1
$\begingroup$

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?

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ 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
    Commented Sep 12, 2020 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
    Commented Sep 25, 2020 at 11:30
  • 1
    $\begingroup$ In principle, but it sounds like no fun at all in practice, and numerically ill-conditioned on top of that. $\endgroup$
    – Thomas Lumley
    Commented Sep 28, 2020 at 6:36

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.