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The Newton Method over $\mathbb{R}$ has the property that the precision is doubled (under some continuous differentiable assumption) in each iteration. For the ring $\mathbb{Z}_p$ of $p$-adic integers, we also have the Hensel's lemma, but the precision is increased by 1 in each iteration. The proof about the double precision in the case over $\mathbb{R}$ uses the Taylor expansion and its remainder, see here (Wayback Machine).

In a special case that is finding square root, one can use $x_{n+1} := \frac{1}{2} x_n - \frac{3}{2} x_n^3 $ ( by considering $x^{-2}-c^{-1}$ instead of $x^2-c$) and one can show that the precision is doubled in the case over $\mathbb{Z}_p$.

In general, is the precision is doubled in each iteration in the case over $\mathbb{Z}_p$, under some conditions?

Sorry, I just found that in fact the precision is doubled in $p$-adic case, without further assumptions. I mentioned the precision increased by 1, since this is written in "A first course in $p$-adic Analysis" by Alain M. Robert.

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But if one examines the proof, we have $p(\hat{x}) \equiv 0 \ \mathrm{mod} \ p^{n-2k}$. And if $k = 0$, the precision is doubled.

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    $\begingroup$ If I remembered the reference, I would make this an answer. I think it was in a text by Ostrovsky that I saw a proof of Hensel's Lemma that proceeds from a congruence modulo $I$ to a congruence mod $I^2$. I managed to reconstruct for myself what I thought the proof was, but the method is interesting only theoretically, being no good for explicit computation. Maybe someone better informed in the literature than I am can give the reference. $\endgroup$
    – Lubin
    Commented Jan 18, 2012 at 21:13
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    $\begingroup$ A bit late to add to the above comment, but just what “Hensel’s Lemma” refers to, is in a state of confusion. For me, H’s L relates not at all to finding the root of a polynomial but rather to lifting a characteristic-$p$ factorization back to characteristic zero (or the appropriate generalization in the equal-characteristic case). $\endgroup$
    – Lubin
    Commented Oct 30, 2012 at 2:51
  • $\begingroup$ Hensel's lemma can be proved by Newton's method or by the contraction mapping theorem. See Theorem 4.1 in math.uconn.edu/~kconrad/blurbs/gradnumthy/hensel.pdf with different proofs in Sections 5 and 6. In the first proof, the inductively proved bound $|f(a_n)|_p \leq |f'(a_1)|_p^2t^{2^{n-1}}$ where $t = |f(a)/f'(a)^2|_p$ explains why Newton's method doubles the number of correct digits at each step. See Examples 5.1 and 5.2 for explicit doubling, and in Example 6.1 is a comparison of doubling in Newton's method with linear improvements by the contraction mapping iteration. $\endgroup$
    – KConrad
    Commented Mar 12, 2015 at 18:59
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    $\begingroup$ @KConrad I agree that the formula visually proves why the iteration doubles the number of correct digits, but I would say that the explanation is that the Newton formula $F_f: x \to x - f(x)/f'(x)$ turns the zeros of $f$ into ramified fixed points of the function $F_f(x)$. Basic dynamics then explains why points in a neighborhood of a ramified fixed point end up approaching the fixed point at this rate. In other words, the convergence isn't really a special property of Newton iteration. (I realize this is philosophy, not mathematics.) $\endgroup$
    – Joe Silverman
    Commented May 23, 2017 at 17:52

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