Hessian Definition and 44 Threads

Hello everybody, I have a question regarding this visualization of a multidimensional function. Given f(u, v) = e^{−cu} sin(u) sin(v). Im confused why the maximas/minimas have half positive Trace and half negative Trace. I thought because its maxima it only has to be negative. 3D vis 2D...

In local coordinates, the hessian of the function ##f## at point ##p## is ##H = \partial_i \partial_k f dx^i \otimes dx^k##. A coordinate-free generalisation is (see) ##H = \nabla df##, or explicitly ##H = \nabla_i (df)_k dx^i \otimes dx^k = \nabla_i \partial_k f dx^i \otimes dx^k##. How is...

Hey! A function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex if for all $x,y\in \mathbb{R}^n$ the inequality $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$$ holds for all $t\in [0,1]$. Show that a twice continuously differentiable funtion $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex iff the...

I would like to demonstrate the equation (1) below in the general form of the Log-likelihood : ##E\Big[\frac{\partial \mathcal{L}}{\partial \theta} \frac{\partial \mathcal{L}^{\prime}}{\partial \theta}\Big]=E\Big[\frac{-\partial^{2} \mathcal{L}}{\partial \theta \partial...

Hi, Is there a way of representing the Laplacian ( Say for 2 variables, to start simple) ##\partial^2(f):= f_{xx}+f_{yy} ## as a "square of Jacobians" ( More precisely, as ##JJ^T ; J^T ## is the transpose of J, for dimension reasons)? I am ultimately trying to use this to show that the...

As part of my work, I'm making use of the familiar properties of function minima/maxima in a way which I can't seem to find in the literature. I was hoping that by describing it here, someone else might recognise it and be able to point me to a citation. I think it's highly unlikely that I'm the...

Wikipedia defines hessian of Difference of Gaussians as and earlier in the page uses D for difference of gaussians, So do i just need D(x,y) or do i need d/dx D(x,y) for the elements? If so how does one go about differentiating DoG? Any help appreciated

I am doing a nonlinear least squares estimation on a function of 14 variables (meaning that, to estimate ##y=f(x)##, I minimize ##\Sigma_i(y_i-(\hat x_i))^2## ). I do this using the quasi-Newton algorithm in MATLAB. This also gives the Hessian (matrix of second derivatives) at the minimizing...

Hey all. Let me just get right to it! Assume you have a function f:\mathbb{R}^n\rightarrow\mathbb{R}^m and we know nothing else except the following equation: \triangledown_x\triangledown_x^Tf(x)^TQy=0 where \triangledown_x is the gradient with respect to vector x (outer product of two gradient...

I have a function $$\displaystyle V(x)=\frac{1}{2}\sum_i \sum_{j \neq i} q_i q_j \frac{1}{\left|r_i - r_j\right|}$$ where ##r_i=\sqrt{x_i^2+y_i^2+z_i^2}## which is the coulomb potential energy of a system of charges. I need to calculate ##\frac{\partial V}{\partial x_k}## and...

Consider a function ##f : U \subseteq \mathbb{R}^{n} -> \mathbb{R}## that is an element of ##C^{2}## which has an minimum in ##p \in U##. According to Taylor's theorem for multiple variable functions, for each ##h \in U## there exists a ##t \in ]0,1[## such that : ##f(p+h)-f(p) =...

Several questions I have been thinking about... let me know if you have thoughts on any of them I added numbers to for coherence and readability. So, the Hessian matrix can be used to determine the stability of critical points of functions that act on \mathbb{R}^{n}, by examining its...

So I'm looking at the hessian of the Newtonian potential: \partial^2\phi / \partial x_i \partial x_j Using the fact that (assuming the mass is constant): F = m \cdot d^2 x / d t^2 = - \nabla \phi This implies: \partial^2\phi / \partial x_i \partial x_j = -m \cdot...

Apparently it is a well-known fact that if G(x)=(G_{ij}(x_1,\ldots,x_n)) is a smooth nxn matrix-valued function such that G_{ij,k}=G_{ik,j} for all i,j,k, then there exists a smooth function g such that Hess(g)=G; i.e. g_{,ij}=G_{ij}. (f_{,k} denotes partial differentiation with respect to the...

How does one derive the second derivative test for three variables? It's clear that D(a,b) = fxx * fyy - (fxy)^2 AND fxx(a,b) Tells us almost all we need to know about local maxima and local minima for a function of 2 variables x and y, but how do I make sense of the second directional...

Hello all, Suppose I have a simple 1-D signal and I want to compute the hessian. In that case, it should generalise for second derivative for normal scalar functions. So, I observe the signal as v = [x_1, x_2, x_3, x_4...]. Then, numerically the hessian is given as (assuming I am only...

Hellow! I was studying matrix calculus and learned new things as: \frac{d\vec{y}}{d\vec{x}}=\begin{bmatrix} \frac{dy_1}{dx_1} & \frac{dy_1}{dx_2} \\ \frac{dy_2}{dx_1} & \frac{dy_2}{dx_2} \\ \end{bmatrix} \frac{d}{d\vec{r}}\frac{d}{d\vec{r}} = \frac{d^2}{d\vec{r}^2} = \begin{bmatrix}...

Hey guys. I am having some trouble visualizing one aspect of the Second derivative test in the 2 variable case (related to #3 below). Essentially, what does the curve look like when f_{xx}f_{yy} > 0, BUT f_{xx}f_{yy} < [f_{xy}]^{2}? To be more detailed, if the function is f(x,y), H(x,y) is the...

Homework Statement Identify and determine the nature of the critical points of the function $$f(x,y,z) = (x^2 + 2y^2 + 1) cos z$$ Homework Equations ##\vec{x}## is a critical point ##\iff Df(\vec{x}) = 0## ##\vec{x}## is a minimum ##\iff## every determinant of upper left submatrix...

Hi everyone: I am rookie in classical physics and first-time PF user so please forgive me if I am making mistakes here. My current project needs some guidance from physics and I am describing the problem, my understanding and question as below. I have an independent electrostatic system...

Homework Statement I was wondering what exactly the difference between a regular (proper? is that the term) hessian is and a bordered hessian. It is difficult to find material in the book or online at this point. I mean mathmatically so that were i to do a problem i would know the layout and...

Hello Everyone! I'm trying to remember a quick method for determining whether a function is concave or convex. There was something that involved finding the Hessian of the function, and then looking at the diagonal elements, then, I completely forgot... What's the rest of this method, I don't...

I am trying to figure out how the least squares formula is derived. With the error function as Ei = yi - Ʃj xij aj the sum of the errors is SSE = Ʃi Ei2 so the 1st partial derivative of SSE with respect to aj is ∂SSE / ∂aj = Ʃi 2 Ei ( ∂Ei / ∂aj ) with the 1st partial derivative of...

I am struggling a bit with the second order conditions of a constrained maximization problem with n variables and k constraints (with k>n). In the equality constraints case we have to check if the (n-k) leading principal minors of the bordered Hessian alternate in sign, starting from the...

Homework Statement For the function f(x, y) = xye^[-(x^2 + y^2)] find all the critical points and classify them each as a relative maximum, a relative minimum, or a saddle point. Homework Equations Partial differentiation and Hessian determinants. The Attempt at a Solution I get how...

Homework Statement Find the critical point(s) of this function and determine if the function has a maxi- mum/minimum/neither at the critical point(s) (semi colons start a new row in the matrix) f(x,y,z) = 1/2 [ x y z ] [3 1 0; 1 4 -1; 0 -1 2] [x;y;z] Homework Equations The...

Homework Statement Consider the function f(a)= 1 ∫ [g(x)-(anxn+an-1xn-1+...+a0)]2 dx 0 where a=(a0,a1,...an) and g is some known function defined on [0,1]. From this, we can show that Thus, the Hessian of f at a = [2/(j+k+1)] j=0,1,2,...n; k=0,1,2,...,n. Fact: This Hessian...

Homework Statement Given a function f: R^2 -> R of class C^3 with a critical point c. Why CANNOT the hessian matrix of f at point c be given by: 1 -2 2 3 Homework Equations The Attempt at a Solution So first i want to clarify this. When it says f: R^2 -> R, that...

hi, I'm kind of new to optimization theory, and I have to maximize a multi-dimensional problem where I know the exact gradient and hessian. In other words, techniques such as BFGS are not sufficient because I don't want to approximate the Hessian (with an initial guess for example of H=I), I...

Can someone direct me to a good deep exposition of Jacobians and Hessians? I am especially looking for stuff that pertains to their being generalizations of derivatives of vector and scalar functions as well as div, grad, curl. Book sources or web links are appreciated.

Homework Statement For what real values of the parameters a,b,c,d does the functiob f(x,y)=ax^3+by^3+cx^4+dy^4-(x+y)^5 have a local minimum at (0,0)Homework Equations I calculated the gradient at (0,0) and it is always zero regardless of parameters. The problem is that the Hessian matrix is...

How do I use Taylor Series to show f(P) is a local maximum at a stationary point P if the Hessian matrix is negative definite. I understand that some of the coefficients of the terms of the taylor series expansion are the coordinates of the Hessian matrix but for the f_xy term there is no...

Hi, Suppose we have a function of two n-dimensional vectors f(\mathbf{x},\mathbf{y}). How can we find the gradient and Hessian of this function? Regards

The formula given by my instructor for a Taylor Series approximation of the second order at point (a,b) is f(a,b) + grad(f(a,b))x + 1/2 H(f(a,b)) x If you recognize this formula, do you know what the x vector is? Note: x is the x-vector, and H represents the Hessian Matrix. Thanks! The...

What is the general approach to take when the Hessian is inconclusive when classifying critical points? ie the determinant = 0?

Can someone tell me what this actually is. So, in the case when the Hessian is positive (or negative) semidefinite, the second derivative test is inconclusive. However, I think I've read that even in the case where the Hessian is positive semidefinite at a stationary point x, we can still...

Hello, I have solved for the critical points using the gradient, and I have solved for the Hession, which yields a 2x2 matrix. I have plugged in my critical points into the gradient. Now, do I apply the same rules as in linear algebra where I find the determinant and trace to calculate...

Homework Statement what can I do if I have hessian = 0? ex. function f(x,y)=x^2+y^4 hessian is 0, what now? this is simply but what can i do in more complicated functions?

What's Hessian matrix ? Here are all my problem ~ 1. What's Hessian matrix ? 2. How Hessian matrix was derived ? 3. Can u recommend some books about this ?

g(x,y) = x^3 - 3x^2 + 5xy -7y^2 Hessian Matrix = 6x-6******5 5********-7 Now I have to find the eigenvalues of this matrix, so I end up with the equation (where a = lambda) (6x - 6 - a)(-7 - a) - 25 = 0 Multiplying out I get: a^2 - 6xa + 13a - 42x + 17 = 0 How am I supposed to solve...

1) f(x,y,z)=x3-3x-y3+9y+z2 Find and classify all critical points. I am confused about the following: The Hessian matrix is diagonal with diagonal entries 6x, -6y, 2. Now, the diagonal entries of a diagonal matrix are the eigenvalues of the matrix. (this has to be true, it is already...

Please,check my solution. Find critical points of the function f(x,y,z)=x^3+y^2+z^2+12xy+2z and determine their types (degenerate or non-degenerate, Morse index for non- degenerate). Attempt \frac{df}{dx}=3x^2+12y=0 \frac{df}{dy}=2y+12x=0 \frac{df}{dz}=2z+2=0 Critical points...

Homework Statement The height of a certain hill (in feet) is given by h(x,y) = 10(2xy-3x^2-4y^2-18x+28y+12) where y is the distance (in miles) north, x the distance east of South Hadley. a) Where is the top of the hill located? b) How high is the hill? c) How steep is the slope (in...

In my calc 3 class, we've taken an alternative(?) route to learning maxes and mins of multivariable equations. By using a Hessian Matrix, we're supposed to be able to find the eigenvalues of a function at the point, and determine whether the point is a max, min, saddle point, or indeterminant...