THERMAL SCIENCE

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Chushen Chen

,

Xiaolian Liao

SOLVING NON-LINEAR EQUATIONS BY FIXED POINT ITERATION METHOD AND ITS ACCELERATING APPROACH

ABSTRACT
Due to the inherent complexity of non-linear equations, finding their analytical solutions is often challenging, leading to a reliance on numerical solutions. This article examines key methods for solving these solutions, beginning with an introduction to the fixed point iteration method. Through examples, we explore the nuances of this technique in solving non-linear equations. We then examine Aitken's accelerated method and Steffensen's iteration approach, and discuss their convergence properties. By applying these methods to a consistent example, we compare their effectiveness in achieving similar error accuracy. The analysis shows that both the Aitken method and the Steffensen method require significant-ly fewer iterations than the fixed point iteration approach for solving non-linear equations, with the Steffensen method showing superior performance over the Aitken method. Finally, we have applied the Steffensen method to the flash evaporation problem, and the computational results are highly efficient.
KEYWORDS
non-linear equation, fixed point iteration method, astringency, Aitken accelerated method, Steffensen iteration method
PAPER SUBMITTED: 2023-12-16
PAPER REVISED: 2024-07-07
PAPER ACCEPTED: 2024-07-07
PUBLISHED ONLINE: 2025-07-06
DOI REFERENCE: https://doi.org/10.2298/TSCI2503031C
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2025, VOLUME 29, ISSUE Issue 3, PAGES [2031 - 2039]
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Solving non-linear equations by fixed point iteration method and its accelerating approach

2025 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence