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

Abstract

In this paper, we present many new one-parameter families of classical Rall's method (modified Newton's method), Schröder's method, Halley's method and super-Halley method for the first time which will converge even though the guess is far away 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, for multiple roots. Further, we also propose an optimal family of iterative methods of fourth-order convergence and converging to a required root in a stable manner without divergence, oscillation or jumping problems. All the methods considered here are found to be more effective than the similar robust methods available in the literature. In their dynamical study, it has been observed that the proposed methods have equal or better stability and robustness as compared to the other methods.

General Local Convergence Theorems about the Picard Iteration in Arbitrary Normed Fields with Applications to Super–Halley Method for Multiple Polynomial Zeros

“…Nowadays, there are a plethora of convergence results about the iterative methods of the type (1) but the proofs are often based on Taylor's expansions or other methods that require the existence of higher-order derivatives of f and ensure only asymptotic error constants where available. Moreover, the most of the existing convergence theorems do not provide exact information about the sets of initial approximations that guarantee the convergence of the iteration (1) (see e.g., [5][6][7] and the references therein).…”

Convergence and Dynamics of a Higher-Order Method

“…Our technique can also be used to extend the applicability of other methods defined in [1][2][3]. The novelty of our work, compared to other such as [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], is that we give weaker conditions, only in the first derivative, to guarantee the convergence of the described method.…”