Solving non-linear equations by fixed point iteration method and its accelerating approach

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.