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The method for roots of a scalar function $f$ alternates Newton steps ($x_k\mapsto z_{k+1}$) and simplified Newton steps ($z_{k+1}\mapsto x_{k+1}$).

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I think I need to Taylor expand all the f, but I do not know how to deal with the $f(z_{k+1})$ term. Seems like Taylor expanding that is very hard. Any help is appreciated.

Lutz Lehmann
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  • Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures. – José Carlos Santos Sep 17 '20 at 06:06
  • There's something odd about the question. What if $f$ has two (or more) roots. How do the $x_k$ know which one they are supposed to converge to? – Gerry Myerson Sep 17 '20 at 07:38
  • @GerryMyerson : When asking for the order, the assumption usually is that the sequence in consideration already converges to a root. Asking which initial points lead to convergence is a different, much more difficult question. – Lutz Lehmann Sep 17 '20 at 07:46
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    Ups, looks like I got one-shot duplicates for [numerical-methods] too. Look at the link and report back if you think your question is different, I'll also vote for re-opening. – Lutz Lehmann Sep 17 '20 at 07:50

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