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  4. Cubic parameter homotopy function for solving nonlinear equations

Cubic parameter homotopy function for solving nonlinear equations

MMC.
2025;
Volume 12, Number 2
: pp. 410–414
https://doi.org/10.23939/mmc2025.02.410
Received: December 20, 2024
Revised: April 09, 2025
Accepted: April 15, 2025

Rahman M. S. A., Nor H. M., Ahmad M. Z., Abd Rahman N. H.  Cubic parameter homotopy function for solving nonlinear equations.  Mathematical Modeling and Computing. Vol. 12, No. 2, pp. 410–414 (2025) 

Authors:
  1. M. S. A. Rahman
  2. H. M. Nor
  3. M. Z. Ahmad
  4. N. H. Abd Rahman
1
Institute of Engineering Mathematics, Pauh Putra Main Campus, Universiti Malaysia Perlis; Department of Computer and Mathematical Sciences, Universiti Teknologi MARA Cawangan Pulau Pinang
2
Institute of Engineering Mathematics, Pauh Putra Main Campus, Universiti Malaysia Perlis
3
Institute of Engineering Mathematics, Pauh Putra Main Campus, Universiti Malaysia Perlis
4
Department of Computer and Mathematical Sciences, Universiti Teknologi MARA Cawangan Pulau Pinang

This study explores the efficacy of the homotopy function, which combines the classical iterative technique with the homotopy continuation method for solving zeros of nonlinear equations and approximating their roots.  While numerous homotopy functions have been examined in previous research, this paper focuses on employing a cubic parameter as the homotopy function within the Newton-HCM framework.  Through a series of numerical examples, we demonstrate that our proposed homotopy function, named cubic parameter homotopy function, consistently outperforms existing alternatives in terms of accuracy and computational efficiency.  The results underscore the effectiveness of leveraging specific homotopy functions within the Newton-HCM approach, offering valuable insights for practitioners and researchers in nonlinear equation solving and computational mathematics.

homotopy continuation method
homotopy function
root-finding
Newton-HCM
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