Editing Newton's method (section)

===Complex functions===
{{main|Newton fractal}}
[[Image:newtroot 1 0 0 0 0 m1.png|thumb|Basins of attraction for {{math|{{var|x}}{{sup|5}} − 1 {{=}} 0}}; darker means more iterations to converge.]]
When dealing with [[complex analysis|complex functions]], Newton's method can be directly applied to find their zeroes.<ref>{{cite book |last=Henrici|author-link=Peter Henrici (mathematician)|first=Peter |title= Applied and Computational Complex Analysis |volume=1 |date=1974 |publisher=Wiley | isbn =9780471598923  }}</ref> Each zero has a [[basin of attraction]] in the complex plane, the set of all starting values that cause the method to converge to that particular zero. These sets can be mapped as in the image shown. For many complex functions, the boundaries of the basins of attraction are [[fractal]]s.

In some cases there are regions in the complex plane which are not in any of these basins of attraction, meaning the iterates do not converge. For example,<ref>{{cite journal|last=Strang |first=Gilbert |title=A chaotic search for {{mvar|i}} |journal=[[The College Mathematics Journal]] |volume=22 |date=January 1991 |issue=1 |pages=3–12 |doi=10.2307/2686733|jstor=2686733 }}</ref> if one uses a real initial condition to seek a root of {{math|{{var|x}}{{sup|2}} + 1}}, all subsequent iterates will be real numbers and so the iterations cannot converge to either root, since both roots are non-real. In this case [[almost all]] real initial conditions lead to [[chaos theory|chaotic behavior]], while some initial conditions iterate either to infinity or to repeating cycles of any finite length.

Curt McMullen has shown that for any possible purely iterative algorithm similar to Newton's method, the algorithm will diverge on some open regions of the complex plane when applied to some polynomial of degree 4 or higher. However, McMullen gave a generally convergent algorithm for polynomials of degree 3.<ref>{{cite journal|last=McMullen |first=Curt |title=Families of rational maps and iterative root-finding algorithms |journal=Annals of Mathematics |series=Second Series |volume=125 |date=1987 |issue=3 |pages=467–493 |doi=10.2307/1971408|jstor=1971408 |url=https://dash.harvard.edu/bitstream/handle/1/9876064/McMullen_FamiliesRationalMap.pdf?sequence=1 }}</ref> Also, for any polynomial, Hubbard, Schleicher, and Sutherland gave a method for selecting a set of initial points such that  Newton's method will certainly converge at one of them at least.<ref>{{Cite journal |last1=Hubbard |first1=John |last2=Schleicher |first2=Dierk |last3=Sutherland |first3=Scott |date=October 2001 |title=How to find all roots of complex polynomials by Newton's method |url=http://dx.doi.org/10.1007/s002220100149 |journal=Inventiones Mathematicae |volume=146 |issue=1 |pages=1–33 |doi=10.1007/s002220100149 |bibcode=2001InMat.146....1H |s2cid=12603806 |issn=0020-9910}}</ref>