Beyond Newton: a new root-finding fixed-point iteration for nonlinear equations

Aggarwal, A. and Pant, S. (2020) Beyond Newton: a new root-finding fixed-point iteration for nonlinear equations. Algorithms, 13(4), 78. (doi: 10.3390/a13040078)

Abstract

Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is Newton’s method. However, its convergence depends heavily on the initial guess, with poor choices often leading to slow convergence or even divergence. In this short note, we seek to enlarge the basin of attraction of the classical Newton’s method. The key idea is to develop a relatively simple multiplicative transform of the original equations, which leads to a reduction in nonlinearity, thereby alleviating the limitation of Newton’s method. Based on this idea, we derive a new class of iterative methods and rediscover Halley’s method as the limit case. We present the application of these methods to several mathematical functions (real, complex, and vector equations). Across all examples, our numerical experiments suggest that the new methods converge for a significantly wider range of initial guesses. For scalar equations, the increase in computational cost per iteration is minimal. For vector functions, more extensive analysis is needed to compare the increase in cost per iteration and the improvement in convergence of specific problems.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Aggarwal, Dr Ankush

Creator Roles

Aggarwal, A.: Conceptualization, Formal analysis, Funding acquisition, Investigation, Writing – original draft, Writing – review and editing

• Authors: Aggarwal, A., and Pant, S.
• College/School: College of Science and Engineering > School of Engineering > Infrastructure and Environment
• Journal Name: Algorithms
• Publisher: MDPI
• ISSN: 1999-4893
• ISSN (Online): 1999-4893
• Copyright Holders: Copyright © 2020 The Authors
• First Published: First published in Algorithms 13(4):78
• Publisher Policy: Reproduced under a Creative Commons Licence

Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
306966		Predicting cardiovascular biomechanical stiffening due to the interplay of tissue layers with focus on calcific aortic valve disease	Ankush Aggarwal	Engineering and Physical Sciences Research Council (EPSRC)	EP/P018912/2	ENG - Infrastructure & Environment