Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations

Aggarwal, Ankush; Pant, Sanjay

doi:10.3390/a13040078

Journal article 1156 views 125 downloads

Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations

Ankush Aggarwal

, Sanjay Pant

Algorithms, Volume: 13, Issue: 4, Start page: 78

Swansea University Authors: Ankush Aggarwal , Sanjay Pant

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DOI (Published version): 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...

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Published in:	Algorithms
ISSN:	1999-4893
Published:	MDPI AG 2020
Online Access:	Check full text
URI:	https://cronfa.swan.ac.uk/Record/cronfa53886

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last_indexed	2020-10-23T03:06:17Z
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spelling	2020-10-22T13:35:11.5296806 v2 53886 2020-03-30 Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations 33985d0c2586398180c197dc170d7d19 0000-0002-1755-8807 Ankush Aggarwal Ankush Aggarwal true false 43b388e955511a9d1b86b863c2018a9f 0000-0002-2081-308X Sanjay Pant Sanjay Pant true false 2020-03-30 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 Journal Article Algorithms 13 4 78 MDPI AG 1999-4893 Newton–Raphson method; Newton’s iteration; nonlinear equations; iterative solution; gradient-based methods 29 3 2020 2020-03-29 10.3390/a13040078 COLLEGE NANME COLLEGE CODE Swansea University 2020-10-22T13:35:11.5296806 2020-03-30T15:15:14.8168681 Faculty of Science and Engineering School of Engineering and Applied Sciences - Uncategorised Ankush Aggarwal 0000-0002-1755-8807 1 Sanjay Pant 0000-0002-2081-308X 2 53886__16975__27535594021e475693b54921ab028dc6.pdf 53886.pdf 2020-03-30T15:18:19.8303471 Output 2544397 application/pdf Version of Record true This is an open access article distributed under the Creative Commons Attribution License (CC-BY). true eng http://creativecommons.org/licenses/by/4.0/
title	Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations
spellingShingle	Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations Ankush Aggarwal Sanjay Pant
title_short	Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations
title_full	Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations
title_fullStr	Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations
title_full_unstemmed	Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations
title_sort	Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations
author_id_str_mv	33985d0c2586398180c197dc170d7d19 43b388e955511a9d1b86b863c2018a9f
author_id_fullname_str_mv	33985d0c2586398180c197dc170d7d19_*_Ankush Aggarwal 43b388e955511a9d1b86b863c2018a9f_*_Sanjay Pant
author	Ankush Aggarwal Sanjay Pant
author2	Ankush Aggarwal Sanjay Pant
format	Journal article
container_title	Algorithms
container_volume	13
container_issue	4
container_start_page	78
publishDate	2020
institution	Swansea University
issn	1999-4893
doi_str_mv	10.3390/a13040078
publisher	MDPI AG
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description	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
published_date	2020-03-29T10:04:22Z
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Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations

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