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...
Hello,
I am using the Lagrange multipliers method to find the extremums of ##f(x,y)## subjected to the constraint ##g(x,y)##, an ellipse.
So far, I have successfully identified several triplets ##(x^∗,y^∗,λ^∗)## such that each triplet is a stationary point for the Lagrangian: ##\nabla...
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...
Hello community!
I am facing a conceptual problem with the correlation matrix between maximum likelihood estimators.
I estimate two parameters (their names are SigmaBin0 and qqzz_norm_0) from a multidimensional likelihood function, actually the number of parameters are larger than the two I am...
I am working on implementing a PDE model that simulates a certain physical phenomenon on the surface of a 3D mesh.
The model involves calculating mixed partial derivatives of a scalar function defined on the vertices of the mesh.
What I tried so far (which is not giving good results), is this...
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) =...
Homework Statement
So the test is to take the determinant (D) of the Hessian matrix of your multivar function.
Then if D>0 & fxx>0 it's a min point, if D>0 & fxx<0 it's a max point.
For D<0 it's a saddle point, and D=0 gives no information.
My question is, what happens if fxx=0? Is that...
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...
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...
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...
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...
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
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...
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...
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...
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...
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...