Editing Newton's method (section)

===Chebyshev's third-order method===
{{Main|Chebyshev iteration}}Since higher-order Taylor expansions offer more accurate local approximations of a function {{mvar|f}}, it is reasonable to ask why Newton’s method relies only on a second-order Taylor approximation. In the 19th century, Russian mathematician Pafnuty Chebyshev explored this idea by developing a variant of Newton’s method that used cubic approximations.<ref>{{cite book |last1=Chebyshev |first1=Pafnutii L'vovich |title=Polnoe sobranie sochinenii |last2=Bernshtein |first2=Sergei Natanovich |publisher=Izd-vo Akademii nauk SSSR |year=1947}}</ref><ref>{{Cite journal |last1=Ahmadi |first1=Amir Ali |last2=Chaudhry |first2=Abraar |last3=Zhang |first3=Jeffrey |date=2024 |title=Higher-order Newton methods with polynomial work per iteration |url=https://linkinghub.elsevier.com/retrieve/pii/S0001870824003232 |journal=Advances in Mathematics |language=en |volume=452 |pages=109808 |arxiv=2311.06374 |doi=10.1016/j.aim.2024.109808}}</ref><ref>{{Cite web |last=Hartnett |first=Kevin |date=2025-03-24 |title=Three Hundred Years Later, a Tool from Isaac Newton Gets an Update |url=https://www.quantamagazine.org/three-hundred-years-later-a-tool-from-isaac-newton-gets-an-update-20250324/ |access-date=2025-04-03 |website=Quanta Magazine |language=en}}</ref>