The reason of newton's method's error having quadratic decrease

[Mahdi1998]

I am learning single variable calculus using MIT OCW's 18.01 course taught by Prof. David Jerison and I can't understand something on lecture 14.
Here is the link:
MIT OCW - Single Variable Calculus (18.01) taught by Prof. David Jerison - Lecture 14

In 5:37, Prof David Jerison says: "the question is how much shorter is E2 than E1?" and then talks about the vertical, horizontal and perpendicular distances between the tangent line and point (x_1, 0) being almost the same in addition to the situation where the tangent line touching the curve. Then, he concludes these things cause the separation (E_2) to be quadratic but I can't understand how he made that conclusion.

I have been trying very hard to understand the relation and even asked the ChatGPT to clarify the problem for me. But I still don't know how this conclusion has been reached.

I know algebraically that why E_2 is almost equal to (E_1)^2 but I am struggling to understand how Prof. Jerison makes this conclusion geometrically.
Any help would heavily be appreciated since this problem has taken me many hours and I still don't have the answer. I am literally struggling ...

 

[blamocur]

I looked at the segment around 5:37 and did not find any "geometrical" explanation for the quadratic order of magnitude for [imath]E_2[/imath]. What do you mean that you know this "algebrically", and what do you consider "geometrical conclusion"?


It seems that the professor simply states that for any smooth curve the distance from the tangent is approximately quadratic for small deviations -- do you find this reasonable?

 

[Mahdi1998]

I meant this proof as an algebraic way to explain E_2 being almost equal to (E_1)^2:



(Image has been taken from 5:02 video at MIT Open Library, section 6: Convergence Rate)

And to answer your question of E_2 being quadratic is reasonable or not, yes visually I find it reasonable that at each step the error is getting smaller in quadratic way.

I understand above proof which I called the algebraic proof totally but in my opinion Prof. Jerison is proving the quadratic decrease in error in some other way.
Here is the transcript of the video at the part in which he's explaining the geometric proof:

"So the first distance, again, is E_1, is this distance here. That's the E_1. And then this shorter
distance, here, this little bit, which I'll mark maybe in green, is E_2. So how much shorter is
E_1 than E_2? Well, the idea is pretty simple. It's that if this distance and this vertical distance,
they are probably about the same as the perpendicular distance. And this is basically the
situation of a curve touching a tangent line. Then the separation is going to be quadratic."

As far as I can understand he is saying since the vertical, horizontal and perpendicular distances on the diagram are nearly the same and the situation of curve touching the tangent line, we can say the separation (E_2) is quadratic. How does the fact of those 3 distance being nearly the same relate to E_2 being quadratic ?

 

[blamocur]

I looked at it again, and I am sure now there is no "geometric" proof there. Instead, I see this as an illustration of the "algebraic" stuff. I.e., this is meant to help with understanding, not to provide another proof.

BTW, when he says "the same" he means "same order of magnitude", and the goal is to show that the error of each step of the Newton algorithm is "the same" as the square of the previous error.

Does this make sense?

 

[Mahdi1998]

Yes it makes sense but I still can't understand his purpose of talking about those distances specially the perpendicular distance which isn't even on the graph. Can you explain it to me please ?

 

[blamocur]

Yes it makes sense but I still can't understand his purpose of talking about those distances specially the perpendicular distance which isn't even on the graph. Can you explain it to me please ?

My understanding is that those three distances, including E_1, have the same order of magnitude, i.e. order of magnitude of the error of the previous step, whereas E_2 is the order of the magnitude of the next step. I would not worry too much about these details as long as the general idea of orders of magnitude in this scheme makes sense to you.