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The convergence balls as well as the dynamical characteristics of two sixth order Jarratt-like methods (JLM1 and JLM2) are compared. First, the ball analysis theorems for these algorithms are proved by applying generalized Lipschitz conditions on derivative of the first order. As a result, significant information on the radii of convergence and the regions of uniqueness for the solution are found along with calculable error distances. Also, the scope of utilization of these algorithms is extended. Then, we compare the dynamical properties, using the attraction basin approach, of these iterative schemes. At the end, standard application problems are considered to demonstrate the efficacy of our theoretical findings on ball convergence. For these problems, the convergence balls are computed and compared. From these comparisons, it is confirmed that JLM1 has the bigger convergence balls than JLM2. Also, the attraction basins for JLM1 are larger in comparison to JLM2. Thus, for numerical applications, JLM1 is better than JLM2.

For the purpose of obtaining solutions to Banach-space-valued nonlinear models, we offer a new extended analysis of the local convergence result for a seventh-order iterative approach without derivatives. Existing studies have used assumptions up to the eighth derivative to demonstrate its convergence. However, in our convergence theory, we only use the first derivative. Thus, in contrast to previously derived results, we obtain conclusions on calculable error estimates, convergence radius, and uniqueness region for the solution. As a result, we are able to broaden the utility of this efficient method. In addition, the convergence regions of this scheme for solving polynomial equations with complex coefficients are illustrated using the attraction basin approach. This study is concluded with the validation of our convergence result on application problems.

In the earlier work, expensive Taylor formula and conditions on derivatives up to the eighthorder have been utilized to establish the convergence of a derivative free class of seventh orderiterative algorithms. Moreover, no error distances or results on uniqueness of the solution weregiven. In this study, extended ball convergence analysis is derived for this class by imposingconditions on the first derivative. Additionally, we offer error distances and convergence radiustogether with the region of uniqueness for the solution. Therefore, we enlarge the practicalutility of these algorithms. Also, convergence regions of a specific member of this class are displayedfor solving complex polynomial equations. At the end, standard numerical applicationsare provided to illustrate the efficacy of our theoretical findings.

The most significant contribution made by this study is that the applicability and convergence domain of a fifth-order convergent nonlinear equation solver is extended. We use Hölder condition on the first Fréchet derivative to study the local analysis, and this expands the applicability of the formula for such problems where the earlier study based on Lipschitz constants cannot be used. This study generalizes the local analysis based on Lipschitz constants. Also, we avoid the use of the extra assumption on boundedness of the first derivative of the involved operator. Finally, numerical experiments ensure that our analysis expands the utility of the considered scheme. In addition, the proposed technique produces a larger convergence domain, in comparison to the earlier study, without using any extra conditions.

Obtaining convergence domain is an important task in the study of iterative schemes. Analysis of local convergence of an iterative procedure provides essential information about its convergence domain around a solution. In this manuscript, we study the local analysis of the uni-parametric Kou’s class of iterative algorithms for addressing nonlinear equations. This approach expands the utility of the methods by preventing the use of Taylor expansion in convergence analysis. In the view of extending the applicability of these methods, the convergence analysis is shown using Lipschitz condition on the first derivative. Our study provides radii of convergence balls and the uniqueness of the solution along with the calculable error distances. The complex dynamical analysis of the family is also presented. Numerical examples are solved to show that our theoretical conclusions work well in the situation where the earlier analysis cannot be implemented.

The main contribution of this study is that the applicability and convergence domain of a fifth-order convergent equation solver is extended. We use $$\omega $$ condition on the first Frechet derivative to study the local analysis, and this expands the applicability of the formula for such problems where the earlier study based on Lipschitz constants cannot be used. Also, we avoid the use of the extra assumption on boundedness of the first derivative of the nonlinear operator. Our idea can be used on other iterative methods. Numerical tests confirmed that the proposed analysis produces a larger convergence domain, in comparison to the earlier study, without using additional conditions.

This paper intends to investigate the dynamical behavior of a parameter based sixth and seventh order convergent family of iterative methods in the complex plane. The fixed points of the rational function obtained from this family are described with their stability. Further to this, the stability regions for the fixed points are displayed using graphical tools. Parameter planes for independent free critical points are presented to determine stable and poor members of the family. Dynamical planes are also provided for various iterative methods of the family having important numerical properties.

In a Banach space setting, we provide the local convergence analysis of a cubically convergent nonlinear system solver assuming that the first-order Frechet derivative belongs to the Lipschitz class. The significance of this study is that it avoids the standard practice of Taylor expansion in the analysis of convergence and extends the applicability of the algorithm by using the theory based on the first-order derivative only. Also, our analysis provides the radii of convergence balls and computable error bounds along with the uniqueness of the solution. Furthermore, the generalization of this analysis using Holder condition is studied. Different numerical tests confirm that the new technique produces better results, and it is useful in solving such problems where previous studies fail to solve.

In this manuscript, the study of local convergence analysis for the cubically convergent Weerakoon–Fernando method is presented to obtain solutions of nonlinear operator equations in Banach spaces. The novelty of our paper is that our analysis only requires the $$\omega $$ continuity of the first Frechet derivative and extends the applicability of the algorithm when both Lipschitz and Holder conditions fail without engaging derivatives of the higher order which do not appear on this method. The local convergence offers radii of balls of convergence, the error bounds and uniqueness of the solution. Furthermore, the convergence region for the scheme to approximate the zeros of various polynomials is studied using basins of attraction tool. Several numerical tests are performed to show the usefulness of our theoretical results.

We present the local and semi-local convergence analysis of the cubically convergent harmonic mean Newton method (HNM) for obtaining solutions of Banach space valued nonlinear operator equations. The significance of this work is that the convergence study needs only the condition that the first-order Frechet derivative obeys $$\omega $$ continuity. Moreover, we avoid using higher order derivatives, which do not occur in this scheme. The dynamical properties of the scheme are also explored using basins of attraction technique for various complex polynomials. Finally, the convergence radii for benchmark numerical problems are computed applying our analytical results.

Based on Lipschitz continuity condition, we study the local convergence analysis for the fifth-order convergent modified Weerakoon’s method for solving nonlinear equations in Banach spaces. Lipschitz continuity condition on the first derivative is assumed to extend the applicability of the scheme. This analysis enables us to solve such problems for which previous studies based on higher-order derivatives unable to find the solution. A theorem showing the existence and uniqueness of the solution along with computable error bounds is established. Standard numerical examples like nonlinear integral equation and system of nonlinear equations are solved to demonstrate the productiveness of our theoretical outcomes.

In this study, we present an improved local analysis using $$\omega$$ condition to enlarge the convergence domain of deformed Halley method. This analysis avoids the use of the extra assumption on the boundedness of the first derivative of the nonlinear operator. Finally, numerical tests confirm that our analysis provides a larger convergence domain, in comparison to the earlier study, without using additional conditions.

Applying the Lipschitz continuity of the first-order Frechet derivative, we describe the local convergence of a sixth-order nonlinear system solver in Banach spaces. This study eliminates the standard practice of Taylor expansion in local analysis and enhances the algorithm applicability through the use of a set of conditions on the first-order derivative. Also, our analysis offers the radii of convergence balls and computable error distances together with the unique result. Various numerical experiments are conducted to demonstrate that our technique is beneficial when prior studies fail to solve problems.

In this paper, we deal with the construction, analysis and applications of a modified uniparametric family of methods to solve nonlinear equations in R. We study the convergence of new methods which shows the order of convergence is at least five and for a particular value 3/2 of the parameter γ, the method is sixth order convergent. We discuss several applications such as Max Planck’s conservative law, chemical equilibrium and multi-factor effect to demonstrate the productiveness and capability of the suggested method (for γ = 3/2 ). At every iteration our method is compared with Maroju et al. method[1] and Parhi and Gupta method[2] in terms of the values |f (xn)| and |xn − xn−1|. From the numerical experiments, advantages of our method is observed. Furthermore, we study the complex dynamics to determine the stability and dynamical properties of the methods.

Using the Lipschitz continuous derivative, we study the local convergence analysis of a third and fourth-order convergent uni-parametric class of iterative schemes. This technique avoids the usual practice of Taylor expansion in convergence analysis and extends the applicability of the family by using the assumption based on the first-order derivative only. Our study provides the radii of balls of convergence and computable error bounds along with the uniqueness of the solution. Also, complex dynamical properties of the family are discussed to select good schemes in the view of numerical stability. Numerical illustrations show that our analysis is useful in solving such problems where previous studies fail to solve.

The significance of this study is that the applicability of a Newton-Simpson-like scheme is extended using weaker conditions for convergence analysis. In earlier works, the local convergence of the scheme was established applying third-order Frechet derivative in finite-dimensional Euclidean spaces. In addition, no information about the error bounds, convergence radii and uniqueness of the solution was provided. Therefore, the applicability of the scheme is restricted. In this work, all these issues are addressed using Lipschitz continuous first derivative in Banach spaces. The convergence radii, uniqueness results and the error estimates are also discussed. Furthermore, the generalization of this analysis is studied under Holder condition. At last, using several problems, it is shown that our findings produce better results and effective enough to address such equations where earlier works fail.