Simultaneous Global Convergence Failure of Newton's and Halley's Methods in Degree Four
Abstract
We construct an explicit quartic polynomial for which Newton's and Halley's root-finding methods both fail to converge to a root on nonempty open sets of initial values. The example can be found through a reproducible computational search within a one-parameter family of quartics designed to exhibit a prescribed Newton two-cycle. We then prove analytically that Newton's method has a superattracting two-cycle and that Halley's method has a distinct attracting extraneous two-cycle, using a two-interval contraction argument. This fills a degree-four gap in the simultaneous Newton--Halley failure problem. The computational approach also identifies another quartic with numerical evidence of simultaneous failure of Newton's, Halley's, and Schröder's methods.
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BibTeX
@misc{basti2026simultaneous,
title = {{Simultaneous Global Convergence Failure of Newton's and Halley's Methods in Degree Four}},
author = {Alessio Basti and Tommaso Cremaschi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03246},
eprint = {2610.03246},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03246},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Simultaneous Global Convergence Failure of Newton's and Halley's Methods in Degree Four
AU - Alessio Basti
AU - Tommaso Cremaschi
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03246
UR - https://arxiv.org/abs/2610.03246
AB - We construct an explicit quartic polynomial for which Newton's and Halley's root-finding methods both fail to converge to a root on nonempty open sets of initial values. The example can be found through a reproducible computational search within a one-parameter family of quartics designed to exhibit a prescribed Newton two-cycle. We then prove analytically that Newton's method has a superattracting two-cycle and that Halley's method has a distinct attracting extraneous two-cycle, using a two-interval contraction argument. This fills a degree-four gap in the simultaneous Newton--Halley failure problem. The computational approach also identifies another quartic with numerical evidence of simultaneous failure of Newton's, Halley's, and Schröder's methods.
ER -