Highly efficient family of two-step simultaneous method for all polynomial roots

Shams M., KAUSAR N., Araci S., Kong L., Carpentieri B.

AIMS Mathematics, vol.9, no.1, pp.1755-1771, 2024 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 9 Issue: 1
  • Publication Date: 2024
  • Doi Number: 10.3934/math.2024085
  • Journal Name: AIMS Mathematics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Directory of Open Access Journals
  • Page Numbers: pp.1755-1771
  • Keywords: CPU-time, fractals, iterative methods, numerical algorithm, polynomial equations
  • Yıldız Technical University Affiliated: Yes


In this article, we constructed a derivative-free family of iterative techniques for extracting simultaneously all the distinct roots of a non-linear polynomial equation. Convergence analysis is discussed to show that the proposed family of iterative method has fifth order convergence. Nonlinear test models including fractional conversion, predator-prey, chemical reactor and beam designing models are included. Also many other interesting results concerning symmetric problems with application of group symmetry are also described. The simultaneous iterative scheme is applied starting with the initial estimates to get the exact roots within the given tolerance. The proposed iterative scheme requires less function evaluations and computation time as compared to existing classical methods. Dynamical planes are exhibited in CAS-MATLAB (R2011B) to show how the simultaneous iterative approach outperforms single roots finding methods that might confine the divergence zone in terms of global convergence. Furthermore, convergence domains, namely basins of attraction that are symmetrical through fractal-like edges, are analyzed using the graphical tool. Numerical results and residual graphs are presented in detail for the simultaneous iterative method. An extensive study has been made for the newly developed simultaneous iterative scheme, which is found to be efficient, robust and authentic in its domain.