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Let's a nonlinear function $ f:[-1,+1]^N\subset\mathbb{R}^N\to\mathbb{R}^N,\; N\in\mathbb{N}, $ such that the the sequence generated by the method of Newton-Raphson $$ x_{n+1}=x_n-[Df(x_n)]^{-1}\cdot f(x_n),\qquad Df(x_n)=\left(\frac{\partial f_j(x_n)}{\partial e_i} \right)_{n\times n} $$ converges to the same limit $\displaystyle\lim_{n\to \infty}x_n$ which independ the chosen starting point $x_0\in[-1,1]^N$ . A sufficient condition for convergence is that $ f $ is of class $ C^2 $, strictly convex and $ f(x) $ assume positive and negative values for $N=1$​​.

Question. Is there a constructive sequence of polynomials $ p_n:[-1,+1]^N\subset\mathbb{R}^N\to\mathbb{R}^N $ such that such that the the sequence generated by the method of Newton-Raphson $$ y_{n+1}=y_n-[Dp_n(y_n)]^{-1}\cdot p_n(y_n) $$ converges to the same limit $\displaystyle\lim_{n\to \infty}y_n=\displaystyle\lim_{n\to \infty}x_n$ which is independent of the chosen starting point $y_0=x_0\in[-1,1]^N$ and $\displaystyle\lim_{n\to\infty}\sup_{u\in [-1,+1]^N} |f(u)-p_n(u)|=0$?

Solutions for $ N = 1 $ or $ N = 2 $ are welcome.

Update: My attempts, at least for the case $N = 1$ has been the use of Bernstein polynomials and theorems permutation of iterated limits. But I'm not sure if my efforts will be fruitful.

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You know of the Newton-Kantorovich theorem? –  LutzL Mar 11 at 16:40
    
@LutzL I know the Newton-Kantorovich theorem and its several versions. The point is that this question calls for a robust version of the Kantorovich theorem. And not only robust robust locally but globally robust. –  Elias Mar 11 at 22:22
    
It is still not clear what you are asking. Are the $p_n$ perturbations of some "exact" $p$ or totally unrelated? The claim of the Kantorovich theorem includes that, in the notation of the wikipedia article (which is taken from the Ortega article), all initial points in $B(x_0, t^*)$ give quadratic convergence, and it was either $B(x_0,t^{**})$ or with the radius in the middle $B(x_0,(t^*+t^{**})/2)$ where initial points lead to at least linear convergence. These claims can be found in one of the article of Gragg/Tapia or the book of Ortega/Rheinboldt. –  LutzL Mar 11 at 22:50

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