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In this contribution, a novel eighth-order scheme is presented for solving nonlinear equations with multiple roots. The proposed scheme comprises of three steps with the modified Newton method as its first step followed by two weighted Newton steps involving one univariate and one bivariate function respectively. Analysis of convergence confirms that the presented scheme obtains optimal computational order of convergence. The efficiency of presented scheme is compared numerically with recent eighth-order methods. Functions like population growth problem, Newton’s beam problem, etc., have been considered for numerical experimentation. For the comparative study in the complex plane, we employed the concept of basins of attraction.

In 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Based on Petković-Ilić-Džunić method [3], we derive a family of three-step eighth-order Steffensen type iterative methods for solving nonlinear equations. The new methods without memory use two suitable parametric functions at the second and third steps and are free from any derivatives. Per iteration the new methods require four functional evaluations, which implies that the efficiency index of the new methods is 1.682. The advantage of the new methods is that the computing speed of the new methods is faster than that of other eighth-order Steffensen type methods without memory. Numerical examples are made to show the performance of our methods and support the developed theory. [ABSTRACT FROM AUTHOR]

A new triparametric family of three-step optimal eighth-order iterative methods free from second derivatives are proposed in this paper, to find a simple root of nonlinear equations. Convergence analysis as well as numerical experiments confirms the eighth-order convergence and asymptotic error constants. [ABSTRACT FROM AUTHOR]

Abstract: Kung and Traub (1974)  conjectured that an iteration method without memory based on evaluations could achieve optimal convergence order . Hence, based on this conjecture, we derive two optimal three-step eighth-order classes of methods in which there are only four evaluations per full cycle. Analytical proofs of the presented derivative-involved classes are provided. Finally, a number of numerical examples are also proposed to illustrate the accuracy of the contributed methods by comparing with the new existing optimal eighth-order methods without memory in the literature. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we derive a new family of eighth-order methods for solving simple roots of nonlinear equations by using weight function methods. Per iteration these methods require three evaluations of the function and one evaluation of its first derivative, which implies that the efficiency indexes are 1.682. Numerical comparisons are made to show the performance of the derived methods, as shown in the illustration examples. [Copyright &y& Elsevier]

Abstract: A multi-parameter family of three-step eighth-order iterative methods free from second derivatives are proposed in this paper to find a simple root of nonlinear algebraic equations. Convergence analysis as well as numerical experiments confirms the eighth-order convergence and asymptotic error constants. [Copyright &y& Elsevier]