I am trying to find preimage of a kth iterated point under Newton method. Is it possible to find an initial point from which the kth iterated is derived?

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    $\begingroup$ The preimages need not be unique points. As a simple example consider finding $x$ to minimise $x^2$. The first iterate from any starting point $x_0$ is $x_0-2x_0/2=0$, so even the first preimage is the whole real line. $\endgroup$ – Thomas Lumley Sep 12 '20 at 3:46
  • $\begingroup$ Yea pre-image is not unique due to that a map defined by jacobean of f (i.e. $x_k-s_k; s_k$ is a solution of linear system $f(x_k)+D(x_k)s_k=0$) is a multi-valued function. So, is it possible to define branch wise inverse of the Newton map? $\endgroup$ – Lakshman Mahto Sep 25 '20 at 11:30
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    $\begingroup$ In principle, but it sounds like no fun at all in practice, and numerically ill-conditioned on top of that. $\endgroup$ – Thomas Lumley Sep 28 '20 at 6:36

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