Talk:Newton's method

message: I'm a little skeptical of two of the examples in Newton's Method#Failure analysis section.
# The example in the "discontinuous derivative" section. I agree with all of it except the key phrase "and Newton's method will diverge almost everywhere in any neighborhood of it," which seems by no means clearly established. The unboundedness of f(x)/f'(x) near the root seems to be, in principle, perfectly compatible with many/most Newton iterations converging to the root. A simulation with randomly chosen initial points seems to back this up.
# The second half of the "zero derivative" section. What is the meaning of "nearly double" in this context; does it mean that the root is close to a point where the derivative vanishes? Six also seems like a very moderate number of iterations to require. In all, I just don't know what this example is trying to illustrate.
Could anyone clarify or provide a relevant reference? If not, I think these examples should be removed.
Gumshoe2 (talk) 00:06, 23 April 2024 (UTC)
replies: text: Also, although I don't have any issue with the "no second derivative" section, I haven't been able to find any standard textbook reference where this example or a similar one is given. Does anyone here know one? Gumshoe2 (talk) 00:12, 23 April 2024 (UTC)

text: Since nobody has spoken up, I've removed these examples. I've also combined (and somewhat rewrote) the three different example sections into a single one, which I think is helpful for reading. Gumshoe2 (talk) 14:19, 28 April 2024 (UTC)
thread_title:  Questionable examples 

message: Does anyone think it's necessary to give a full proof of this as in the present version? It's possible to summarize it reasonably and convincingly in about two lines, and there are plenty of standard books to reference where the full proof appears. Gumshoe2 (talk) 14:28, 28 April 2024 (UTC)
replies: text: I agree. Also, the preceding subsection is also confusing because of too much technical details. Imo, one must reduce the proofs and explanations to functions of class C2, with the simple mention that there are generalizations to slighty larger classes of functions. D.Lazard (talk) 16:12, 28 April 2024 (UTC)
thread_title:  Proof of quadratic convergence for Newton's iterative method 

message: The following paragraph was added in 2016:
:A large error in the initial estimate can contribute to non-convergence of the algorithm. To overcome this problem one can often linearise the function that is being optimized using calculus, logs, differentials, or even using evolutionary algorithms, such as the Stochastic Funnel Algorithm.  Good initial estimates lie close to the final globally optimal parameter estimate.  In Nonlinear Regression the SSE equation is only "close to" parabolic in the region of the final parameter estimates.  Initial estimates found here will allow the Newton-Raphson method to quickly converge. It is only here that the Hessian of the SSE is positive and the first derivative of the SSE is close to zero.
It is both very unclear and, based on the username of the editor, seems to be added as an advertisement for their own research. For now I've removed it, possibly someone can salvage quality material from it. Gumshoe2 (talk) 15:09, 28 April 2024 (UTC)
replies: text: Remove this paragraph: if the initial estimate induce non-convergence or convergence to a wrong local optimum, the best method if to choose another initial estimate. It is unclear whether this paragraph is about the choice of a better initial estimate. On the opposite, is seems to suggest to replace the function with an approximation of it, which is a very bad idea, as it may change the optimum dramatically. Moreover, the choice a good initial estimate depends on the nature of the problem, and I doubt that there exist general methods for that, that are better than a random choice. D.Lazard (talk) 15:48, 28 April 2024 (UTC)
thread_title:  "Poor initial estimate"