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

Chein-Shan Liu, Essam R. El-Zahar and Chih-Wen Chang ()
Additional contact information
Chein-Shan Liu: Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan
Essam R. El-Zahar: Department of Mathematics, College of Sciences and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia
Chih-Wen Chang: Department of Mechanical Engineering, National United University, Miaoli 36063, Taiwan

Mathematics, 2023, vol. 11, issue 21, 1-21

Abstract: 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 x n + 1 = x n − f ( x n ) / [ a + b f ( x n ) ] , 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.

Keywords: nonlinear equation; two-dimensional approach; fractional iterative scheme; modified derivative-free Newton method; quadratures; fourth-order optimal iterative scheme; three-point generalized Hermite interpolation; eighth-order optimal iterative scheme (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/11/21/4529/pdf (application/pdf)
https://www.mdpi.com/2227-7390/11/21/4529/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:11:y:2023:i:21:p:4529-:d:1273502

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().