An optimal family of eighth-order simple-root finders with weight functions dependent on function-to-function ratios and their dynamics underlying extraneous fixed points

Abstract

b s t r a c tWe extend in this paper an optimal family of three-step eighth-order methods developed by Džunić et al. (2011) with higher-order weight functions employed in the second and third sub-steps and investigate their dynamics under the relevant extraneous fixed points among which purely imaginary ones are specially treated for the analysis of the rich dynamics. Their theoretical and computational properties are fully investigated along with a main theorem describing the order of convergence and the asymptotic error constant as well as proper choices of special cases. A wide variety of relevant numerical examples are illustrated to confirm the underlying theoretical development. In addition, this paper investigates the dynamics of selected existing optimal eighth-order iterative maps with the help of illustrative basins of attraction for various polynomials.

Modification of Newton-Househölder Method for Determining Multiple Roots of Unknown Multiplicity of Nonlinear Equations

“…where s = f (y n )/ f (x n ) and u = f (z n )/ f (y n ). Next, following Lee et al [25], we approximate η(s, u) by the rational function η(s, u) = p 0 + p 1 s + p 2 s 2 + p 3 s 3 + u(p 4 + p 5 s + p 6 s 2 ) q 0 + q 1 s + q 2 s 2 + q 3 s 3 + u(q 4 + q 5 s + q 6 s 2 ) , (7) where p n , q n (0 ≤ n ≤ 6) ∈ R. Substituting ( 6) and ( 7) into (5), we get…”

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

“…This representation can be found in many previous researches. [18][19][20] On the other hand, the stability plane is used when we have a biparametric family whose parameters are real or a uniparametric family whose parameter is complex. Given a strange fixed point depending on the involved parameters, its stability plane represents the values of the parameters where the strange fixed point is repelling, attracting, or neutral.…”

Generating Root-Finder Iterative Methods of Second Order: Convergence and Stability

“…In this paper, we use the weight function technique (see, for example [5][6][7]) to design a general class of second order iterative methods including Newton's scheme and also other known and new schemes. These methods appear when specific weight functions (that may depend on one or more parameters) are selected.…”