequences Research Articles

On the Implementation of Iterative Methods Without Inverse Updating for Solving Equations in Banach Spaces

The implementation of iterative methods using inverses to solve equations is a computationally expensive or impossible task in general. This is the case, since the analytical form of the inverse is difficult to find in practice. That is why, we replace the inverse by a sum of linear operators which is well defined. The convergence of the resulting hybrid methods is studied based on majorizing sequences under generalized continuity assumptions on the operators involved and in the setting of a Banach space. It is demonstrated by numerical experimentations that the convergence order as well as the number of iterations required to obtain a predetermined error tolerance when comparing the original to the hybrid method is essentially the same.

An extended study of two multistep higher order convergent methods for solving nonlinear equations

The local analysis of convergence for two competing methods of order seven or eight is developed to solve Banach space‐valued equations. Previous studies have used the eighth or ninth derivative of the operator involved, which do not appear on the methods, to show the convergence of these methods on the finite‐dimensional Euclidean space. In addition, no computable error distances or isolation of the solution results are provided in this study. These problems limit the applicability of this method to solving equations with operators that are at least nine times differentiable. In the current study, only conditions on the first derivative, appearing in these methods, are employed to show the convergence of these methods. Moreover, computable error bounds depend on the distance in the world as well as the isolation of the solution. Results are provided based on generalized continuity conditions on the first derivative. Furthermore, the more interesting semi‐local analysis of convergence not given before these methods is presented using a majorizing sequence. Finally, a great deal of impressive numerical results has been shown on real‐world problems.

On the Kantorovich Theory for Nonsingular and Singular Equations

We develop a new Kantorovich-like convergence analysis of Newton-type methods to solve nonsingular and singular nonlinear equations in Banach spaces. The outer or generalized inverses are exchanged by a finite sum of linear operators making the implementation of these methods easier than in earlier studies. The analysis uses relaxed generalized continuity of the derivatives of operators involved required to control the derivative and on real majorizing sequences. The same approach can also be implemented on other iterative methods with inverses. The examples complement the theory by verifying the convergence conditions and demonstrating the performance of the methods.

Hybrid Newton-like Inverse Free Algorithms for Solving Nonlinear Equations

Iterative algorithms requiring the computationally expensive in general inversion of linear operators are difficult to implement. This is the reason why hybrid Newton-like algorithms without inverses are developed in this paper to solve Banach space-valued nonlinear equations. The inverses of the linear operator are exchanged by a finite sum of fixed linear operators. Two types of convergence analysis are presented for these algorithms: the semilocal and the local. The Fréchet derivative of the operator on the equation is controlled by a majorant function. The semi-local analysis also relies on majorizing sequences. The celebrated contraction mapping principle is utilized to study the convergence of the Krasnoselskij-like algorithm. The numerical experimentation demonstrates that the new algorithms are essentially as effective but less expensive to implement. Although the new approach is demonstrated for Newton-like algorithms, it can be applied to other single-step, multistep, or multipoint algorithms using inverses of linear operators along the same lines.

Symmetric-Type Multi-Step Difference Methods for Solving Nonlinear Equations

Symmetric-type methods (STM) without derivatives have been used extensively to solve nonlinear equations in various spaces. In particular, multi-step STMs of a higher order of convergence are very useful. By freezing the divided differences in the methods and using a weight operator a method is generated using m steps (m a natural number) of convergence order 2 m. This method avoids a large increase in the number of operator evaluations. However, there are several problems with the conditions used to show the convergence: the existence of high order derivatives is assumed, which are not in the method; there are no a priori results for the error distances or information on the uniqueness of the solutions. Therefore, the earlier studies cannot guarantee the convergence of the method to solve nondifferentiable equations. However, the method may converge to the solution. Thus, the convergence conditions can be weakened. These problems arise since the convergence order is determined using the Taylor series which requires the existence of high-order derivatives which are not present in the method, and they may not even exist. These concerns are our motivation for authoring this article. Moreover, the novelty of this article is that all the aforementioned problems are addressed positively, and by using conditions only related to the divided differences in the method. Furthermore, a more challenging and important semi-local analysis of convergence is presented utilizing majorizing sequences in combination with the concept of the generalized continuity of the divided difference involved. The convergence is also extended from the Euclidean to the Banach space. We have chosen to demonstrate our technique in the present method. But it can be used in other studies using the Taylor series to show the convergence of the method. The applicability of other single- or multi-step methods using the inverses of linear operators with or without derivatives can also be extended with the same methodology along the same lines. Several examples are provided to test the theoretical results and validate the performance of the method.

Convergence of High-Order Derivative-Free Algorithms for the Iterative Solution of Systems of Not Necessarily Differentiable Equations

In this study, we extended the applicability of a derivative-free algorithm to encompass the solution of operators that may be either differentiable or non-differentiable. Conditions weaker than the ones in earlier studies are employed for the convergence analysis. The earlier results considered assumptions up to the existence of the ninth order derivative of the main operator, even though there are no derivatives in the algorithm, and the Taylor series on the finite Euclidian space restricts the applicability of the algorithm. Moreover, the previous results could not be used for non-differentiable equations, although the algorithm could converge. The new local result used only conditions on the divided difference in the algorithm to show the convergence. Moreover, the more challenging semi-local convergence that had not previously been studied was considered using majorizing sequences. The paper included results on the upper bounds of the error estimates and domains where there was only one solution for the equation. The methodology of this paper is applicable to other algorithms using inverses and in the setting of a Banach space. Numerical examples further validate our approach.

Generalized Convergence for Multi-Step Schemes under Weak Conditions

We have developed a local convergence analysis for a general scheme of high-order convergence, aiming to solve equations in Banach spaces. A priori estimates are developed based on the error distances. This way, we know in advance the number of iterations required to reach a predetermined error tolerance. Moreover, a radius of convergence is determined, allowing for a selection of initial points assuring the convergence of the scheme. Furthermore, a neighborhood that contains only one solution to the equation is specified. Notably, we present the generalized convergence of these schemes under weak conditions. Our findings are based on generalized continuity requirements and contain a new semi-local convergence analysis (with a majorizing sequence) not seen in earlier studies based on Taylor series and derivatives which are not present in the scheme. We conclude with a good collection of numerical results derived from applied science problems.

Enhancing Equation Solving: Extending the Applicability of Steffensen-Type Methods

Local convergence analysis is mostly carried out using the Taylor series expansion approach, which requires the utilization of high-order derivatives, not iterative methods. There are other limitations to this approach, such as the following: the analysis is limited to finite-dimensional Euclidean spaces; no a priori computable error bounds on the distance or uniqueness of the solution results are provided. The local convergence analysis in this paper positively addresses these concerns in the more general setting of a Banach space. The convergence conditions involve only the operators in the methods. The more important semi-local convergence analysis not studied before is developed by using majorizing sequences. Both types of convergence analyses are based on the concept of generalized continuity. Although we study a certain class of methods, the same approach applies to extend the applicability of other schemes along the same lines.

Extending the Domain with Application of Four-Step Nonlinear Scheme with Average Lipschitz Conditions

A novel local and semi-local convergence theorem for the four-step nonlinear scheme is presented. Earlier studies on local convergence were conducted without particular assumption on Lipschitz constant. In first part, the main local convergence theorems with a weak ϰ-average (assuming it as a positively integrable function and dropping the essential property of ND) are obtained. In comparison to previous research, in another part, we employ majorizing sequences that are more accurate in their precision along with the certain form of ϰ average Lipschitz criteria. A finer local and semi-local convergence criteria, boosting its utility, by relaxing the assumptions is derived. Applications in engineering to a variety of specific cases, such as object motion governed by a system of differential equations, are illustrated.

Extended Semilocal Convergence for Chebyshev-Halley-Type Schemes for Solving Nonlinear Equations under Weak Conditions

The application of the Chebyshev-Halley type scheme for nonlinear equations is extended with no additional conditions. In particular, the purpose of this study is two folds. The proof of the semi-local convergence analysis is based on the recurrence relation technique in the first fold. In the second fold, the proof relies on majorizing sequences. Iterates are shown to belong to a larger domain resulting in tighter Lipschitz constants and a finer convergence analysis than in earlier works. The convergence order of these methods is at least three. The numerical example further validates the theoretical results.

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.

On the Semi-Local Convergence of a Noor–Waseem-like Method for Nonlinear Equations

The significant feature of this paper is that the semi-local convergence of high order methods for solving nonlinear equations defined on abstract spaces has not been studied extensively as done for the local convergence by a plethora of authors which is certainly a more interesting case. A process is developed based on majorizing sequences and the notion of restricted Lipschitz condition to provide a semi-local convergence analysis for the third convergent order Noor–Waseem method. Due to the generality of our technique, it can be used on other high order methods. The convergence analysis is enhanced. Numerical applications complete are used to test the convergence criteria.

On the semi-local convergence of the Homeier method in Banach space for solving equations

In this paper we consider the semi-local convergence analysis of the Homeier method for solving nonlinear equation in Banach space. As far as we know no semi-local convergence has been given for the Homeier under Lipschitz conditions. Our goal is to extend the applicability of the Homeier method in the semi-local convergence under these conditions. We use majorizing sequences and conditions only on the first derivative which appear on the method for proving our results. Numerical experiments are provided in this study.

Super-Halley method under majorant conditions in Banach spaces

In this paper, we have studied local convergence of Super-Halley method in Banach spaces under the assumption of second order majorant conditions. This approach allows us to obtain generalization of earlier convergence analysis under majorizing sequences. Two important special cases of the convergence analysis based on the premises of Kantorovich and Smale type conditions have also been concluded. To show efficacy of our approach we have given three numerical examples.

Construction of simple majorizing sequences for iterative methods

The interest of the majorizing (scalar) sequences lies in that, from their convergence, we can deduce the convergence of an iterative method in Banach spaces. We propose a new technique to construct majorizing sequences that generalizes that given by Kantorovich for Newton’s method.

On a two-step optimal Steffensen-type method: Relaxed local and semi-local convergence analysis and dynamical stability

In this paper, we study on the local and semi-local convergence and dynamic of an iterative optimal two-step Steffensen's method. We study local convergence by using Lipschitz conditions and continuity of differentiable function F. We also obtain semi-local convergence by using majorizing sequences. By this approach, we can obtain the convergence of the method by minimum of auxiliary sequences and few initial conditions. Moreover, we analyze the stability of the method via Julia sets and parameter planes that is a novelty for Steffensen-type methods. We extend this method for obtaining the sign function of a matrix. Numerical examples are given to confirm our theoretical results.

Expanding the Applicability of Stirling’s Method under Weaker Conditions and Restricted Convergence Regions

Abstract In this paper we have provided sufficient conditions to study semilocal and local convergence of the Stirling’s method. The method is used to find fixed points of nonlinear operator equation. We assume Lipschtiz continuity type conditions on the first Fréchet derivative of the operator but no contractive conditions as in earlier works. This way expand the applicability of this method. Here we introduce a new type of majorizing sequences instead of usual majorizing sequences and recurrence relations. Finally the paper will be concluded with numerical examples and a favorable comparison with known results.

On the complexity of choosing majorizing sequences for iterative procedures

The aim of this paper is to introduce general majorizing sequences for iterative procedures which may contain a non-differentiable operator in order to solve nonlinear equations involving Banach valued operators. A general semi-local convergence analysis is presented based on majorizing sequences. The convergence criteria, if specialized can be used to study the convergence of numerous procedures such as Picard’s, Newton’s, Newton-type, Stirling’s, Secant, Secant-type, Steffensen’s, Aitken’s, Kurchatov’s and other procedures. The convergence criteria are flexible enough, so if specialized are weaker than the criteria given by the aforementioned procedures. Moreover, the convergence analysis is at least as tight. Furthermore, these advantages are obtained using Lipschitz constants that are least as tight as the ones already used in the literature. Consequently, no additional hypotheses are needed, since the new constants are special cases of the old constants. These ideas can be used to study, the local convergence, multi-step multi-point procedures along the same lines. Some applications are also provided in this study.

Local convergence analysis for Chebyshev’s method

In this work, we are working to present a local convergence analysis for Chebyshev’s method by using majorizing sequence. The given method is a third order iterative process, used in order to approximate a zero of an nonlinear operator equation in a Banach space. Here we are using a new type of majorant conditions to prove the convergence. We will also try to establish relations between this majorant conditions with results of based on Kantorovich-type and Smale-type assumptions.

Directional k-Step Newton Methods in n Variables and its Semilocal Convergence Analysis

The directional k-step Newton methods (k a positive integer) is developed for solving a single nonlinear equation in n variables. Its semilocal convergence analysis is established by using two different approaches (recurrent relations and recurrent functions) under the assumption that the first derivative satisfies a combination of the Lipschitz and the center-Lipschitz continuity conditions instead of only Lipschitz condition. The convergence theorems for the existence and uniqueness of the solution for each of them are established. Numerical examples including nonlinear Hammerstein-type integral equations are worked out and significantly improved results are obtained. It is shown that the second approach based on recurrent functions solves problems failed to be solved by first one using recurrent relations. This demonstrates the efficacy and applicability of these approaches. This work extends the directional one and two-step Newton methods for solving a single nonlinear equation in n variables. Their semilocal convergence analysis using majorizing sequences are studied in Levin (Math Comput 71(237):251–262, 2002) and Ioannis (Num Algorithms 55(4):503–528, 2010) under the assumption that the first derivative satisfies the Lipschitz and the combination of the Lipschitz and the center-Lipschitz continuity conditions, respectively. Finally, the computational order of convergence and the computational efficiency of developed method are studied.