A Two-Dimensional Variant of Newton’s Method and a Three-Point Hermite Interpolation: Fourth- and Eighth-Order Optimal Iterative Schemes

A nonlinear equation f(x)=0 is mathematically transformed to a coupled system of quasi-linear equations in the two-dimensional space. Then, a linearized approximation renders a fractional iterative scheme xn+1=xn−f(xn)/[a+bf(xn)], which requires one evaluation of the given function per iteration. A...

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Bibliographic Details
Published in:Mathematics (Basel) Vol. 11; no. 21; p. 4529
Main Authors: Liu, Chein-Shan, El-Zahar, Essam R., Chang, Chih-Wen
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.11.2023
Subjects:
Basketball players
Finite difference method
fourth-order optimal iterative scheme
fractional iterative scheme
Interpolation
Iterative methods
Linear equations
Mathematical analysis
Methods
modified derivative-free Newton method
Newton methods
nonlinear equation
Nonlinear equations
Quadratures
two-dimensional approach
Weighting functions
Serbia
ISSN:2227-7390, 2227-7390
Online Access:Get full text
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Summary:A nonlinear equation f(x)=0 is mathematically transformed to a coupled system of quasi-linear equations in the two-dimensional space. Then, a linearized approximation renders a fractional iterative scheme xn+1=xn−f(xn)/[a+bf(xn)], which requires one evaluation of the given function per iteration. A local convergence analysis is adopted to determine the optimal values of a and b. Moreover, upon combining the fractional iterative scheme to the generalized quadrature methods, the fourth-order optimal iterative schemes are derived. The finite differences based on three data are used to estimate the optimal values of a and b. We recast the Newton iterative method to two types of derivative-free iterative schemes by using the finite difference technique. A three-point generalized Hermite interpolation technique is developed, which includes the weight functions with certain constraints. Inserting the derived interpolation formulas into the triple Newton method, the eighth-order optimal iterative schemes are constructed, of which four evaluations of functions per iteration are required.
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ISSN:2227-7390
2227-7390
DOI:10.3390/math11214529