Multipoint Iteration Research Articles

On general families of multipoint iterations by inverse interpolation and their applications

In this paper, a general family of derivative-free n+1-point iterative methods with n+1 parameters is constructed by inverse interpolation for solving nonlinear equations. A general family of n-point iterative methods with the first derivative and n parameters is also constructed by inverse interpolation. They satisfy the conjecture of Kung and Traub (1974) that an iterative method based on n+1 evaluations per iteration without memory would have optimal order 2n. Furthermore, the two families are accelerated by divided difference expressions for the parameters with one memory f(xk−1,n) per iteration to achieve higher orders of convergence 2n+2n−1 and 2n+2n−2, respectively. Finally, the proposed families are verified by solving nonlinear equations and applied to solve nonlinear ODEs.

General approach to constructing optimal multipoint families of iterative methods using Hermite’s rational interpolation

We discuss accelerating convergence of multipoint iterative methods for solving scalar equations, using particular type rational interpolant. Both derivative-free and Newton-type methods are investigated simultaneously. As a conclusion a Theorem of König’s type for multipoint iterations is stated. A new optimal multipoint family of methods based on rational interpolation is constructed. The iteration uses n function evaluations per cycle and O(j) operations in jth step of a single iteration to obtain 2n−1 order of convergence. Several equivalent forms of the obtained iterates and development techniques are presented.

Another Newton-type method with (k+2) order convergence for solving quadratic equations

In this paper we define another Newton-type method for finding simple root of quadratic equations. It is proved that the new one-point method has the convergence order of k  2 requiring only three function evaluations per full iteration,where k is the number of terms in the generating series. The Kung and Traub conjecture states that the multipoint iteration methods, without memory based on n function evaluations, could achieve maximum convergence order 1 2nï€ ­but, the new method produces convergence order of nine, which is better than the expected maximum convergence order. Finally, we have demonstrated that our present method is very competitive with the similar methods.

Modified Newton method to determine multiple zeros of nonlinear equations

New one-point iterative method for solving nonlinear equations is constructed. It is proved that the new method has the convergence order of three. Per iteration the new method requires two evaluations of the function. Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve maximum convergence order2n-1 but, the new method produces convergence order of three, which is better than expected maximum convergence order of two. Hence, we demonstrate that the conjecture fails for a particular set of nonlinear equations. Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed method using only a few function evaluations.

On improved Steffensen type methods with optimal eighth-order of convergence

This paper presents an improvement of the existing eighth-order derivative involved method [14] into derivative-free scheme, holding the order of convergence of the original method. Each member of the family requires only four function evaluations per iteration to achieve the eighth-order of convergence, while they are totally free from derivative evaluation. Hence, they agree with the optimality conjecture of Kung-Traub for providing multipoint iterations without memory. The proposed methods are compared with their closest competitors in a series of numerical experiments. Numerical experiments show that such derivative-free, high order schemes offer significant advantages over the derivative involved methods.

A new family of multi-point iterative methods for finding multiple roots of nonlinear equations

In this paper, new three-point eighth-order iterative methods for solving nonlinear equations are constructed. It is proved that these methods have the convergence order of eight requiring only four function evaluations 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 \(n\) evaluations, could achieve optimal convergence order \(2^{n-1} .\) Thus, we present new iterative methods which agree with the Kung and Traub conjecture for \(n=4\). Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed methods using only a few function evaluations.

New modifications of Newton-type methods with eighthorder convergence for solving nonlinear equations

The aims of this paper are, firstly, to define a new family of the Thukral and Petkovic type methods for finding zeros of nonlinear equations and secondly, to introduce new formulas for approximating the order of convergence of the iterative method. It is proved that these methods have the convergence order of eight requiring only four function evaluations per iteration. In fact, the optimal order of convergence which supports the Kung and Traub conjecture have been obtained. Kung and Traub conjectured that the multipoint iteration methods, without memory based on n evaluations, could achieve optimal convergence order 1 2. n  Thus, new iterative methods which agree with the Kung and Traub conjecture for 4 n  have been presented. It is observed that our proposed methods are competitive with other similar robust methods and very effective in high precision computations.

A New Optimal Eighth-Order Ostrowski-Type Family of Iterative Methods for Solving Nonlinear Equations

Based on Ostrowski's method, a new family of eighth-order iterative methods for solving nonlinear equations by using weight function methods is presented. Per iteration the new methods require three evaluations of the function and one evaluation of its first derivative. Therefore, this family of methods has the efficiency index which equals 1.682. Kung and Traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2n−1. Thus, we provide a new class which agrees with the conjecture of Kung-Traub for n=4. Numerical comparisons are made to show the performance of the presented methods.

A multi-step class of iterative methods for nonlinear systems

In this article, the numerical solution of nonlinear systems using iterative methods are dealt with. Toward this goal, a general class of multi-point iteration methods with various orders is constructed. The error analysis is presented to prove the convergence order. Also, a thorough discussion on the computational complexity of the new iterative methods will be given. The analytical discussion of the paper will finally be upheld through solving some application-oriented problems.

A family of Three-point Methods of Eighth-order for Finding Multiple Roots of Nonlinear Equations

In this paper, two new three-point eighth-order iterative methods for solving nonlinear equations are constructed.� It is proved that these methods have the convergence order of eight requiring only four function evaluations 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 \(n\) evaluations, could achieve optimal convergence order \(2^{n-1}\).� Thus, we present new iterative methods which agree with the Kung and Traub conjecture for \(n=4\) Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed methods using only a few function evaluations.

An Efficient Family of Root-Finding Methods with Optimal Eighth-Order Convergence

We derive a family of eighth-order multipoint methods for the solution of nonlinear equations. In terms of computational cost, the family requires evaluations of only three functions and one first derivative per iteration. This implies that the efficiency index of the present methods is 1.682. Kung and Traub (1974) conjectured that multipoint iteration methods without memory based on n evaluations have optimal order . Thus, the family agrees with Kung-Traub conjecture for the case . Computational results demonstrate that the developed methods are efficient and robust as compared with many well-known methods.

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

New four-point derivative-free sixteenth-order iterative methods for solving nonlinear equations are constructed. It is proved that these methods have the convergence order of sixteen requiring only five function evaluations 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 <i>n</i> evaluations, could achieve optimal convergence order Thus, we present new derivative-free methods which agree with the Kung and Traub conjecture for Numerical comparisons are made with other existing methods to show the performance of the presented methods.

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.

An optimized derivative-free form of the Potra–Pták method

In this paper, we discuss iterative methods for solving univariate nonlinear equations. First of all, we construct a family of methods with optimal convergence rate 4 based upon the Potra–Pták scheme and provide its error equation theoretically. Second, by using this derivative-involved family, a novel derivative-free family of two-step iterations without memory is derived. This derivative-free family agrees with the Kung–Traub conjecture (1974) for building optimal multi-point iterations without memory, since it is proven that each derivative-free method of the family reaches the convergence rate 4 requiring only three function evaluations per full iteration. Finally, numerical test problems are also provided to confirm the theoretical results.

Optimal Steffensen-type methods with eighth order of convergence

This paper proposes two classes of three-step without memory iterations based on the well known second-order method of Steffensen. Per computing step, the methods from the developed classes reach the order of convergence eight using only four evaluations, while they are totally free from derivative evaluation. Hence, they agree with the optimality conjecture of Kung–Traub for providing multi-point iterations without memory. As things develop, numerical examples are employed to support the underlying theory developed for the contributed classes of optimal Steffensen-type eighth-order methods.

An Improvement of Ostrowski’s and King’s Techniques with Optimal Convergence Order Eight

In this paper, we first establish a new class of three-point methods based on the two-point optimal method of Ostrowski. Analysis of convergence shows that any method of our class arrives at eighth order of convergence by using three evaluations of the function and one evaluation of the first derivative per iteration. Thus, this order agrees with the conjecture of Kung and Traub (J. ACM 643–651, 1974) for constructing multipoint optimal iterations without memory. We second present another optimal eighth-order class based on the King’s fourth-order family and the first attained class. To support the underlying theory developed in this work, we examine some methods of the proposed classes by comparison with some of the existing optimal eighth-order methods in literature. Numerical experience suggests that the new classes would be valuable alternatives for solving nonlinear equations.

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.

An optimal Steffensen-type family for solving nonlinear equations

In this paper, a general family of Steffensen-type methods with optimal order of convergence for solving nonlinear equations is constructed by using Newton’s iteration for the direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub [H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21 (1974) 634–651] that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2 m−1 . Its error equations and asymptotic convergence constants are obtained. Finally, it is compared with the related methods for solving nonlinear equations in the numerical examples.

Robust cubically and quartically iterative techniques free from derivative

Constructing of a technique which is both accurate and derivative-free is one of the most important tasks in the field of iterative processes. Hence in this study, convergent iterative techniques are suggested for solving single variable nonlinear equations. Their error equations are given theoretically to show that they have cubic and quartical convergence. Per iteration the novel schemes include three evaluations of the function while they are free from derivative as well. In viewpoint of optimality, the developed quartically class reaches the optimal efficiency index 41/3 ≈ 1.587 based on the Kung-Traub Hypothesis regarding the optimality of multi-point iterations without memory. In the end, the theoretical results are supported by numerical examples to elucidate the accuracy ofthe developed schemes.

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.