On some modified families of multipoint iterative methods for multiple roots of nonlinear equations

The rate of multiplicity of the roots of nonlinear equations and its application to iterative methods

“…There is an extensive literature about methods for multiple roots (for instance, [1,2,[8][9][10]15,19,20]). In general, when m is known, one step methods for multiple roots are formulated in this form: (5) σ (x; m) depends on f(x) and, eventually, its derivatives.…”

Higher-Order Families of Multiple Root Finding Methods Suitable for Non-Convergent Cases and Their Dynamics

“…The first most striking feature of this contribution is that we have developed families of Rall's, Schröder's, super-Halley and Halley's methods for the first time which will converge even though the guess is far from the desired root or the derivative is small in the vicinity of the root and have the same error equations as those of their original methods respectively. It is also interesting to note that for γ = 0, these formulas reduce to Rall's [12], Schröder's [13], super-Halley and Halley's [5] methods for multiple roots respectively.…”

“…where f : I ⊆ R → R is a nonlinear sufficiently differentiable function in an interval I. To solve nonlinear equation (1.1), one can use classical iterative methods such as Rall's method (modified Newton's method) [12], [11], Schröder's method [13], Halley's and super-Halley method [5]. Perhaps, the most celebrated of all such iterative methods is the classical Rall's method (also known as modified Newton's method) for finding multiple roots of nonlinear equation (1.1), given by…”

A sixth order transformation method for finding multiple roots of nonlinear equations and basin attractors for various methods

“…In order to improve the order of convergence of (1), several higher-order methods have been proposed in the literature with known multiplicity m, for example, [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. On the other hand, if multiplicity m is not known explicitly, Traub [6] suggested a simple transformation:…”