Abstract
Many optimal order multiple root techniques involving derivatives have been proposed in literature. On the contrary, optimal order multiple root techniques without derivatives are almost nonexistent. With this as a motivational factor, here we develop a family of optimal fourth-order derivative-free iterative schemes for computing multiple roots. The procedure is based on two steps of which the first is Traub–Steffensen iteration and second is Traub–Steffensen-like iteration. Theoretical results proved for particular cases of the family are symmetric to each other. This feature leads us to prove the general result that shows the fourth-order convergence. Efficacy is demonstrated on different test problems that verifies the efficient convergent nature of the new methods. Moreover, the comparison of performance has proven the presented derivative-free techniques as good competitors to the existing optimal fourth-order methods that use derivatives.
Highlights
IntroductionHave been derived and analyzed in literature (see, for example, [2,3,4,5,6,7,8,9,10,11,12,13,14,15] and references cited therein)
We consider derivative-free methods for finding the multiple root with multiplicity m of a nonlinear equation f (t) = 0, i.e., f ( j) (α) = 0, j = 0, 1, 2, . . . , m − 1 and f (m) (α) 6= 0.Several higher order methods, with or without the use of modified Newton’s method [1]t k +1 = t k − m f, f 0 (1)have been derived and analyzed in literature.In such methods, one requires determining the derivatives of either first order or both first and second order
We propose a family of fourth-order derivative-free numerical methods for obtaining multiple roots of nonlinear equations
Summary
Have been derived and analyzed in literature (see, for example, [2,3,4,5,6,7,8,9,10,11,12,13,14,15] and references cited therein) In such methods, one requires determining the derivatives of either first order or both first and second order. Higher-order derivative-free methods to compute multiple roots are yet to be investigated. These methods are important in the problems where derivative f 0 is complicated to process or is costly to evaluate. The basic derivative-free method is the Traub–Steffensen method [16], which uses the approximation f 0 (tk ) '
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