Ljiljana D. Petković

Three-point methods with and without memory for solving nonlinear equations

Abstract A new family of three-point derivative free methods for solving nonlinear equations is presented. It is proved that the order of convergence of the basic family without memory is eight requiring four function-evaluations, which means that this family is optimal in the sense of the Kung–Traub conjecture. Further accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step. This self-accelerating parameter is calculated using information from the current and previous iteration so that the presented methods may be regarded as the methods with memory. The self-correcting parameter is calculated applying the secant-type method in three different ways and Newton’s interpolatory polynomial of the second degree. The corresponding R -order of convergence is increased from 8 to 4 ( 1 + 5 / 2 ) ≈ 8.472 , 9, 10 and 11. The increase of convergence order is attained without any additional function calculations, providing a very high computational efficiency of the proposed methods with memory. Another advantage is a convenient fact that these methods do not use derivatives. Numerical examples and the comparison with existing three-point methods are included to confirm theoretical results and high computational efficiency.

Multipoint methods for solving nonlinear equations: A survey

Multipoint iterative methods belong to the class of the most efficient methods for solving nonlinear equations. Recent interest in the research and development of this type of methods has arisen from their capability to overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. This survey paper is a mixture of theoretical results and algorithmic aspects and it is intended as a review of the most efficient root-finding algorithms and developing techniques in a general sense. Many existing methods of great efficiency appear as special cases of presented general iterative schemes. Special attention is devoted to multipoint methods with memory that use already computed information to considerably increase convergence rate without additional computational costs. Some classical results of the 1970s which have had a great influence to the topic, often neglected or unknown to many readers, are also included not only as historical notes but also as genuine sources of many recent ideas. To a certain degree, the presented study follows in parallel main themes shown in the recently published book (Petkovic et al., 2013) [53], written by the authors of this paper.

A note on some recent methods for solving nonlinear equations

Abstract In this note we present some comments on the recent results concerning iterative methods for solving nonlinear equations. In the first part we compare a new method [N. Ujevic, A method for solving nonlinear equations, Appl. Math. Comput. 174 (2006) 1416–1426] with the Newton and Ostrowski method, including an extensive analysis of numerical results together with the choice of initial approximations, and the computational efficiency of the considered methods. In the second part we give a list of recently derived root-finding methods and we study the question of priority; namely, we find out that all reviewed methods were already derived at the beginning of the 60s of the last century, one of them dates from the 18th century. Some remarks on the comparison analysis and the application of iteration methods are included.

A class of three-point root-solvers of optimal order of convergence

The construction of a class of three-point methods for solving nonlinear equations of the eighth order is presented. These methods are developed by combining fourth order methods from the class of optimal two-point methods and a modified Newtons method in the third step, obtained by a suitable approximation of the first derivative based on interpolation by a nonlinear fraction. It is proved that the new three-step methods reach the eighth order of convergence using only four function evaluations, which supports the Kung-Traub conjecture on the optimal order of convergence. Numerical examples for the selected special cases of two-step methods are given to demonstrate very fast convergence and a high computational efficiency of the proposed multipoint methods. Some computational aspects and the comparison with existing methods are also included.

On Schröder's families of root-finding methods

Schroders methods of the first and second kind for solving a nonlinear equation f(x)=0, originally derived in 1870, are of great importance in the theory and practice of iteration processes. They were rediscovered several times and expressed in different forms during the last 130 years. It was proved in the paper of Petkovic and Herceg (1999) [7] that even seven families of iteration methods for solving nonlinear equations are mutually equivalent. In this paper we show that these families are also equivalent to another four families of iteration methods and find that all of them have the origin in Schroders generalized method (of the second kind) presented in 1870. In the continuation we consider Smales open problem from 1994 about possible link between Schroders methods of the first and second kind and state the link in a simple way.

Accelerating generators of iterative methods for finding multiple roots of nonlinear equations

Two accelerating generators that produce iterative root-finding methods of arbitrary order of convergence are presented. Primary attention is paid to algorithms for finding multiple roots of nonlinear functions and, in particular, of algebraic polynomials. First, two classes of algorithms for solving nonlinear equations are studied: those with a known order of multiplicity and others with no information on multiplicity. We also demonstrate the acceleration of iterative methods for the simultaneous approximations of multiple roots of algebraic polynomials. A discussion about the computational efficiency of the root-solvers considered and three numerical examples are given.

A one parameter square root family of two-step root-finders ☆

A one parameter family of iterative methods for solving nonlinear equations is constructed. All the methods of the proposed family are cubically convergent for a simple root, except one particular method which attains the fourth order without the increase of computational cost. These methods belong to the class of two-step methods and require three function evaluations per iteration. The square-root structure of the family provides finding a complex zero of real functions in some cases. A comparison analysis shows that the presented family generates the methods which are comparable or even superior than the existing two-step iterative methods of the third order.

On the k-th root in circular arithmetic

AbstractThe representation of thek-th root of a complex circular intervalZ={c;r} is considered in this paper. Thek-th root is defined by the circular intervals which include the exact regionZ1/k={z:zk∈Z}. Two representations are given: (i) the centered inclusive disks

Point estimation of a family of simultaneous zero-finding methods

\cup \{ c^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} ; \mathop {\max }\limits_{z \in Z} |z^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} - c^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} |\}