Ramandeep Behl

On developing fourth-order optimal families of methods for multiple roots and their dynamics

There are few optimal fourth-order methods for solving nonlinear equations when the multiplicity m of the required root is known in advance. Therefore, the first focus of this paper is on developing new fourth-order optimal families of iterative methods by a simple and elegant way. Computational and theoretical properties are fully studied along with a main theorem describing the convergence analysis. Another main focus of this paper is the dynamical analysis of the rational map associated with our proposed class for multiple roots; as far as we know, there are no deep study of this kind on iterative methods for multiple roots. Further, using Mathematica with its high precision compatibility, a variety of concrete numerical experiments and relevant results are extensively treated to confirm the theoretical development.

An optimal fourth-order family of methods for multiple roots and its dynamics

There are few optimal fourth-order methods for solving nonlinear equations when the multiplicity m of the required root is known in advance. Therefore, the principle focus of this paper is on developing a new fourth-order optimal family of iterative methods. From the computational point of view, the conjugacy maps and the strange fixed points of some iterative methods are discussed, their basins of attractions are also given to show their dynamical behavior around the multiple roots. Further, using Mathematica with its high precision compatibility, a variety of concrete numerical experiments and relevant results are extensively treated to confirm the theoretical development.

An eighth-order family of optimal multiple root finders and its dynamics

There is a very small number of higher-order iteration functions for multiple zeros whose order of convergence is greater than four. Some scholars have tried to propose optimal eighth-order methods for multiple zeros. But, unfortunately, they did not get success in this direction and attained only sixth-order convergence. So, as far as we know, there is not a single optimal eighth-order iteration function in the available literature that will work for multiple zeros. Motivated and inspired by this fact, we present an optimal eighth-order iteration function for multiple zeros. An extensive convergence study is discussed in order to demonstrate the optimal eighth-order convergence of the proposed scheme. In addition, we also demonstrate the applicability of our proposed scheme on real-life problems and illustrate that the proposed methods are more efficient among the available multiple root finding techniques. Finally, dynamical study of the proposed schemes also confirms the theoretical results.

Stable high-order iterative methods for solving nonlinear models

There are several problems of pure and applied science which can be studied in the unified framework of the scalar and vectorial nonlinear equations. In this paper, we propose a sixth-order family of Jarratt type methods for solving nonlinear equations. Further, we extend this family to the multidimensional case preserving the order of convergence. Their theoretical and computational properties are fully investigated along with two main theorems describing the order of convergence. We use complex dynamics techniques in order to select, among the elements of this class of iterative methods, those more stable. This process is made by analyzing the conjugacy class, calculating the fixed and critical points and getting conclusions from parameter and dynamical planes. For the implementation of the proposed schemes for system of nonlinear equations, we consider some applied science problems namely, Van der Pol problem, kinematic syntheses, etc. Further, we compare them with existing sixth-order methods to check the validity of the theoretical results. From the numerical experiments, we find that our proposed schemes perform better than the existing ones. Further, we also consider a variety of nonlinear equations to check the performance of the proposed methods for scalar equations.

A family of higher order iterations free from second derivative for nonlinear equations in R

Abstract The aim of this article is to develop a family of higher order iterations free from second derivative for solving nonlinear equations in R . Their theoretical, computational and dynamical aspects are fully investigated and theorems are established to provide their order of convergence and asymptotic error constant. It is observed that the family includes sixth order methods and for a particular case its eighth order can be achieved. In this family, methods use three functions and one first derivative evaluations. The family of methods can be shown to be optimal by using Kung–Traub conjecture. A number of numerical examples are worked out to demonstrate the applicability of these methods. The improved results are obtained in comparison to some of the existing robust methods on the considered test examples. Local convergence analysis and dynamical study of the proposed family of methods are also carried out.

An Optimal Family of Eighth-Order Iterative Methods with an Inverse Interpolatory Rational Function Error Corrector for Nonlinear Equations

AbstractThe main motivation of this study is to propose an optimal scheme with an inverse interpolatory rational function error corrector in a general way that can be applied to any existing optimal multi-point fourth-order iterative scheme whose first sub step employs Newtons method to further produce optimal eighth-order iterative schemes. In addition, we also discussed the theoretical and computational properties of our scheme. Variety of concrete numerical experiments and basins of attraction are extensively treated to confirm the theoretical development.

Some novel and optimal families of King's method with eighth and sixteenth-order of convergence

In this study, our principle aim is to provide some novel eighth and sixteenth-order families of Kings method for solving nonlinear equations which should be superior than the existing schemes of same order. The relevant optimal orders of the proposed families satisfy the classical Kung-Traub conjecture which was made in 1974. The derivations of the proposed schemes are based on the weight function and rational approximation approaches, respectively. In addition, convergence properties of the proposed families are fully investigated along with one lemma and two main theorems describing their order of convergence. We consider a concrete variety of real life problems e.g. the trajectory of an electron in the air gap between two parallel plates, chemical engineering problem, Van der Waals equation which explains the behavior of a real gas by introducing in the ideal gas equations and fractional conversion in a chemical reactor, in order to check the validity, applicability and effectiveness of our proposed methods. Further, it is found from the numerical results that our proposed methods perform better than the existing ones of the same order when the accuracy is checked in the multi precision digits.

A family of second derivative free fourth order continuation method for solving nonlinear equations

In this paper, we present a parameter based iterative method free from the second derivative for solving nonlinear equations of type f ( x ) = 0 . Ezquerro and Hernandez (1999) discussed the convergence analysis of a uniparametric family of an iterative method in Banach space. Based on this idea, we propose a uniparametric family of second derivative free iterative method in R . We discuss the convergence analysis and observe that it possesses a fourth-order convergence for solving nonlinear equations in α ź R . Several numerical examples are worked out with our proposed methods for parameter α = 1 , α = 2 and the existing fourth-order iterative method proposed in King (1973), Chun (2007), Soleymani etźal. (2012) and Maheshwari (2009). Finally, from the comparison, we observe that our method is more efficient than existing methods. Finally, we compare basins of attraction of our methods with the second derivative free fourth-order iterative method proposed in King (1973), Chun (2007), Soleymani etźal. (2012) and Maheshwari (2009) observe that the proposed scheme is more efficient.

Ball Convergence for a Family of Quadrature-Based Methods for Solving Equations in Banach Space

We present a local convergence analysis for a family of quadrature-based predictor–corrector methods in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as Howk [2016] [Howk, C. L. [2016] “A classs of efficient quadrature-based predictor–corrector methods for solving nonlinear systems, Appl. Math. Comput. 276, 394–406] the 1 + 2 order of convergence was shown on the m-dimensional Euclidean space using Taylor series expansion and hypotheses reaching up to the third-order Frechet-derivative of the operator involved although only the first-order Frechet-derivative appears in these methods, which restrict the applicability of these methods. In this paper, we expand the applicability of these methods in a Banach space setting and using hypotheses only on the first Frechet-derivative. Moreover, we provide computable radii of convergence as well as error bounds on the distances involved using Lipschitz constants. Numerical examples are also presented to solve equations in cases where earlier results cannot apply.

Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative

In this paper, we propose a local convergence analysis of an eighth order three-step method to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Further, we also study the dynamic behaviour of that scheme. In an earlier study, Sharma and Arora (2015) did not discuss these properties. Furthermore, the order of convergence was shown using Taylor series expansions and hypotheses up to the fourth order derivative or even higher of the function involved which restrict the applicability of the proposed scheme. However, only the first order derivatives appear in the proposed scheme. To overcome this problem, we present the hypotheses for the proposed scheme maximum up to first order derivative. In this way, we not only expand the applicability of the methods but also suggest convergence domain. Finally, a variety of concrete numerical examples are proposed where earlier studies can not be applied to obtain the solutions of nonlinear equations on the other hand our study does not exhibit this type of problem/restriction.