# Inexact Newton method.

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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