[casualrandomcom]

Maybe I had my Gell-Mann effect moment (https://en.wikipedia.org/wiki/Gell-Mann_amnesia_effect)

The illustration of the Newton method is wrong, isn't it? The approximating second order polynomials that were drawn are not really graphs of second order polynomials, they are not even functions... those are parables in the 2D plane, but Newton's method won't work like that... Besides, the convexity of the target function looks negative near to the first guess, the Newton method shall probably miserably fail here

[magicalhippo]

Perhaps the graphics designer didn't get the memo. You can see in the panels that it's the same parabola, they've just rotated it in the first panels so it seems to fit the shape of the local curve "better".

[casualrandomcom]

Yes, that's what I guess has happened.

[jonlong]

For late arrivals to this thread: note that the illustration being discussed has been updated (see the Wayback Machine for the old version). The new version probably still does not show the best second-order approximations, but the obvious qualitative errors have been corrected.

[mrkeen]

I'm not following your objection. To my eye those approximation graphs are indeed 2nd order functions in the 2d plane. But they are perhaps not parabolic for lack of symmetry?

[casualrandomcom]

They look symmetric to me, but that's not point anyway, I guess they are parables, only they are rotated, their axis is not parallel to the y axis.

If those are 2nd order functions in the 2d plane, then we don't agree on terminology.

The reason why I feel an expert is that it is clear at first sight to me that if you really implement the Newton method in that situation, the approximating functions that you will get are totally different from those that were drawn in the illustration.

The third parable from the left on the lower left figure, is definitely not a second order approximation to the target function: the convexity is reversed!

[MITSardine]

Well, now that you mention it, even the first one is on the wrong side of the graph. The second derivative is apparently (to my eyes) negative there, so the approximating parabola should be turning downwards. Whichever the sign, its value is close to zero anyways, the approximation would look very flat there.

Anyways, it's a little graph to illustrate the idea of a second order approximation, I think it does the job.

[ckemere]

The function is also not convex…

[seanhunter]

When you say they are not even functions, are you saying because they are not everywhere bijective? Because they look like rotated parabolas to me which means they will have continuous first and second derivatives which is (I think) all you need for Newton’s method isn’t it?

[casualrandomcom]

You are thinking of those parables as parametric curves, I guess, but Newton's method is about approximating an arbitrary function by its Taylor expansion truncated to the second degree, which is a polynomial function of second degree. The graphs of these functions can be thought as parametric curves, but (my point is also) these are not those that were drawn in the illustration.

[MITSardine]

I think the person meant that, to a single x, they map more than one value. A function needs to have at most one image to the antecedent. In other words, it can't have "vertical stretches". This is different from bijectivity (specifically, injectivity here) which asks that two different x values never map to the same y.