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Is there a general form of equations where we can say Newton's method does not converge?

Also, can we find values where Newton's method does not converge if we guess those points for an equation?

Mathrix
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  • Generally speaking the answer to both of your questions is no. The structural behavior of Newton's method as a dynamical system is often quite complicated. – Ian Dec 03 '19 at 18:14
  • If you work in the complex plane, you can make wild plots showing which starting points converge to which roots - at the boundary of these regions, the method fails to converge. These regions are pretty wild, which is good evidence that it'd be hard to write any satisfying formula. (See, for instance, Google image results for this) – Milo Brandt Dec 03 '19 at 18:16
  • If you do not have any assumption on you objective $f$, then in principle you can pretty much design your function in any way, so that if you have $x_{t+1} = x_{t} - \Delta_{t}$, and $\Delta_{t}$ is from a Newton step, then just pick the desired values for $f(x_t)$ and the gradients and Hessians accordingly to get what you want. So you can make the function $f$ even getting larger at each next $f(x_{t+1})$, by selecting just the right values that lead Newton method a stray. Of cause, this depends on you knowing the starting $x_0$, if it is random then this approach would not work. – them Dec 03 '19 at 18:16
  • See https://math.stackexchange.com/a/3455757/589 – lhf Dec 03 '19 at 18:19
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    What do you mean with "does not converge"? If the function is differentiable and the derivative is Lipschitz, then around every simple root there is some small ball where the Newton method converges quadratically. It will need some effort to negate at least one of these assumptions with computer generable functions, you will need fractal functions at least if you want some general situation. But for instance it is known that the signed root functions $f(x)={\rm sign}(x)\sqrt[k]{|x|}$ are usually very bad for Newton's method. – Lutz Lehmann Dec 03 '19 at 22:51

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