Abstract
In this paper, we introduce a new family of efficient and optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity ( m ≥ 1 ) . We use the weight function approach involving one and two parameters to develop the new family. A comprehensive convergence analysis is studied to demonstrate the optimal eighth-order convergence of the suggested scheme. Finally, numerical and dynamical tests are presented, which validates the theoretical results formulated in this paper and illustrates that the suggested family is efficient among the domain of multiple root finding methods.
Highlights
The problem of solving nonlinear equation is recognized to be very old in history as many practical problems which arise are nonlinear in nature
The above-cited methods are designed for the simple root of nonlinear equations but the behavior of these methods is not similar when dealing with multiple roots of nonlinear equations
As stated by Ostrowski [1], if an iterative method possess order of convergence as O and total number of functional evaluations is n per iterative step, the index defined by E = O1/n is recognized as efficiency index of an iterative method
Summary
The problem of solving nonlinear equation is recognized to be very old in history as many practical problems which arise are nonlinear in nature. The well known Newton’s method with quadratic convergence for simple roots of nonlinear equations decays to first order when dealing with multiple roots of nonlinear equations. A two-step sixth-order non-optimal family for multiple roots presented by Geum et al [9] is given by:. Q : C2 → C is holomorphic in a neighborhood of (0, 0) Another non-optimal family of three-point sixth-order methods for multiple roots by. Behl et al [20] presented a multiple root finding family of iterative methods possessing convergence order eight given as: yn − un Q( hn ) 0.
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