Timeline for Newton Method in $p$-adic case
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10 events
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| Dec 9, 2017 at 9:05 | history | edited | YCor |
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| S Dec 9, 2017 at 8:48 | history | suggested | jeq | CC BY-SA 3.0 |
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| Dec 9, 2017 at 1:32 | review | Suggested edits | |||
| S Dec 9, 2017 at 8:48 | |||||
| May 23, 2017 at 17:52 | comment | added | Joe Silverman | @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.) | |
| Mar 12, 2015 at 18:59 | comment | added | KConrad | 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. | |
| Oct 30, 2012 at 2:51 | comment | added | Lubin | 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). | |
| Jan 18, 2012 at 21:13 | comment | added | Lubin | 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. | |
| Jan 18, 2012 at 11:08 | history | edited | user565739 | CC BY-SA 3.0 |
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| Jan 18, 2012 at 9:53 | history | edited | user565739 | CC BY-SA 3.0 |
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| Jan 18, 2012 at 7:53 | history | asked | user565739 | CC BY-SA 3.0 |