Editing Newton's method (section)

====Maehly's procedure====
A nonlinear equation has multiple solutions in general. But if the initial value is not appropriate, Newton's method may not converge to the desired solution or may converge to the same solution found earlier. When we have already found {{mvar|N}} solutions of <math>f(x)=0</math>, then the next root can be found by applying Newton's method to the next equation:<ref>{{harvnb|Press|Teukolsky|Vetterling|Flannery|2007}}</ref><ref>{{cite book | url=https://archive.org/details/introductiontonu0000stoe/mode/2up | page=279 | last1=Stoer|last2=Bulirsch|date=1980| first1= Josef | first2= Roland | oclc= 1244842246 | title= Introduction to numerical analysis | url-access= registration}}</ref>

<math display="block">F(x) = \frac{f(x)}{\prod_{i=1}^N(x-x_i)} = 0 .</math>

This method is applied to obtain zeros of the [[Bessel function]] of the second kind.<ref>{{cite book <!-- Citation bot: don't add identifiers. there is a bogus doi at the s2cid for this title that actually points to a review of the book, and i know from experience that bots will conflate the two and bugger up the reference -->| last1= Zhang | first1= Shanjie | last2= Jin | first2= Jianming | date= 1996 | publisher= Wiley | title= Computation of Special Functions|isbn=9780471119630}}{{page needed |date=June 2024}}</ref>