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method converges to a local minimum or maximum often the
Hessian matrix is shifted sufficiently to pass the Second partial derivative test as desired; unfortunately, there is nothing mentioned in this article about how to do that, its not as simple as H = H + k*diagonalMatrix (or identity matrix), is it? Regardless, there should be some discussion on this.
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Its never mentioned in the article that this can be used as a critical step in Newton's Method (amongst other numerical methods of unconstrained optimization). Further, its an important issue that Newton's method not converge to a saddle point when this is not desired. In order to ensure that the
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I think "overdoing it" was meant more for people analyzing simple surfaces (planes, paraboloids, the like), in which case it would be ridiculous to bother using the second derivative test. Nonetheless, it's not all that helpful and I don't find it contributes to the article; it's strictly about the
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Well, I have resolved this, but perhaps in a way you will not appreciate: I removed the section. Someone with access to a couple decent multivariable calc books might be able to find a version of this with some sourcing, but what was written there struck me as mostly handwaving.
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In the section "Geometric interpretation in the two-variable case" a figure is desperately needed for the case where D<0 and both fxx and fyy have the same sign. This is the most interesting case and the most difficult to understand. A figure would really help here.
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It's also POV; for many applications -- say, writing code, a numerical computation of the
Hessian is more feasible than writing an algorithm to qualitatively analyze the level curves. Saying that one method is "overdoing it" compared to another is a bias. --Heath
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Yes it has one. The determinant of the hessian at a particular point is the same as the
Gaussian curvature of the graph at that point. If it is positively curved (locally spherical), there are two cases to consider concave up or concave down (fxx:
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I’d wonder what geometric interpretation the determinant has in this case. Does it even have one? Suppose we have a function in which M : -->
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It says in the article that in higher dimensions you need to look at the eigenvalues. However, as is clearly explained in the article on
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discusses the same thing as this article and even does it better. This page and that section should be somehow merged, I think. -
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Yes, it had previously been M for half the article and D for half, and when I standardized it seems I missed a couple of Ms.
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In the geometric intuition example, what does it mean when it says "between the x and y axis"? How can you be between axes?
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599:{\displaystyle {\frac {\partial z}{\partial y}}=x\left(3(-2x)^{2}+2(-2x)(x+1)+x\right)=x^{2}(8x-3)\neq 4x^{2}(2x-1)}
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0 alone? Perhaps this should be further explained in the article!?. Sorry for the bad
English! --
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Please explain what M stands for in the geometric interpretation explantion of a 2-D hessian.
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Added a quick interpretation of it. Please help improve if it needs more clarification!
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0 or fxx<0). If it is negative, there is only such surface, a saddle.
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If I'm not mistaken M=D, so this is probably just a typo.--
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203:"don't overdo it"? lame...
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