Editing Newton's method (section)

===In a Banach space===
Another generalization is Newton's method to find a root of a [[Functional (mathematics)|functional]] {{mvar|F}} defined in a [[Banach space]]. In this case the formulation is

<math display="block">X_{n+1}=X_n-\bigl(F'(X_n)\bigr)^{-1}F(X_n),\,</math>

where {{math|{{var|{{prime|F}}}}({{var|X}}{{sub|{{var|n}}}})}} is the [[Fréchet derivative]] computed at {{math|{{var|X}}{{sub|{{var|n}}}}}}. One needs the Fréchet derivative to be boundedly invertible at each {{math|{{var|X}}{{sub|{{var|n}}}}}} in order for the method to be applicable. A condition for existence of and convergence to a root is given by the [[Kantorovich theorem|Newton–Kantorovich theorem]].<ref>{{cite book |first=Tetsuro |last=Yamamoto |chapter=Historical Developments in Convergence Analysis for Newton's and Newton-like Methods |pages=241–263 |editor-first=C. |editor-last=Brezinski |editor2-first=L. |editor2-last=Wuytack |title=Numerical Analysis: Historical Developments in the 20th Century |publisher=North-Holland |year=2001 |isbn=0-444-50617-9 }}</ref>

====Nash–Moser iteration====
{{details|Nash–Moser theorem}}
In the 1950s, [[John Forbes Nash Jr.|John Nash]] developed a version of the Newton's method to apply to the problem of constructing [[isometric embedding]]s of general [[Riemannian manifold]]s in [[Euclidean space]]. The ''loss of derivatives'' problem, present in this context, made the standard Newton iteration inapplicable, since it could not be continued indefinitely (much less converge). Nash's solution involved the introduction of [[smoothing]] operators into the iteration. He was able to prove the convergence of his smoothed Newton method, for the purpose of proving an [[implicit function theorem]] for isometric embeddings. In the 1960s, [[Jürgen Moser]] showed that Nash's methods were flexible enough to apply to problems beyond isometric embedding, particularly in [[celestial mechanics]]. Since then, a number of mathematicians, including [[Mikhael Gromov (mathematician)|Mikhael Gromov]] and [[Richard S. Hamilton|Richard Hamilton]], have found generalized abstract versions of the Nash–Moser theory.<ref>{{cite journal|first=Richard S.|last=Hamilton|mr=0656198|title=The inverse function theorem of Nash and Moser|journal=[[Bulletin of the American Mathematical Society]] |series=New Series |volume=7|year=1982|issue=1|pages=65–222|doi-access=free|doi=10.1090/s0273-0979-1982-15004-2|zbl=0499.58003|author-link1=Richard S. Hamilton}}</ref><ref>{{cite book|last1=Gromov|first1=Mikhael|title=Partial differential relations|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3)|volume=9|publisher=[[Springer-Verlag]]|location=Berlin|year=1986|isbn=3-540-12177-3|mr=0864505|author-link1=Mikhael Gromov (mathematician)|doi=10.1007/978-3-662-02267-2}}</ref> In Hamilton's formulation, the Nash–Moser theorem forms a generalization of the Banach space Newton method which takes place in certain [[Fréchet space]]s.