On developing fourth-order optimal families of methods for multiple roots and their dynamics

Abstract

Motsa, SS.; Torregrosa Sánchez, JR. (2015). On developing fourth-order optimal families of methods for multiple roots and their dynamics. Applied Mathematics and Computation. 265:520-532. doi:10.1016/j.amc.2015 AbstractThere are few optimal fourth-order methods for solving nonlinear equations when the multiplicity m of the required root is known in advance. Therefore, the first focus of this paper is on developing new fourth-order optimal families of iterative methods by a simple and elegant way. Computational and theoretical properties are fully studied along with a main theorem describing the convergence analysis. Another main focus of this paper is the dynamical analysis of the rational map associated with our proposed class for multiple roots;as far as we know, there are no deep study of this kind on iterative methods for multiple roots. Further, using Mathematica with its high precision compatibility, a variety of concrete numerical experiments and relevant results are extensively treated to confirm the theoretical development.

Dynamical Analysis via Möbius Conjugacy Map on a Uniparametric Family of Optimal Fourth-Order Multiple-Zero Solvers with Rational Weight Functions

“…The result of Theorem 3 enables us to discover that = 0 (corresponding to fixed point of or root of ( ) = [( − )( − )] ) and = ∞ (corresponding to fixed point of or root of ( )) are clearly two of their fixed points of the conjugate map ( ; ), regardless of -values. Besides, by direct computation, we find that = 1 is a strange fixed point [13][14][15] of (that is not a root of ( ) = [( − )( − )] ) due to the fact that (1; ) = 1, regardless of -values.…”

A new two‐point scheme for multiple roots of nonlinear equations

“…Construction of higher‐order iterative methods for multiple roots having prior knowledge of multiplicity ( m >1) is one of the most important and challenging task in computational mathematics. No doubts, we have a small number of fourth‐order iterative methods for multiple roots, which were proposed by Neta and Johnson in (2008), Li et al in (2009), Neta, Sharma and Sharma, and Li et al in (2010), Zhou et al in (2011), Sharifi et al in (2012), Soleymani et al, Soleymani and Babajee, Liu and Zhou, and Zhou et al in (2013), Thukral in (2014), Behl et al and Hueso et al in (2015), Behl et al in (2016), and Zafar et al in (2018).…”

“…Construction of higher-order iterative methods for multiple roots having prior knowledge of multiplicity (m > 1) is one of the most important and challenging task in computational mathematics. No doubts, we have a small number of fourth-order iterative methods for multiple roots, which were proposed by Neta 8 Soleymani and Babajee, 9 Liu and Zhou, 10 and Zhou et al 11 in (2013), Thukral 12 in (2014), Behl et al 13 Out of them, iterative functions proposed by Li et al 5 (expect two of them are optimal), Neta and Johnson, 1 and Neta 3 are nonoptimal schemes of fourth order. On the other hand, rest of them are optimal according to classical Kung-Traub conjecture.…”

Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders

“…Newton's method has been widely accepted in view of its simplicity and quadratic convergence. Development of higher-order iterative root-finding schemes [1][2][3][4][5][6][7][8] has been a primary topic among the scholars and researchers working in this area.…”