On an Efficient Family of Steffensen-Like Methods for Finding Nonlinear Equations

Torkashvand, Vali; Kazemi, Manochehr

doi:10.1201/9780429288739-6

Cited by 1 publication

(3 citation statements)

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“…In Table 13, the presented COC2 = 9.157 for f 4 (x) = x 3 + 4x 2 − 10 = 0 is better than that computed in [52]. The presented COC2 = 15.634 for f 3 (x) = (x − 1) 3 − 1 = 0 is better than that computed in [50], where COC = 6.9041 was obtained.…”

Section: Enhancing the Updating Speed Of Parametersmentioning

confidence: 70%

“…The COC2 = 21.009 is abnormal for the solution of f 3 (x) = 0. In Table 12, the presented COC2 = 8.53 for f 4 (x) = x 3 + 4x 2 − 10 = 0 with the root r 4 = 1.365230013414097 is better than that computed in [52], where COC=7 is smaller than 8.53. Even if we do not use the full memory information, the performance is better than that in [52], where…”

Section: Functionsmentioning

confidence: 80%

“…To further accelerate the updating speed of the parameters A n and B n , according to [52], we can take…”

Section: Enhancing the Updating Speed Of Parametersmentioning

confidence: 99%

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New Memory-Updating Methods in Two-Step Newton’s Variants for Solving Nonlinear Equations with High Efficiency Index

Liu,

Chang

2024

Mathematics

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In the paper, we iteratively solve a scalar nonlinear equation f(x)=0, where f∈C(I,R),x∈I⊂R, and I includes at least one real root r. Three novel two-step iterative schemes equipped with memory updating methods are developed; they are variants of the fixed-point Newton method. A triple data interpolation is carried out by the two-degree Newton polynomial, which is used to update the values of f′(r) and f′′(r). The relaxation factor in the supplementary variable is accelerated by imposing an extra condition on the interpolant. The new memory method (NMM) can raise the efficiency index (E.I.) significantly. We apply the NMM to five existing fourth-order iterative methods, and the computed order of convergence (COC) and E.I. are evaluated by numerical tests. When the relaxation factor acceleration technique is combined with the modified Dzˇunic´’s memory method, the value of E.I. is much larger than that predicted by the paper [Kung, H.T.; Traub, J.F. J. Assoc. Comput. Machinery 1974, 21]. for the iterative method without memory.

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Section: Enhancing the Updating Speed Of Parametersmentioning

confidence: 70%

Section: Functionsmentioning

confidence: 80%