Search

This paper deals with the study of relaxed conditions for semi-local convergence for a general iterative method, k-step Newton's method, using majorizing sequences. Dynamical behavior of the mentioned method is also analyzed via Julia set and basins of attraction. Numerical examples of nonlinear systems of equations will be examined to verify the given theory. [ABSTRACT FROM AUTHOR]

In this paper, there has been developed an adaptive method with memory based on the first two steps of the general optimal method without memory by Kung and Traub. Included is the convergence analysis along with numerical implementations. The proposed adaptive method with memory can increase the efficiency index of the optimal method without memory as high as possible while reusing all the available information of the functional evaluations in the previous iteration for this purpose. Also, we discuss the numerical stability from the point of basins of attraction. [ABSTRACT FROM AUTHOR]

In this study, based on the optimal free derivative without memory methods proposed by Cordero et al. [A. Cordero, J.L. Hueso, E. Martinez, J.R. Torregrosa, Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolation, Mathematical and Computer Modeling. 57 (2013) 1950-1956], we develop two new iterative with memory methods for solving a nonlinear equation. The first has two steps with three self-accelerating parameters, and the second has three steps with four self-accelerating parameters. These parameters are calculated using information from the current and previous iteration so that the presented methods may be regarded as the with memory methods. The self-accelerating parameters are computed applying Newton's interpolatory polynomials. Moreover, they use three and four functional evaluations per iteration and corresponding R-orders of convergence are increased from 4 ad 8 to 7.53 and 15.51, respectively. It means that, without any new function calculations, we can improve convergence order by 93% and 96%. We provide rigorous theories along with some numerical test problems to confirm theoretical results and high computational efficiency. [ABSTRACT FROM AUTHOR]

First, it is attempted to derive an optimal derivative-free Steffensen–King’s type family without memory for computing a simple zero of a nonlinear function with efficiency index 4 1 / 3 ≈ 1.587 . Next, since our without memory family includes a parameter in which it is still possible to increase the convergence order without any new function evaluations. Therefore, we extract a new method with memory so that the convergence order rises to six without any new function evaluation and therefore reaches efficiency index 6 1 / 3 ≈ 1.817 . Consequently, derivative-free and high efficiency index would be the substantial contributions of this work as opposed to the classical Steffensen’s and King’s methods. Finally, we compare some of the convergence planes with different weight functions in order to show which are the best ones. [ABSTRACT FROM AUTHOR]

In this paper, we present a fourth order method for computing simple roots of nonlinear equations by using suitable Taylor and weight function approximation. The method is based on Weerakoon-Fernando method [S. Weerakoon, G.I. Fernando, A variant of Newton's method with third-order convergence, Appl. Math. Lett. 17 (2000) 87-93]. The method is optimal, as it needs three evaluations per iterate, namely one evaluation of function and two evaluations of first derivative. So, Kung and Traub's conjecture is fulfilled. We also perform some numerical tests that confirm the theoretical results and allow us to compare the proposed method with some existing methods of the same type. [ABSTRACT FROM AUTHOR]

A class of derivative-free methods without memory for approximating a simple zero of a nonlinear equation is presented. The proposed class uses four function evaluations per iteration with convergence order eight. Therefore, it is an optimal three-step scheme without memory based on Kung-Traub conjecture. Moreover, the proposed class has an accelerator parameter with the property that it can increase the convergence rate from eight to twelve without any new functional evaluations. Thus, we construct a with memory method that increases considerably efficiency index from 81/4 1.681 to 121/4 1.861. Illustrations are also included to support the underlying theory. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we use multi-projection operators to solve the linear Fredholm integral equation of the second kind with wavelets. Moreover, a numerical convergence and computation of a bound for the condition number of the multi-projection operators method is obtained. [Copyright &y& Elsevier]