Multipoint Iterative Methods Research Articles

A Class of Three‐Step Derivative‐Free Root Solvers with Optimal Convergence Order

A class of three‐step eighth‐order root solvers is constructed in this study. Our aim is fulfilled by using an interpolatory rational function in the third step of a three‐step cycle. Each method of the class reaches the optimal efficiency index according to the Kung‐Traub conjecture concerning multipoint iterative methods without memory. Moreover, the class is free from derivative calculation per full iteration, which is important in engineering problems. One method of the class is established analytically. To test the derived methods from the class, we apply them to a lot of nonlinear scalar equations. Numerical examples suggest that the novel class of derivative‐free methods is better than the existing methods of the same type in the literature.

New Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based onnevaluations could achieve optimal convergence order of . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for . Numerical comparisons are made to demonstrate the performance of the methods presented.

Another Simple Way of Deriving Several Iterative Functions to Solve Nonlinear Equations

We present another simple way of deriving several iterative methods for solving nonlinear equations numerically. The presented approach of deriving these methods is based on exponentially fitted osculating straight line. These methods are the modifications of Newton′s method. Also, we obtain well‐known methods as special cases, for example, Halley′s method, super‐Halley method, Ostrowski′s square‐root method, Chebyshev′s method, and so forth. Further, new classes of third‐order multipoint iterative methods free from a second‐order derivative are derived by semidiscrete modifications of cubically convergent iterative methods. Furthermore, a simple linear combination of two third‐order multipoint iterative methods is used for designing new optimal methods of order four.

Eighth-Order Iterative Methods without Derivatives for Solving Nonlinear Equations

A new family of eighth-order derivative-free methods for solving nonlinear equations is presented. It is proved that these methods have the convergence order of eight. These new methods are derivative-free and only use four evaluations of the function per iteration. In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on evaluations, could achieve optimal convergence order . Thus, we present new derivative-free methods which agree with Kung and Traub conjecture for . Numerical comparisons are made to demonstrate the performance of the methods presented.

A family of optimal sixteenth-order multipoint methods with a linear fraction plus a trivariate polynomial as the fourth-step weighting function

A new family of four-step optimal multipoint iterative methods of order sixteen for solving nonlinear equations are developed along with their convergence properties. Numerical experiments with comparison to some existing methods are demonstrated to support the underlying theory.

A biparametric family of four-step sixteenth-order root-finding methods with the optimal efficiency index

A biparametric family of four-step multipoint iterative methods of order sixteen to solve nonlinear equations are developed and their convergence properties are established. The optimal efficiency indices are all found to be 1 6 1 / 5 ≈ 1.741101 . Numerical examples as well as comparison with existing methods are demonstrated to verify the developed theory.

A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots

A uniparametric family of three-step eighth-order multipoint iterative methods requiring only a first derivative are proposed in this paper to find simple roots of nonlinear equations. Development and convergence analysis on the proposed methods is described along with numerical experiments including comparison with existing methods.

A biparametric family of optimally convergent sixteenth-order multipoint methods with their fourth-step weighting function as a sum of a rational and a generic two-variable function

A biparametric family of four-step multipoint iterative methods of order sixteen to numerically solve nonlinear equations are developed and their convergence properties are investigated. The efficiency indices of these methods are all found to be 16 1/5≈1.741101, being optimally consistent with the conjecture of Kung–Traub. Numerical examples as well as comparison with existing methods developed by Kung–Traub and Neta are demonstrated to confirm the developed theory in this paper.

A biparametric family of eighth-order methods with their third-step weighting function decomposed into a one-variable linear fraction and a two-variable generic function

This paper proposes a biparametric family of three-step eighth-order multipoint iterative methods with optimal efficiency index in the sense of Kung-Traub for simple roots of nonlinear equations. We employ their third-step weighting function decomposed into King’s linear fractional function and a two-variable function to construct such a family of optimal methods. Development and convergence analysis on the proposed methods is fully described in addition to numerical experiments including comparison with existing methods.

A penta-parametric family of fifteenth-order multipoint methods for nonlinear equations

A penta-parametric family of four-step multipoint iterative methods of order fifteen for nonlinear algebraic equations are developed and their convergence properties are established. The efficiency indices are all found to be 15 1/5 ≈ 1.71877, better than 14 1/5 ≈ 1.69522 of a family of fourteenth-order methods suggested by Neta [9]. Numerical examples are demonstrated to verify the developed theory.

Iterative methods for use with nonlinear discrete algebraic models

A family of multi-point iterative methods for solving nonlinear equations were described in Cordero and Torregrosa (2008) [4], and a general error analysis was given, always for a simple root. Here we study the order of convergence of such methods when we have multiple roots. We prove that the order of convergence goes down to 1 but, when the multiplicity n is known, it may be raised to 2 by using different types of correction. For n unknown, we show that some methods of this family converge faster than the classical Newton’s method. In addition, we provide various numerical tests which confirm or improve on theoretical results and allow us to compare some methods of the aforementioned family.

Modified Efficient Families of Two and Three-Step Predictor-Corrector Iterative Methods for Solving Nonlinear Equations

In this paper, we present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting near the root. The main advantage of our methods is that they perform better and moreover, have the same efficiency indices as that of existing multipoint iterative methods. Furthermore, the convergence analysis of the new methods is discussed and several examples are given to illustrate their efficiency.

A new family of modified Ostrowski’s methods with accelerated eighth order convergence

Based on Ostrowski’s fourth order method, we derive a family of eighth order methods for the solution of nonlinear equations. In terms of computational cost the family requires three evaluations of the function and one evaluation of first derivative. Therefore, the efficiency index of the present methods is 1.682 which is better than the efficiency index 1.587 of Ostrowski’s method. Kung and Traub conjectured that multipoint iteration methods without memory based on n evaluations have optimal order 2n − 1. Thus, the family agrees with Kung–Traub conjecture for the case n = 4. The efficacy of the present methods is tested on a number of numerical examples. It is observed that our methods are competitive with other similar robust methods and very effective in high precision computations.

Exponentially fitted variants of Newton's method with quadratic and cubic convergence

In this paper, we present some new families of Newton-type iterative methods, in which f′(x)=0 is permitted at some points. The presented approach of deriving these iterative methods is different. They have well-known geometric interpretation and admit their geometric derivation from an exponential fitted osculating parabola. Cubically convergent methods require the use of the first and second derivatives of the function as Euler's, Halley's, Chebyshev's and other classical methods do. Furthermore, new classes of third-order multipoint iterative methods free from second derivative are derived by semi-discrete modifications of cubically convergent iterative methods. Further, the approach has been extended to solve a system of non-linear equations.

A family of multi-point iterative methods for solving systems of nonlinear equations

We extend to n -dimensional case a known multi-point family of iterative methods for solving nonlinear equations. This family includes as particular cases some well known and also some new methods. The main advantage of these methods is they have order three or four and they do not require the evaluation of any second or higher order Fréchet derivatives. A local convergence analysis and numerical examples are provided.

Modified families of multi-point iterative methods for solving nonlinear equations

We further present some semi-discrete modifications to the cubically convergent iterative methods derived by Kanwar and Tomar (Modified families of Newton, Halley and Chebyshev methods, Appl. Math. Comput. http://dx.doi.org/10.1016/j.amc.2007.02.119) and derived a number of interesting new classes of third-order multi-point iterative methods free from second derivatives. Furthermore, several functions have been tested and all the methods considered are found to be effective and compared to the well-known existing third and fourth-order multi-point iterative methods.

A class of multi-point iterative methods for nonlinear equations

A family of multi-point iterative methods for solving nonlinear equations is described and the order of convergence of its elements is studied. The computational efficiency of some elements of this family, in terms of the number of function evaluations, is also provided. In addition, we present different numerical tests, which confirm or improve the theoretical results and allow us to compare some methods of this family.

On some families of multi-point iterative methods for solving nonlinear equations

Some semi-discrete analogous of well known one-point family of iterative methods for solving nonlinear scalar equations dependent on an arbitrary constant are proposed. The new families give multi-point iterative processes with the same or higher order of convergence. The convergence analysis and numerical examples are presented.

Multipoint iterative method with cubic convergence

A new third-order multipoint method without using second derivative for finding simple zeros of non-linear functions, permitting f′( x) = 0 in the vicinity of the required root is presented in this paper. Furthermore, numerical tests supporting the theory and comparison of the results with respect to the existing classical methods are also given.

On a family of iterative methods for simultaneous extraction of all roots of algebraic polynomial

A family of multi-point iterative methods for simultaneous determination of all roots of polynomial equation is obtained. This family is corresponding to a family of multi-point methods for approximation of only one root of nonlinear equations. The convergence analysis and numerical examples are given.