A Convex Combination Approach for Mean-Based Variants of Newton’s Method

Cordero, Alicia; Franceschi, Jonathan; Torregrosa, Juan R.; Chiara, Zagati, Anna

doi:10.3390/sym11091106

Cited by 4 publications

(3 citation statements)

References 10 publications

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“…Observe that for α = 1.0 (RNM) the following values of parameters were obtained: ANI = 179.9, CAI = 0.98, time = 0.68 s, whereas for α = 20.47 ANI parameter obtained minimal value 5.41 when CAI = 1.0 and time = 0.025 s. This means that the M-RNM for α = 20.47 has attained the best improvement in comparison to the RNM-the number of iterations was reduced 33 times and time reduction was 27 times with CAI close to 1.0. Similar behaviour occurs on the flat parts of ANI and CAI plots for α values roughly in the range [10,30]. When α > 40.0 random fading oscillations are seen, what means that the M-RNM loses its stability and stops working properly.…”

Section: Variable α Sequencesupporting

confidence: 57%

“…By looking at this formula, we can observe that at the (i +1)-th iteration the M-RNM moves z i into the direction given by vector After including (10) into (16), we obtain the following:…”

Section: Algorithm 2: Computation Of N Pmentioning

confidence: 99%

“…Here, we would like to point out some of the most recent results in this area. In [10] various modifications of Newton's method with arithmetic, harmonic, and contraharmonic means are investigated. In [3,8,14] Newton's method was modified by using fractional derivatives instead of the classical ones.…”

Section: Introductionmentioning

confidence: 99%

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On the robust Newton’s method with the Mann iteration and the artistic patterns from its dynamics

2021

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There are two main aims of this paper. The first one is to show some improvement of the robust Newton’s method (RNM) introduced recently by Kalantari. The RNM is a generalisation of the well-known Newton’s root finding method. Since the base method is undefined at critical points, the RNM allows working also at such points. In this paper, we improve the RNM method by applying the Mann iteration instead of the standard Picard iteration. This leads to an essential decrease in the number of root finding steps without visible destroying the sharp boundaries among the basins of attractions presented in polynomiographs. Furthermore, we investigate visually the dynamics of the RNM with the Mann iteration together with the basins of attraction for varying Mann’s iteration parameter with the help of polynomiographs for several polynomials. The second aim of this paper is to present the intriguing polynomiographs obtained from the dynamics of the RNM with the Mann iteration under various sequences used in this iteration. The obtained polynomiographs differ considerably from the ones obtained with the RNM and are interesting from the artistic perspective. Moreover, they can easily find applications in wallpaper or fabric design.

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Section: Variable α Sequencesupporting

confidence: 57%

“…By looking at this formula, we can observe that at the (i +1)-th iteration the M-RNM moves z i into the direction given by vector After including (10) into (16), we obtain the following:…”

Section: Algorithm 2: Computation Of N Pmentioning

confidence: 99%

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On the robust Newton’s method with the Mann iteration and the artistic patterns from its dynamics

2021

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Two-point generalized Hermite interpolation: Double-weight function and functional recursion methods for solving nonlinear equations

Liu

2022

Mathematics and Computers in Simulation

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A new family of generalized quadrature methods for solving nonlinear equations

Liu

Lee

2021

Asian-European J. Math.

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Weerakoon and Fernando [A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000) 87–93] were resorted on a trapezoidal quadrature rule to derive an arithmetic mean Newton method with third-order convergence of the iterative scheme to solve nonlinear equations. Different quadrature methods have been developed, which form a special class of third-order iterative schemes requiring three evaluations of functions on [Formula: see text] per iteration, where [Formula: see text] is generated from the first Newton step. As an extension of these methods, we derive a new family of iterative schemes by using a new weight function [Formula: see text] to generalize the quadrature methods, of which [Formula: see text] signifies an approximate area under the curve [Formula: see text] between [Formula: see text] and [Formula: see text]. Then, a generalization of the midpoint Newton method is obtained by using another weight function, which is based on three evaluations of functions on [Formula: see text] per iteration. The sufficient conditions of these two weight functions are derived, which guarantee that the convergence order of the proposed iterative schemes is three.

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A Convex Combination Approach for Mean-Based Variants of Newton’s Method

Cited by 4 publications

References 10 publications

On the robust Newton’s method with the Mann iteration and the artistic patterns from its dynamics

On the robust Newton’s method with the Mann iteration and the artistic patterns from its dynamics

Two-point generalized Hermite interpolation: Double-weight function and functional recursion methods for solving nonlinear equations

A new family of generalized quadrature methods for solving nonlinear equations

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