On the choice of the best members of the Kim family and the improvement of its convergence

Abstract

The best members of the Kim family, in terms of stability, are obtained by using complex dynamics. From this elements, parametric iterative methods with memory are designed. A dynamical analysis of the methods with memory is presented in order to obtain information about the stability of them. Numerical experiments are shown for confirming the theoretical results. KEYWORDSbasin of attraction, iterative methods with memory, low-dimensional dynamical systems, nonlinear algebraic or transcendental equations, parameter plane, stability MSC CLASSIFICATION 65H05 INTRODUCTIONTo find a solution of the nonlinear equation f(x) = 0 is a common problem that appears frequently in different fields of science and engineering. These nonlinear problems require iterative methods to be solved. In the last decades, there has been an extensive literature focused on the generation of iterative procedures to find a solution . We can see a good overview on classical and recent results in Amat and Busquier 1 or Petković et al. 2 The iterative methods are fixed-point schemes that, starting from one or more initial estimations, obtain a new value that approaches our solution asIf we use only the last iteration (p = 0), we have an iterative method without memory and, if we use more than one previous iterations (p > 0), the iterative scheme is with memory.Related with the order of convergence of iterative methods without memory, the Kung-Traub conjecture 3 establishes that the order of a scheme without memory is always lower than 2 d−1 , where d is the number of functional evaluations per iteration. When this bound is reached, the iterative method is called optimal. One technique for improving the order of convergence without increasing the number of functional evaluations is to introduce memory in the scheme, that is, the new iterate depend on more than one previous iterates.

Parametric Iterative Method for Addressing an Embedded-Steel Constitutive Model with Multiple Roots

“…The asymptotic behavior of the critical points is a key fact in analyzing the stability of the method. Previous results [25] state that at least one critical point appears in each immediate basin of attraction, that is, in the connected component of the basin of attraction containing the attractor. Let us also remark that superattracting fixed points are indeed critical points.…”

“…In this case, each point of the plane refers to a value of G3 ∈ Ĉ, that is, a member of Family (4). Instead of representing the 11 parameter planes, we just represent the unified parameter plane [25] for the sake of simplicity. White points represent convergence to any of the roots, whereas black points represent convergence to a different point or even divergence.…”

“…E s (MPa) [195,000,205,000] f y (MPa) [400,500] f ct (MPa) [25,50] φ(mm) [6,40] Tables 2-4 show the numerical performance of the known Rall and Schröder's schemes and the proposed method (9) in solving Model (10). The stopping criterion set was | f (x k+1 )| < 10 −16 , taking as an initial guess, the value x 0 = 9 4000 .…”

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

“…Cordero et al constructed a multi-step method based on Steffensen's method [18]. Also, in [19], some methods to find the solutions of nonlinear systems were investigated. Kanwar et al [20] presented a new method to compute the multiple roots of problems.…”

A New High-Order Jacobian-Free Iterative Method with Memory for Solving Nonlinear Systems

“…Using this technique, the stability of the fixed and critical points of secant, Steffensen' and Kurchatov's methods (among others) were studied in [19]. It was also used to analyze other procedures, such as those described in [20], the one defined by Choubey et al in [21] and those by Chicharro et al in [22][23][24].…”