Asymptotic Error Constant Research Articles

A general approach to the study of the convergence of Picard iteration with an application to Halley's method for multiple zeros of analytic functions

In this paper, we define a new wide class of iteration functions and then we use it to prove a general convergence theorem that provides exact domain of initial approximations to guarantee the high Q-order of convergence of Picard iteration generated by this class of functions. As an application of this theorem, we prove some local convergence theorems about the famous Halley's method for simple and multiple zeros of analytic functions. All obtained results are supplied with a priori and a posteriori error estimates as well as with assessments of the asymptotic error constants.

Derivation of a Numerical Method with Free Second-order Derivatives

We have proposed the second-derivative-free numerical method and determined the control parameters to converge cubically. In addition, we have developed the order of convergence and the asymptotic error constant. Applying this iterative scheme to a variety of examples, numerical results have shown a successful asymptotic error constants with cubic convergence.

Development of a New Multi‐step Iteration Scheme for Solving Non‐Linear Models with Complex Polynomiography

The appearance of nonlinear equations in science, engineering, economics, and medicine cannot be denied. Solving such equations requires numerical methods having higher‐order convergence with cost‐effectiveness, for the equations do not have exact solutions. In the pursuit of efficient numerical methods, an attempt is made to devise a modified strategy for approximating the solution of nonlinear models in either scalar or vector versions. Two numerical methods of second‐and sixth‐order convergence are carefully merged to obtain a hybrid multi‐step numerical method with twelfth‐order convergence while using seven function evaluations per iteration, resulting in the efficiency index of about 1.4262. The convergence is also ascertained theoretically, and the asymptotic error constant is computed. Furthermore, various numerical methods of varying orders are used to compare the numerical results obtained with the proposed hybrid method in approximate solution, number of iterations, absolute error, absolute functional value, and the machine time measured in seconds. The obtained results outperformed the chosen methods when applied models arising from physical and natural fields were solved. Finally, to observe the convergence graphically, some complex polynomials are plotted as polynomiographs, wherein the rapid convergence of the proposed method is guaranteed.

An iteration solver for the Poisson–Nernst–Planck system and its convergence analysis

In this paper, we provide a theoretical analysis for an iteration solver to implement a finite difference numerical scheme for the Poisson-Nernst–Planck (PNP) system, based on the Energetic Variational Approach (EnVarA), in which a non-constant mobility H−1 gradient flow is formulated. In particular, the nonlinear and singular nature of the logarithmic energy potentials has always been the essential difficulty. In the numerical design, the mobility function is explicitly updated, for the sake of unique solvability analysis. The logarithmic and the electric potential diffusion terms, which come from the gradient of convex energy functional parts, are implicitly computed. The positivity-preserving property for all the concentrations, an unconditional energy stability, and the optimal rate error estimate have been established in a recent work. A modified Newton iteration for the nonlinear and logarithmic part, combined with a linear iteration for the electric potential part, is proposed to implement the given numerical scheme, in which a non-constant linear elliptic equation needs to be solved at each iteration stage. A theoretical analysis is presented in this article, and a linear convergence is proved for such an iteration, with an asymptotic error constant in the same order of the time step size. A numerical test is also presented in this article, which demonstrates the linear convergence rate of the proposed iteration solver.

A generalized multivariable Newton method

It is well known that the Newton method may not converge when the initial guess does not belong to a specific quadratic convergence region. We propose a family of new variants of the Newton method with the potential advantage of having a larger convergence region as well as more desirable properties near a solution. We prove quadratic convergence of the new family, and provide specific bounds for the asymptotic error constant. We illustrate the advantages of the new methods by means of test problems, including two and six variable polynomial systems, as well as a challenging signal processing example. We present a numerical experimental methodology which uses a large number of randomized initial guesses for a number of methods from the new family, in turn providing advice as to which of the methods employed is preferable to use in a particular search domain.

A new higher-order optimal derivative free scheme for multiple roots

In this paper, we presented a novel and efficient fourth order derivative free optimal family of iterative methods for approximating the multiple roots of nonlinear equations. Initially the convergence analysis is performed for particular values of multiple roots afterward it concludes in general form. In addition, we study several numerical experiments on real life problems in order to confirm the efficiency and accuracy of our methods. We illustrate the applicability and comparisons of our methods on eigenvalue problem, Van der Waals equation of state, continuous stirred tank reactor (CSTR), Plank’s radiation and clustering problem of roots with earlier robust iterative methods. Finally, on the basis of obtained computational results, we conclude that our methods perform better than the existing ones in terms of CPU timing, absolute residual errors, asymptotic error constants, absolute error difference between two last consecutive iterations and approximated roots compared to the existing ones.

A new two‐point scheme for multiple roots of nonlinear equations

In most of the earlier research for multiple zeros, in order to obtain a new iteration function from the existing scheme, the usual practice is to make no change at the first substep. In this paper, we explore the idea that what are the advantages if the flexibility of choice is also given at the first substep. Therefore, we present a new two‐point sixth‐order scheme for multiple roots (m>1). The main advantages of our scheme over the existing schemes are flexibility at both substeps, simple body structure, smaller residual error, smaller error difference between two consecutive iterations, and smaller asymptotic error constant. The development of the scheme is based on midpoint formula and weight functions of two variables. We compare our methods with the existing methods of the same order with real‐life applications as well as standard test problems. From the numerical results, we find that our methods can be considered as better alternates for the existing methods of the same order. Finally, dynamical study of the proposed schemes is presented that confirms the theoretical results.

An accelerated version of Newton’s method with convergence order [formula omitted

A root-finding method is developed that, like Newton’s Method, evaluates both the function and its first derivative once per iteration, but the new method converges at the rate 3+1, and moreover, it’s asymptotic error constant is proportional to the function’s fourth order derivative. By contrast, Newton’s Method converges quadratically with the asymptotic error constant being proportional to the function’s second order derivative. Each iteration (except the first) of our Accelerated Newton’s Method (ANM) uses the values of both the function and its first derivative at the previous iteration in order to estimate the function’s second derivative. For the initial iteration we develop and recommend the use of a modified version of Jarratt’s Method; a method that calculates the derivative of the function twice in each iteration. Like Jarratt’s Method, our modification of it converges with the fourth power of the initial error, but our asymptotic error constant depends primarily on the product of the function’s second and third order derivatives rather than depending separately on the value of the second derivative. The efficient performance of our Accelerated Newton’s Method (ANM) is illustrated using nine test functions and a range of initial values for each test function. These tests indicate that our Accelerated Newton’s Method requires on average 30% fewer function and derivative evaluations than the straightforward Newton’s Method to achieve the same accuracy; noting again that the function and its derivative are evaluated once per iteration, exactly as in Newton’s Method. Moreover, we find that our Accelerated Newton’s Method is more robust than Newton’s Method in that it converges to the root over a wider range of initial conditions.

Support and approximation properties of Hermite splines

In this paper, we formally investigate two mathematical aspects of Hermite splines that are relevant to practical applications. We first demonstrate that Hermite splines are maximally localized, in the sense that the size of their support is minimal among pairs of functions with identical reproduction properties. Then, we precisely quantify the approximation power of Hermite splines for the reconstruction of functions and their derivatives. It is known that the Hermite and B-spline approximation schemes have the same approximation order. More precisely, their approximation error vanishes as O(T4) when the step size T goes to zero. In this work, we show that they actually have the same asymptotic approximation error constants, too. Therefore, they have identical asymptotic approximation properties. Hermite splines combine optimal localization and excellent approximation power, while retaining interpolation properties and closed-form expression, in contrast to existing similar functions. These findings shed a new light on the convenience of Hermite splines in the context of computer graphics and geometrical design.

A General Way to Construct a New Optimal Scheme with Eighth-Order Convergence for Nonlinear Equations

In this paper, we present a new and interesting optimal scheme of order eight in a general way for solving nonlinear equations, numerically. The beauty of our scheme is that it is capable of producing further new and interesting optimal schemes of order eight from every existing optimal fourth-order scheme whose first substep employs Newton’s method. The construction of this scheme is based on rational functional approach. The theoretical and computational properties of the proposed scheme are fully investigated along with a main theorem which establishes the order of convergence and asymptotic error constant. Several numerical examples are given and analyzed in detail to demonstrate faster convergence and higher computational efficiency of our methods.

Some Real-Life Applications of a Newly Constructed Derivative Free Iterative Scheme

In this study, we present a new higher-order scheme without memory for simple zeros which has two major advantages. The first one is that each member of our scheme is derivative free and the second one is that the present scheme is capable of producing many new optimal family of eighth-order methods from every 4-order optimal derivative free scheme (available in the literature) whose first substep employs a Steffensen or a Steffensen-like method. In addition, the theoretical and computational properties of the present scheme are fully investigated along with the main theorem, which demonstrates the convergence order and asymptotic error constant. Moreover, the effectiveness of our scheme is tested on several real-life problems like Van der Waal’s, fractional transformation in a chemical reactor, chemical engineering, adiabatic flame temperature, etc. In comparison with the existing robust techniques, the iterative methods in the new family perform better in the considered test examples. The study of dynamics on the proposed iterative methods also confirms this fact via basins of attraction applied to a number of test functions.

New Iterative Methods for Solving Nonlinear Problems with One and Several Unknowns

In this manuscript, a new type of study regarding the iterative methods for solving nonlinear models is presented. The goal of this work is to design a new fourth-order optimal family of two-step iterative schemes, with the flexibility through weight function/s or free parameter/s at both substeps, as well as small residual errors and asymptotic error constants. In addition, we generalize these schemes to nonlinear systems preserving the order of convergence. Regarding the applicability of the proposed techniques, we choose some real-world problems, namely chemical fractional conversion and the trajectory of an electron in the air gap between two parallel plates, in order to study the multi-factor effect, fractional conversion of species in a chemical reactor, Hammerstein integral equation, and a boundary value problem. Moreover, we find that our proposed schemes run better than or equal to the existing ones in the literature.

Eighth order family of iterative methods for nonlinear equations and their basins of attraction

We present a new family of eighth order methods for solving nonlinear equations. The order of convergence of the considered methods is proved and corresponding asymptotic error constants are expressed in terms of four parameters. Numerical examples, obtained using Mathematica with high precision arithmetic, demonstrate convergence and efficacy of our family of methods. For some combinations of parameter values, the new eighth order methods produce very good results on tested examples compared to the results produced by some of the eighth order methods existing in the related literature. An exploration of the relevant dynamics of the proposed methods is presented along with illustrative basins of attraction for various polynomials.

Multiplicity anomalies of an optimal fourth-order class of iterative methods for solving nonlinear equations

There is a few number of optimal fourth-order iterative methods for obtaining the multiple roots of nonlinear equations. But, in most of the earlier studies, scholars gave the flexibility in their proposed schemes only at the second step (not at the first step) in order to explore new schemes. Unlike what happens in existing methods, the main aim of this manuscript is to construct a new fourth-order optimal scheme which will give the flexibility to the researchers at both steps as well as faster convergence, smaller residual errors and asymptotic error constants. The construction of the proposed scheme is based on the mid-point formula and weight function approach. From the computational point of view, the stability of the resulting class of iterative methods is studied by means of the conjugacy maps and the analysis of strange fixed points. Their basins of attractions and parameter planes are also given to show their dynamical behavior around the multiple roots. Finally, we consider a real-life problem and a concrete variety of standard test functions for numerical experiments and relevant results are extensively treated to confirm the theoretical development.

An optimal eighth-order class of three-step weighted Newton's methods and their dynamics behind the purely imaginary extraneous fixed points

ABSTRACTIn this paper, we not only develop an optimal class of three-step eighth-order methods with higher order weight functions employed in the second and third sub-steps, but also investigate their dynamics underlying the purely imaginary extraneous fixed points. Their theoretical and computational properties are fully described along with a main theorem stating the order of convergence and the asymptotic error constant as well as extensive studies of special cases with rational weight functions. A number of numerical examples are illustrated to confirm the underlying theoretical development. Besides, to show the convergence behaviour of global character, fully explored is the dynamics of the proposed family of eighth-order methods as well as an existing competitive method with the help of illustrative basins of attraction.

A Family of Combined Iterative Methods for Solving Nonlinear Equations

In this article we construct some higher-order modifications of Newton’s method for solving nonlinear equations, which is based on the undetermined coefficients. This construction can be applied to any iteration formula. It can be found that per iteration the resulting methods add only one additional function evaluation, their order of convergence can be increased by two or three units. Higher order convergence of our methods is proved and corresponding asymptotic error constants are expressed. Numerical examples, obtained using Matlab with high precision arithmetic, are shown to demonstrate the convergence and efficiency of the combined iterative methods. It is found that the combined iterative methods produce very good results on tested examples, compared to the results produced by the existing higher order schemes in the related literature.

An optimal and efficient general eighth-order derivative free scheme for simple roots

The main motivation of this study is to present an optimal scheme in a general way that can be applied to any existing optimal multipoint fourth-order iterative scheme whose first substep employs Steffensen’s method or Steffensen like method to further produce optimal eighth-order iterative schemes. A rational function approximation approach is used in the construction of proposed scheme. In addition, we also discussed the theoretical and computational properties of our scheme. Each member of the presented scheme satisfies the optimality conjecture for multipoint iterative methods without memory which was given by Kung and Traub in 1970. Finally, we also concluded on the basis of obtained numerical results that our methods have faster convergence in contrast to the existing methods of same order because they have minimum residual errors, minimum error difference between two consecutive iterations and minimum asymptotic error constants corresponding to the considered test function.

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

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 simple-root finders with weight functions dependent on function-to-function ratios and their dynamics underlying extraneous fixed points

We extend in this paper an optimal family of three-step eighth-order methods developed by Džunić et al. (2011) with higher-order weight functions employed in the second and third sub-steps and investigate their dynamics under the relevant extraneous fixed points among which purely imaginary ones are specially treated for the analysis of the rich dynamics. Their theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence and the asymptotic error constant as well as proper choices of special cases. A wide variety of relevant numerical examples are illustrated to confirm the underlying theoretical development. In addition, this paper investigates the dynamics of selected existing optimal eighth-order iterative maps with the help of illustrative basins of attraction for various polynomials.

References On a Variant of Newton’s Method for Simple and Multiple Roots of Non Linear Equations

ABSTRACT In this paper the convergence behavior of a variant of Newton’s method based on the root mean square for solving nonlinear equations is proposed. The convergence properties of this method for solving non linear equations which have simple or multiple roots have been discussed and it is shown that it converges cubically to simple roots and linearly to multiple roots. Moreover, the values of the corresponding asymptotic error constants of convergence are determined. Theoretical results have been verified on the relevant numerical problems. A comparison of the efficiency of this method with other mean-based Newton’s methods is also included. Convergence behavior and error equations are also exhibited graphically for comparison on considering a particular example.