Abstract
Many optimal order multiple root techniques, which use derivatives in the algorithm, have been proposed in literature. But contrarily, derivative free optimal order techniques for multiple root are almost nonexistent. By this as an inspirational factor, here we present a family of optimal fourth order derivative-free techniques for computing multiple roots of nonlinear equations. At the beginning the convergence analysis is executed for particular values of multiplicity afterwards it concludes in general form. Behl et. al derivative-free method is seen as special case of the family. Moreover, the applicability and comparison is demonstrated on different nonlinear problems that certifies the efficient convergent nature of the new methods. Finally, we conclude that our new methods consume the lowest CPU time as compared to the existing ones. This illuminates the theoretical outcomes to a great extent of this study.
Highlights
Construction of optimal higher-order methods, in the sense of Kung-Traub conjecture [1], free from the derivatives, is always required for the multiple roots of nonlinear function of the form χ( x ) = 0 with multiplicity θ, i.e., χ( j) (α) = 0, j = 0, 1, 2, . . . , θ − 1 and χ(θ ) (α) 6= 0
In order to validate of theoretical results that have been proven in previous sections, the new methods BM, NM1, NM2, NM3 and NM4 are checked numerically by imposing them on some nonlinear equations
They are compared with some existing derivative free optimal fourth order methods
Summary
Numerous higher order methods, have been developed in literature by Dong [3], Geum et al [4], Hansen [5], Li et al [6,7], Neta [8], Osada [9], Sharifi et al [10], Sharma and Sharma [11], Zhou et al [12], Victory and Neta [13], Agarwal et al [14] and Soleymani et al [15] Such methods require the evaluations of derivatives.
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