Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity

Abstract

In this paper, we propose, to the best of our knowledge, the first iterative scheme with memory for finding roots whose multiplicity is unknown existing in the literature. It improves the efficiency of a similar procedure without memory due to Schröder and can be considered as a seed to generate higher order methods with similar characteristics. Once its order of convergence is studied, its stability is analyzed showing its good properties, and it is compared numerically in terms of their basins of attraction with similar schemes without memory for finding multiple roots.

Modifying Kurchatov's method to find multiple roots of nonlinear equations

“…We also use a maximum of 100 iterations as a stopping criterion. We compare the proposed methods with the method coming from [4], which we denote by gTM.…”

“…In a similar way, in paper [4], the authors construct an iterative method with memory for approximating the multiple roots, that avoids the need to know a priori the multiplicity. In this manuscript, we apply several techniques to Kurchatov's scheme in order to obtain an iterative method with memory and without derivatives for finding multiple roots .…”

An optimal eighth order derivative free multiple root finding numerical method and applications to chemistry

“…Sharma et al in [14,16] presented eight order scheme for computing multiple root of nonlinear equations. nevertheless, other derivative-free methods for multiple roots have been generated by using different approaches, such as [4].…”

Constructing a Matrix Mid-Point Iterative Method for Matrix Square Roots and Applications

“…Apart from such standard procedures to compute matrix functions, iterative methods are significant as long as it is required to compute them when the entries are changing over time or when a sharp initial approximation is available in some applications (for more information, see [2,3]).…”

“…As can be observed from Equation (2), both the principal square root and its inverse are required to find the numerical solution. Note that M ∈ C n×n has a square root if no two terms are the same odd integer in the ascent sequence of integers t 1 , t 2 , • • • , given in [4]:…”