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Peer-reviewedCC BY 4.0PLOS One

Hybrid accelerated iterative scheme based on the Picard–Ishikawa approach for systems of nonlinear equations

  • Abubakar Sani Halilu1
  • Hatem E. Semary2
  • Mohamad Afendee Mohamed3
  • Asmaa S. Al-Moisheer4
  • Kabiru Ahmed5
  • Sulaiman M. Ibrahim6
  • Mohammed Yusuf Waziri7
  • Arunava Majumder
7 affiliations
  1. 1Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Campus Besut, Terengganu, Malaysia
  2. 2Department of Mathematics, Sule Lamido University Kafin Hausa, Kafin Hausa, Nigeria
  3. 3Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
  4. 4Department of Mathematics, Federal University, Dutse, Nigeria
  5. 5College of Applied and Health Sciences, A’Sharqiyah University, Ibra, Sultanate of Oman
  6. 6Department of Mathematical Sciences, Bayero University, Kano, Nigeria
  7. 7Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, Tamil Nadu, India
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Abstract

Systems of nonlinear equations arise in numerous scientific, engineering, and optimization applications, where efficient and reliable numerical methods are required to obtain accurate solutions. Hybridization techniques have recently attracted considerable attention because of their ability to improve the convergence behavior of iterative schemes. Motivated by this development, this paper proposes a novel hybrid accelerated iterative method for solving systems of nonlinear equations by integrating the Picard–Ishikawa process with an accelerated derivative-free framework. The proposed approach employs a diagonal approximation of the Jacobian matrix through an acceleration parameter, thereby reducing computational cost while maintaining numerical accuracy. The novelty of the method lies in combining the three-step Picard–Ishikawa process with an accelerated scheme to enhance convergence performance and robustness. Under standard assumptions, the global convergence of the proposed method is established. Extensive numerical experiments on a collection of benchmark problems demonstrate that the proposed method outperforms several existing approaches in terms of iteration count, computational time, and robustness. These results confirm the effectiveness and suitability of the proposed scheme for solving large-scale systems of nonlinear equations arising in practical applications.

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Cite

BibTeXRIS
BibTeX
@article{halilu2026hybrid,
  title = {{Hybrid accelerated iterative scheme based on the Picard–Ishikawa approach for systems of nonlinear equations}},
  author = {Abubakar Sani Halilu and Hatem E. Semary and Mohamad Afendee Mohamed and Asmaa S. Al-Moisheer and Kabiru Ahmed and Sulaiman M. Ibrahim and Mohammed Yusuf Waziri and Arunava Majumder},
  journal = {PLOS One},
  year = {2026},
  month = sep,
  doi = {10.1371/journal.pone.0355734},
  url = {https://doi.org/10.1371/journal.pone.0355734},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Hybrid accelerated iterative scheme based on the Picard–Ishikawa approach for systems of nonlinear equations
AU  - Abubakar Sani Halilu
AU  - Hatem E. Semary
AU  - Mohamad Afendee Mohamed
AU  - Asmaa S. Al-Moisheer
AU  - Kabiru Ahmed
AU  - Sulaiman M. Ibrahim
AU  - Mohammed Yusuf Waziri
AU  - Arunava Majumder
PY  - 2026
DA  - 2026/09/09
JO  - PLOS One
DO  - 10.1371/journal.pone.0355734
UR  - https://doi.org/10.1371/journal.pone.0355734
AB  - Systems of nonlinear equations arise in numerous scientific, engineering, and optimization applications, where efficient and reliable numerical methods are required to obtain accurate solutions. Hybridization techniques have recently attracted considerable attention because of their ability to improve the convergence behavior of iterative schemes. Motivated by this development, this paper proposes a novel hybrid accelerated iterative method for solving systems of nonlinear equations by integrating the Picard–Ishikawa process with an accelerated derivative-free framework. The proposed approach employs a diagonal approximation of the Jacobian matrix through an acceleration parameter, thereby reducing computational cost while maintaining numerical accuracy. The novelty of the method lies in combining the three-step Picard–Ishikawa process with an accelerated scheme to enhance convergence performance and robustness. Under standard assumptions, the global convergence of the proposed method is established. Extensive numerical experiments on a collection of benchmark problems demonstrate that the proposed method outperforms several existing approaches in terms of iteration count, computational time, and robustness. These results confirm the effectiveness and suitability of the proposed scheme for solving large-scale systems of nonlinear equations arising in practical applications.
SN  - 1932-6203
ER  -