Higher-order Iterative Methods Research Articles

On Traub–Steffensen-Type Iteration Schemes With and Without Memory: Fractal Analysis Using Basins of Attraction

This paper investigates the design and stability of Traub–Steffensen-type iteration schemes with and without memory for solving nonlinear equations. Steffensen’s method overcomes the drawback of the derivative evaluation of Newton’s scheme, but it has, in general, smaller sets of initial guesses that converge to the desired root. Despite this drawback of Steffensen’s method, several researchers have developed higher-order iterative methods based on Steffensen’s scheme. Traub introduced a free parameter in Steffensen’s scheme to obtain the first parametric iteration method, which provides larger basins of attraction for specific values of the parameter. In this paper, we introduce a two-step derivative free fourth-order optimal iteration scheme based on Traub’s method by employing three free parameters and a weight function. We further extend it into a two-step eighth-order iteration scheme by means of memory with the help of suitable approximations of the involved parameters using Newton’s interpolation. The convergence analysis demonstrates that the proposed iteration scheme without memory has an order of convergence of 4, while its memory-based extension achieves an order of convergence of at least 7.993, attaining the efficiency index 7.9931/3≈2. Two special cases of the proposed iteration scheme are also presented. Notably, the proposed methods compete with any optimal j-point method without memory. We affirm the superiority of the proposed iteration schemes in terms of efficiency index, absolute error, computational order of convergence, basins of attraction, and CPU time using comparisons with several existing iterative methods of similar kinds across diverse nonlinear equations. In general, for the comparison of iterative schemes, the basins of iteration are investigated on simple polynomials of the form zn−1 in the complex plane. However, we investigate the stability and regions of convergence of the proposed iteration methods in comparison with some existing methods on a variety of nonlinear equations in terms of fractals of basins of attraction. The proposed iteration schemes generate the basins of attraction in less time with simple fractals and wider regions of convergence, confirming their stability and superiority in comparison with the existing methods.

Local convergence study of tenth-order iterative method in Banach spaces with basin of attraction

<abstract><p>Many applications from computational mathematics can be identified for a system of non-linear equations in more generalized Banach spaces. Analytical methods do not exist for solving these type of equations, and so we solve these equations using iterative methods. We introduced a new numerical technique for finding the roots of non-linear equations in Banach space. The method is tenth-order and it is an extension of the fifth-order method which is developed by Arroyo et.al. <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. We provided a convergence analysis to demonstrate that the method exhibits tenth-order convergence. Also, we discussed the local convergence properties of the suggested method which depends on the fundamental supposition that the first-order Fréchet derivative of the involved function $ \Upsilon $ satisfies the Lipschitz conditions. This new approach is not only an extension of prior research, but also establishes a theoretical concept of the radius of convergence. We validated the efficacy of our method through various numerical examples. Our method is comparable with the methods of Tao Y et al. <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>. We also compared it with higher-order iterative methods, and we observed that it either performs similarly or better for the numerical examples. We also gave the basin of attraction to demonstrate the behaviour in the complex plane.</p></abstract>

Generalized high-order iterative methods for solutions of nonlinear systems and their applications

<abstract><p>In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order $ p $ of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order $ p+3 $ (for $ p\geq3 $) and have extended it to a generalized $ (m+2) $-step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to $ p+3m $, $ m\in\mathbb{N} $. We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations.</p></abstract>

A Unified Local-Semilocal Convergence Analysis of Efficient Higher Order Iterative Methods in Banach Spaces

To deal with the estimation of the locally unique solutions of nonlinear systems in Banach spaces, the local as well as semilocal convergence analysis is established for two higher order iterative methods. The given methods do not involve the computation of derivatives of an order higher than one. However, the convergence analysis was carried out in earlier studies by using the assumptions on the higher order derivatives as well. Such types of assumptions limit the applicability of techniques. In this regard, the convergence analysis is developed in the present study by imposing the conditions on first order derivatives only. The central idea for the local analysis is to estimate the bounds on convergence domain as well as the error approximations of the iterates along with the formulation of sufficient conditions for the uniqueness of the solution. Based on the choice of initial estimate in the given domain, the semilocal analysis is established, which ensures the convergence of iterates to a unique solution in that domain. Further, some applied problems are tested to certify the theoretical deductions.

On the complexity of convergence for high order iterative methods

Lipschitz-type conditions on the second derivative or conditions on higher than two derivatives not appearing on these methods have been employed to prove convergence. But these restrictions limit the applicability of high convergence order iterative methods although they may converge. That is why a new semi-local analysis is presented using only information taken from the derivatives on these methods. The new results compare favorably to the earlier ones even if the earlier conditions are used, since the latter use tighter Lipschitz parameters. Special cases and applications test convergence criteria.

Higher-Order Iterative Method to Solve Non-Linear Equations

This paper introduces a new tenth-order iterative method for solving non-linear equations. Modification of Newton’s method with higher-order convergence is presented. Analysis of convergence finalized that the order of convergence is ten. Some numerical examples illustrate that the algorithm is more efficient and performs better than classical Newton’s method and other methods with the same order.

An optimal derivative-free King's family for multiple zeros and its dynamics

PurposeThe purpose of this article is to develop and analyze a new derivative-free class of higher-order iterative methods for locating multiple roots numerically.Design/methodology/approachThe scheme is generated by using King-type iterative methods. By employing the Traub-Steffensen technique, the proposed class is designed into the derivative-free family.FindingsThe proposed class requires three functional evaluations at each stage of computation to attain fourth-order convergency. Moreover, it can be observed that the theoretical convergency results of family are symmetrical for particular cases of multiplicity of zeros. This further motivates the authors to present the result in general, which confirms the convergency order of the methods. It is also worth mentioning that the authors can obtain already existing methods as particular cases of the family for some suitable choice of free disposable parameters. Finally, the authors include a wide variety of benchmark problems like van der Waals's equation, Planck's radiation law and clustered root problem. The numerical comparisons are included with several existing algorithms to confirm the applicability and effectiveness of the proposed methods.Originality/valueThe numerical results demonstrate that the proposed scheme performs better than the existing methods in terms of CPU timing and absolute residual errors.

A Family of Derivative Free Algorithms for Multiple-Roots of Van Der Waals Problem

There are a good number of higher-order iterative methods for computing multiple zeros of nonlinear equations in the available literature. Most of them required first or higher-order derivatives of the involved function. No doubt, high-order derivative-free methods for multiple zeros are more difficult to obtain in comparison with simple zeros and with first order derivatives. This study presents an optimal family of fourth order derivative-free techniques for multiple zeros that requires just three evaluations of function ϕ, per iteration. The approximations of the derivative/s are based on symmetric divided differences. We also demonstrate the application of new algorithms on Van der Waals, Planck law radiation, Manning for isentropic supersonic flow and complex root problems. Numerical results reveal that the proposed derivative-free techniques are more efficient in comparison terms of CPU, residual error, computational order of convergence, number of iterations and the difference between two consecutive iterations with other existing methods.

A New Family of High-Order Ehrlich-Type Iterative Methods

One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.

Some High-Order Iterative Methods for Nonlinear Models Originating from Real Life Problems

We develop a sixth order Steffensen-type method with one parameter in order to solve systems of equations. Our study’s novelty lies in the fact that two types of local convergence are established under weak conditions, including computable error bounds and uniqueness of the results. The performance of our methods is discussed and compared to other schemes using similar information. Finally, very large systems of equations (100×100 and 200×200) are solved in order to test the theoretical results and compare them favorably to earlier works.

A Family of Arbitrary High-Order Iterative Methods for Approximating Inverse and the Moore–Penrose Inverse

In this work, a family of iterative algorithms for approximating the inverse of a square matrix and the Moore-Penrose inverse of a non-square one is proposed. These methods are based on arbitrary high-order iterative techniques which are used for computing roots of a nonlinear function. Therefore the presented techniques occupy any high-order convergence. The proposed methods are convenient and self-explanatory, achieve satisfactory results, and also require less and easy computations compared to some current schemes. Experimental results are provided to illustrate the reliability and robustness of the techniques.

Efficient optimal families of higher-order iterative methods with local convergence

The main aim of this manuscript is to propose two new schemes having three and four substeps of order eight and sixteen, respectively. Both families are optimal in the sense to Kung-Traub conjecture. The derivation of them are based on the weight function approach. In addition, theoretical and computational properties are fully investigated along with two main theorems describing the order of convergence. Further, we also provide the local convergence of them in Banach space setting under weak conditions. From the numerical experiments, we find that they perform better than the existing ones when we checked the performance of them on a concrete variety of non-linear scalar equations. Finally, we analyze the complex dynamical behavior of them which also provide a great extent to this.

Visualization of polynomiography from new higher order iterative methods

Visualization of polynomiography from new higher order iterative methods

Ball Convergence for Combined Three-Step Methods Under Generalized Conditions in Banach Space

Problems from numerous disciplines such as applied sciences, scientific computing, applied mathematics, engineering to mention some can be converted to solving an equation. That is why, we suggest higher-order iterative method to solve equations with Banach space valued operators. Researchers used the suppositions involving seventh-order derivative by Chen, S.P. and Qian, Y.H. But, here, we only use suppositions on the first-order derivative and Lipschitz constrains. In addition, we do not only enlarge the applicability region of them but also suggest computable radii. Finally, we consider a good mixture of numerical examples in order to demonstrate the applicability of our results in cases not covered before.

Approximate Schur-Block ILU Preconditioners for Regularized Solution of Discrete Ill-Posed Problems

High order iterative methods with a recurrence formula for approximate matrix inversion are proposed such that the matrix multiplications and additions in the calculation of matrix polynomials for the hyperpower methods of orders of convergence p=4k+3, where k≥1 is integer, are reduced through factorizations and nested loops in which the iterations are defined using a recurrence formula. Therefore, the computational cost is lowered from κ=4k+3 to κ=k+4 matrix multiplications per step. An algorithm is proposed to obtain regularized solution of ill-posed discrete problems with noisy data by constructing approximate Schur-Block Incomplete LU (Schur-BILU) preconditioner and by preconditioning the one step stationary iterative method. From the proposed methods of approximate matrix inversion, the methods of orders p=7,11,15,19 are applied for approximating the Schur complement matrices. This algorithm is applied to solve two problems of Fredholm integral equation of first kind. The first example is the harmonic continuation problem and the second example is Phillip’s problem. Furthermore, experimental study on some nonsymmetric linear systems of coefficient matrices with strong indefinite symmetric components from Harwell-Boeing collection is also given. Numerical analysis for the regularized solutions of the considered problems is given and numerical comparisons with methods from the literature are provided through tables and figures.

A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots

The aim of this paper is to introduce new high order iterative methods for multiple roots of the nonlinear scalar equation; this is a demanding task in the area of computational mathematics and numerical analysis. Specifically, we present a new Chebyshev–Halley-type iteration function having at least sixth-order convergence and eighth-order convergence for a particular value in the case of multiple roots. With regard to computational cost, each member of our scheme needs four functional evaluations each step. Therefore, the maximum efficiency index of our scheme is 1.6818 for α = 2 , which corresponds to an optimal method in the sense of Kung and Traub’s conjecture. We obtain the theoretical convergence order by using Taylor developments. Finally, we consider some real-life situations for establishing some numerical experiments to corroborate the theoretical results.

Existence, localization and approximation of solution of symmetric algebraic Riccati equations

In this paper we consider a family of high-order iterative methods which is more efficient than the Newton method to approximate a solution of symmetric algebraic Riccati equations. In fact, this paper is devoted to the convergence study of a k-steps iterative scheme with low operational cost and high order of convergence. We analyze their accessibility and computational efficiency. We also obtain results about the existence and localization of solution. Numerical experiments confirm the advantageous performance of the iterative scheme analyzed.

High order iterative methods for matrix inversion and regularized solution of fredholm integral equation of first kind with noisy data

The motivation of the present work is to propose high order iterative methods with a recurrence formula for approximate matrix inversion and provide regularized solution of Fredholm integral equation of first kind with noisy data by an algorithm using the proposed methods. From the given family of methods of orders p = 7,11,15,19 are applied to solve problems of Fredholm integral equation of first kind. From the literature, iterative methods of same orders are used to solve the considered problems and numerical comparisons are shown through tables and figures.

Generating Function Approach to the Derivation of Higher-Order Iterative Methods for Solving Nonlinear Equations

In this paper we propose a generating function method for constructing new two and three-point iterations withp(p= 4, 8) order of convergence. This approach allows us to derive a new family of optimal order iterative methods that include well known methods as special cases. Necessary and sufficient conditions forp-th (p= 4, 8) order convergence of the proposed iterations are given in terms of parameters τnand αn. We also propose some generating functions for τnand αn. We develop a unified representation of all optimal eighth-order methods. The order of convergence of the proposed methods is confirmed by numerical experiments.

A Family of High Order Numerical Methods for Solving Nonlinear Algebraic Equations with Simple and Multiple Roots

In this paper, we present, analyze, and test a family of high order iterative methods for finding simple and multiple roots of nonlinear algebraic equations of the form $$f(x)=0$$ . The proposed method can achieve convergence of order p, where $$p\ge 2$$ is a positive integer. The standard Newton–Raphson method ( $$p=2$$ ) and the Chebyshev’s method ( $$p=3$$ ) are both special cases of this family of methods. The pth order method requires evaluation of the function and its derivative up to order $$p-1$$ at each step. Several numerical experiments are provided to validate the theoretical order of convergence for nonlinear functions with simple and multiple roots.