Third-order Halley-type iterative method with positive and bounded correction function

Abstract

In this paper, we propose a new one-point iterative method for finding simple roots of nonlinear equations by modifying Halley's method. Its correction function is positive and bounded on ℝ and matches the local expansion of Halley's correction function near a simple root. The method avoids the singularity and sign reversal of Halley's correction function while retaining at least third-order local convergence. We also establish sufficient conditions for global convergence. Numerical experiments show that the proposed method achieves high convergence success rates and favorable iteration counts over a wide range of initial guesses. Further experiments near the singularity of Halley's correction function demonstrate robust convergence behavior of the proposed method. An application to the van der Waals equation illustrates the effectiveness of the proposed method.

Iterative Bounded Distance Decoding With 2-Bit Reliability Marker For Product-Like Codes
Xiaowei Wu, Min Qiu, Yueying Zhan +1
arXiv15 Oct 2026PreprintarXiv licenceFull textGeneral mathematicsarXiv
ICL-SEC: Iterative Cross-Layer Semantic Error Correction
Yirun Wang, Soung Chang Liew, Yuyang Du
arXiv19 Sep 2026PreprintarXiv licenceFull textGeneral mathematicsarXiv
Progressive Reliability-Guided Refinement with Cross-Modal Consistency and Iterative Correction for Weakly Supervised PET/CT Tumor Segmentation
Zhaoshuo Diao, Huiyan Jiang
Mathematics26 Oct 2026Peer-reviewedCC BY 4.0General mathematicsCrossref
Adaptive Reliability-Aware Nonlinear State Estimation Framework for GNSS/INS Integrated Navigation with Bounded Position Correction
Shizhong Yang, Zhen Liu, Liuyang Jiang +1
Mathematics10 Oct 2026Peer-reviewedCC BY 4.0General mathematicsCrossref
Positive strongly Kreiss bounded operators on L^p-spaces have logarithmic power growth
Loris Arnold
arXiv1 Sep 2026PreprintCC BY 4.0Full textAnalysis & PDEsarXiv

Cite

BibTeX RIS

BibTeX

@misc{mukasa2026thirdorder,
  title = {{Third-order Halley-type iterative method with positive and bounded correction function}},
  author = {Hirai Mukasa},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08110},
  eprint = {2609.08110},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08110},
}

RIS

TY  - UNPB
TI  - Third-order Halley-type iterative method with positive and bounded correction function
AU  - Hirai Mukasa
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08110
UR  - https://arxiv.org/abs/2609.08110
AB  - In this paper, we propose a new one-point iterative method for finding simple roots of nonlinear equations by modifying Halley's method. Its correction function is positive and bounded on ℝ and matches the local expansion of Halley's correction function near a simple root. The method avoids the singularity and sign reversal of Halley's correction function while retaining at least third-order local convergence. We also establish sufficient conditions for global convergence. Numerical experiments show that the proposed method achieves high convergence success rates and favorable iteration counts over a wide range of initial guesses. Further experiments near the singularity of Halley's correction function demonstrate robust convergence behavior of the proposed method. An application to the van der Waals equation illustrates the effectiveness of the proposed method.
ER  -

Abstract

Related

Cite