Editing Newton's method (section)

==History==
In the [[Babylonian mathematics|Old Babylonian]] period (19th–16th century BCE), the side of a square of known area could be effectively approximated, and this is conjectured to have been done using a special case of Newton's method, [[#Use of Newton's method to compute square roots|described algebraically below]], by iteratively improving an initial estimate; an equivalent method can be found in [[Heron of Alexandria]]'s ''Metrica'' (1st–2nd century CE), so is often called ''[[Heron's method]]''.<ref>{{cite journal |last1=Fowler |first1=David |last2=Robson |first2=Eleanor |year=1998 |title=Square root approximations in Old Babylonian mathematics: YBC 7289 in context |journal=Historia Mathematica |volume=25 |number=4 |pages=366–378 |doi=10.1006/hmat.1998.2209 |doi-access=free}}</ref> [[Jamshīd al-Kāshī]] used a method to solve {{math|{{var|x}}{{sup|{{var|P}}}} − {{var|N}} {{=}} 0}} to find roots of ''{{mvar|N}}'', a method that was algebraically equivalent to Newton's method, and in which a similar method was found in ''Trigonometria Britannica'', published by [[Henry Briggs (mathematician)|Henry Briggs]] in 1633.<ref>{{Cite journal |last=Ypma |first=Tjalling J. |date=1995 |title=Historical Development of the Newton-Raphson Method |url=https://www.jstor.org/stable/2132904 |journal=SIAM Review |volume=37 |issue=4 |pages=531–551 |doi=10.1137/1037125 |issn=0036-1445 |jstor=2132904}}</ref>

The method first appeared roughly in [[Isaac Newton]]'s work in ''[[De analysi per aequationes numero terminorum infinitas]]'' (written in 1669, published in 1711 by [[William Jones (mathematician)|William Jones]]) and in ''De metodis fluxionum et serierum infinitarum'' (written in 1671, translated and published as ''[[Method of Fluxions]]'' in 1736 by [[John Colson]]).<ref name=":0" /><ref>{{Cite book |last=Guicciardini |first=Niccolò |url=https://books.google.com/books?id=uqnuDwAAQBAJ&pg=PA158 |title=Isaac Newton on Mathematical Certainty and Method |date=2009 |publisher=[[MIT Press]] |isbn=978-0-262-01317-8 |series=Transformations |location=Cambridge, Mass |pages=158–159 |language=en |oclc=282968643}}</ref> However, while Newton gave the basic ideas, his method differs from the modern method given above. He applied the method only to polynomials, starting with an initial root estimate and extracting a sequence of error corrections. He used each correction to rewrite the polynomial in terms of the remaining error, and then solved for a new correction by neglecting higher-degree terms. He did not explicitly connect the method with derivatives or present a general formula. Newton applied this method to both numerical and algebraic problems, producing [[Taylor series]] in the latter case.

Newton may have derived his method from a similar, less precise method by mathematician [[François Viète]], however, the two methods are not the same.<ref name=":0">{{Cite journal |last=Cajori |first=Florian |date=1911 |title=Historical Note on the Newton-Raphson Method of Approximation |url=https://www.jstor.org/stable/2973939 |journal=The American Mathematical Monthly |volume=18 |issue=2 |pages=29–32 |doi=10.2307/2973939 |jstor=2973939 |issn=0002-9890}}</ref> The essence of Viète's own method can be found in the work of the mathematician [[Sharaf al-Din al-Tusi]].<ref>{{Cite journal |last=Ypma |first=Tjalling J. |date=1995 |title=Historical Development of the Newton-Raphson Method |url=https://www.jstor.org/stable/2132904 |journal=SIAM Review |volume=37 |issue=4 |pages=531–551 |doi=10.1137/1037125 |jstor=2132904 |issn=0036-1445}}</ref>

The Japanese mathematician [[Seki Kōwa]] used a form of Newton's method in the 1680s to solve single-variable equations, though the connection with calculus was missing.<ref>{{Cite web |title=Takakazu Seki - Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Seki/#:~:text=In%201685,%20he%20solved%20the,b%20a,b%20are%20integers. |access-date=2024-11-27 |website=Maths History |language=en}}</ref>

Newton's method was first published in 1685 in ''A Treatise of Algebra both Historical and Practical'' by [[John Wallis]].<ref>{{cite book |first=John |last=Wallis |author-link=John Wallis |title=A Treatise of Algebra, both Historical and Practical |publisher=Richard Davis |location=Oxford |date=1685 |url=http://www.e-rara.ch/zut/content/titleinfo/2507537 |doi=10.3931/e-rara-8842}}</ref> In 1690, [[Joseph Raphson]] published a simplified description in ''Analysis aequationum universalis''.<ref>{{cite book |last=Raphson |first=Joseph |author-link=Joseph Raphson |title=Analysis Æequationum Universalis |edition=2nd |language=la |date=1697 |publisher=Thomas Bradyll |location=London |url=https://archive.org/details/bub_gb_4nlbAAAAQAAJ |doi=10.3931/e-rara-13516}}</ref> Raphson also applied the method only to polynomials, but he avoided Newton's tedious rewriting process by extracting each successive correction from the original polynomial. This allowed him to derive a reusable iterative expression for each problem. Finally, in 1740, [[Thomas Simpson]] described Newton's method as an iterative method for solving general nonlinear equations using calculus, essentially giving the description above. In the same publication, Simpson also gives the generalization to systems of two equations and notes that Newton's method can be used for solving optimization problems by setting the gradient to zero.

[[Arthur Cayley]] in 1879 in ''The Newton–Fourier imaginary problem'' was the first to notice the difficulties in generalizing Newton's method to complex roots of polynomials with degree greater than 2 and complex initial values. This opened the way to the study of the [[Julia set|theory of iterations]] of rational functions.