Parametric Family of Root-Finding Iterative Methods: Fractals of the Basins of Attraction

Abstract

Research interest in iterative multipoint schemes to solve nonlinear problems has increased recently because of the drawbacks of point-to-point methods, which need high-order derivatives to increase the order of convergence. However, this order is not the only key element to classify the iterative schemes. We aim to design new multipoint fixed point classes without memory, that improve or bring together the existing ones in different areas such as computational efficiency, stability and also convergence order. In this manuscript, we present a family of parametric iterative methods, whose order of convergence is four, that has been designed by using composition and weight function techniques. A qualitative analysis is made, based on complex discrete dynamics, to select those elements of the class with best stability properties on low-degree polynomials. This stable behavior is directly related with the simplicity of the fractals defined by the basins of attraction. In the opposite, particular methods with unstable performance present high-complexity in the fractals of their basins. The stable members are demonstrated also be the best ones in terms of numerical performance of non-polynomial functions, with special emphasis on Colebrook-White equation, with wide applications in Engineering.

From Fractal Behavior of Iteration Methods to an Efficient Solver for the Sign of a Matrix

“…Iterative methods could be compared in terms of their sensitivity of initial approximations, when they possess the same rate of convergence and the same structure [5,6]. Let us now draw the attraction basins of the methods ( 5) and ( 2) in Figures 1-4 for different polynomials in a square of the complex plane using double-precision arithmetic.…”

“…using the initial value (6). Note that, similarly, one may obtain the reciprocal version of (23) as follows:…”

“…Theorem 2. For computing the sign of A with no eigenvalues on the axis of imaginary, let Z 0 be selected via (6). The iterative method (24) (or (23)) converges to the M, and the convergence order is four.…”

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

“…So, our aim is to design a fourth-order optimal iterative method able to find multiple roots with known multiplicity but with low computational cost. Based on the previous study [13] of the multipoint fixed-point class of fourth-order iterative schemes…”