Jacobi Definition and 60 Threads

I need to implement a routine for finding the eigenvalues of a symmetrical matrix (for computing the vibrational frequencies from Hessian). I had already implemented a Jacobi diagonalization algorithm, and in most cases it works properly, but sometimes it crashes. In particular, it crashes when...

(p-6/p)=(-1/p)(2/p)(3/p) Make a table, so at the head row you have p(mod24), (-1/p), (2/p), QRL+-, (p/3) and finally (p-6/p), with in the head column below p (mod 24): 1,5,7,11

program r_jacobi implicit none !!!!Variables!!! real*8 V, V_1, V_2, Lx, Ly integer n ,i , j, k, nx, ny real*8, allocatable :: arrx(:), arry(:), phi(:,:,:) real*8 x, xi, xf, y, yi, yf, dx, dy real*8 d, q, bx, by V=1 V_1=V V_2=-V Lx = 2 Ly = 1 nx = 200 ny = nx/2...

My intuition about the Lie algebra is that it tries to capture how infinitestimal group generators fails to commute. This means ##[a, a] = 0## makes sense naturally. However the Jacobi identity ##[a,[b,c]]+[b,[c,a]]+[c,[a,b]] = 0## makes less sense. After some search, I found this article...

Hey! :giggle: Question 1 : Let $g(x)-=x-x^3$. The point $x=0$ is a fixed point for $g$. Show that if $x^{\star}$ is a fixed point of $g$, $g(x^{\star})=x^{\star}$, then $x^{\star}=0$. If $(x_k)$ the sequence $x_{k+1}=g(x_k)$, $k=0,1,2,\ldots$ show that if $0>x_0>-1$ then $(x_k)$ is...

Given a one parameter family of geodesics, the variation vector field is a Jacobi field. Mathematically this means that the field, ##J##, satisfies the differential equation ## ∇_{V}∇_{V}J =- R(V,J,)V## where ##V## is the tangent vector field and ##R## is the curvature tensor and ##∇## is the...

Are there any useful references or resources that intuitively show how Jacobi Elliptic functions [sn, cn, dn, etc] are geometrically interpreted from properties of ellipses? And how the Jacobi Elliptic functions and integrals can be shown to be generalizations of circular trig functions? Thanks!

I understand how to reach $$\int_0^\phi \frac{d\theta}{\sqrt{1-k^{2}sin^{2}\theta}}=\sqrt \frac g l t$$ from physics but from there I don't get how to turn that into this new (for me) sn(u) form.

Homework Statement Suppose the potential in a problem of one degree of freedom is linearly dependent upon time such that $$H = \frac{p^2}{2m} - mAtx $$ where A is a constant. Solve the dynamical problem by means of Hamilton's principal function under the initial conditions t = 0, x = 0, ##p =...

<Moderator's note: Moved from a homework forum.> Homework Statement From this paper. Let ##L## be the Jacobian operator of a two-sided compact surface embedded in a three-maniold ##(M,g)##, ##\Sigma \subset M##, and defined by $$L(t)=\Delta_{\Sigma(t)}+ \text{Ric}( ν_{t} , ν_{t}...

When doing a problem on a pendulum undergoing elliptical motion, I came across sn(z), which is apparently a "Jacobi Elliptic Function". When I looked into it further, I saw that these functions are essentially circular trigonometric functions but about an ellipse instead of a perfect circle. Can...

Homework Statement Hamiltonian of charged particle in magnetic field in 2D is ##H(x,y,p_x,p_y)=\frac{(p_x-ky)^2+(p_y+kx)^2}{2m}## where ##k## and ##m## are constant parameters. For separation of this system use ##S=U(x)+W(y)+kxy+S_t(t)##. Solve Hamilton - Jacobi equation to get ##x(t), y(t)## ...

Homework Statement The problem is attached. Homework Equations Isolating each x_i. The Attempt at a Solution I watched this video for the Jacobi method.: I also watched this video for the Gauss-Seidel method.: At least based on the videos mentioned above, it seems that the difference...

An isolated mechanical system can be represented by a point in a high-dimensional configuration space. This point evolves along a line. The variational principle of Jacobi says that, among many imagined trajectories between two points, only the SHORTEST is real and is associated with situations...

Homework Statement I have that ##(\psi(z)-e_j)^{1/2}=e^{\frac{-n_jz}{2}}\frac{\sigma(z+\frac{w_j}{2})}{\sigma(\frac{w_j}{2})\sigma(z)}## has period ##w_i## if ##i=j## and period ##2w_i## if ##i\neq j## where ##i,j=1,2,3## and ##w_3=w_1+w_2## (*) where ##e_j=\psi(\frac{w_j}{2})## I have...

Homework Statement I have the Jacobi theta series: ##\theta^{m}(\tau) = \sum\limits^{\infty}_{n=0} r_{m}(\tau) q^{n} ##, where ##q^{n} = e^{2\pi i n \tau} ## and I want to show that ##\theta^{m}(\tau + 1) = \theta^{m}(\tau) ## (dont think its needed but) where ##r_{m} = ## number of ways of...

Hi everyone, I'm new to Physics Forums and to Mathematica, as well as Jacobi Identity. In any case, I was wondering on how I may use Mathematica to solve various Quantum Mechanics related problems through commutators. Like if it's possible to find out what is the form of a particular commutator...

Homework Statement The motion of a free particle on a plane has hamiltonian $$H =E = \text{const} = \frac{1}{2m} (p_r^2 + \frac{p_{\theta}^2}{r^2})$$ Set up and find a complete integral for ##W##, the time independent generating function to canonical coordinates such that new coordinates are...

I am trying to solve a Duffing's equation ##\ddot{x}(t)+\alpha x(t)+\beta x^3(t)=0## where ##\alpha## is a complex number with ##Re \alpha<0## and ##\beta>0##. The solution can be written as Jacobi elliptic function ##cn(\omega t,k)##. Then both ##\omega## and ##k## are complex. The solution to...

I have been asked to solve the actual load flow distribution in a given power network using two iterative methods. I have chosen Jacobi and Gauss Seidel. we have to use MATLAB to find where the solution converges. I am fine with all of this, but we have been tasked with providing graphical...

For the Matrix 1 2 -2 1 1 1 2 2 1 What is the spectrum for the Jacobi iteration matrix and the Gauss-Seidel iteration matrix. And are the methods convergent?

Hey guys so I have this Calc 3 project and the end is throwing me for a loop. I've done the encoding part, and I've coded the standard iterative methods, but I don't see how the two correlate so I can use the iterative methods to decode a "y stream" with the inputs specified...

Hi I am working on a programming assignment that requires me to implement the successive over-relaxation algorithm. We are given the wikipedia page for this: http://en.wikipedia.org/wiki/Successive_over-relaxation. I have read through the wikipedia page for this numerous times but am still...

I have some questions with regards to conjugate points on a congruence of time-like geodesics (will be referring to Wald 9.3 throughout). First, we define ##\gamma## to be a time-like geodesic with tangent ##\xi^a## parametrized by ##\tau## and with ##p\in\gamma##. We consider the "congruence of...

Am I the only one who sees the resemblance between these two identities? Schouten: <p q> <r s> +<p r> <s q>+ <p s > <q r> =0 Jacobi: [A,[B,C]]+[C,[A,B]]+[B,[C,A]]=0 In Schouten the p occours in each term in the three terms, so we can regard it as dumby variable, and somehow get a...

I found this identity in the wiki https://de.wikipedia.org/wiki/Jacobische_elliptische_Funktion#Abstrakte_Definition_als_spezielle_meromorphe_Funktionen One propertie of the ellipitc integral is: K(k') = K'(k), all this set of ideia seems answer an old doubt, ie, exist a complementary...

Suppose we have a torsion free connection. Does anyone here know of a slick way to prove that covariant derivatives satisfy the Jacobi identity? I.e. that $$([\nabla_X,[\nabla_Y,\nabla_Z]] + [\nabla_Z,[\nabla_X,\nabla_Y]] +[\nabla_Y,[\nabla_Z,\nabla_X]])V = 0$$ without going into...

Homework Statement Use the Jacobi identity in the form $$ \left[e_i, \left[e_j,e_k\right]\right] + \left[e_j, \left[e_k,e_i\right]\right] + \left[e_k, \left[e_i,e_j\right]\right] $$ and ## \left[e_i,e_j\right] = c^k_{ij}e_k ## to show that the structure constants ## c^k_{ij} ## satisfy the...

Jacobi identity in local coordinates?!? Apparently (i.e. according to an article written by physicists), the Jacobi identity for the Poisson bracket associated to a Poisson bivector \pi = \sum\pi^{ij}\partial_i\wedge\partial_j is equivalent to...

Homework Statement A particle of mass m moves on the surface of a paraboloidal bowl with position given by r=rcosθi+rsinθj+\frac{r^{2}}{a}k with a>0 constant. The particle is subject to a gravitational force F=-mgk but no other external forces. Show that a suitable Lagrangian for the system is...

Dear all, I am reading R.A. Sharipov's Quick Introduction to Tensor Analysis, and I am stuck on the following issue, on pages 38-39. The text is freely available here: http://arxiv.org/abs/math/0403252. If my understanding is correct, then the Jacobi matrices for the direct and inverse...

Homework Statement for part c , it asked for showing both 2 method converge for any initial condition. I think we can show that by using $$ρ(T_{j}), ρ(T_{g}) <1 $$ I want to know whether it's correct or not , and is there any faster method? Homework Equations $$ρ(A)$$ means spectral...

Would someone be kind enough to explain Jacobi sums in a simple manner using actual numbers. I have read over the math jingo 100 times and have no clue what it actually does. Thanks! Edit: Here is a link to the wiki of the Jacobi sums. http://en.wikipedia.org/wiki/Jacobi_sum

Hello! General Question about the H-J equation. What are the steps to be followed if we are in a conservative system? And while answering my question, please in the step after we find S, and when you derive S wrt alpha and place it equals to β. When is alpha Energy? When it is not? i.e is it...

Homework Statement Let A be a squared, hermitian positive definite matrix. Let D denote the diagonal matrix composed of the diagonal elements of A, i.e. D = diag((A)11,(A)22,...(A)nn). Prove that if the Jacobi iterative method converges for A, then 2D - A must also be hermitian positive...

"Proving" the Jacobi identity from invariance Hi all, In an informal and heuristic manner, I have heard that the "change" in something is the commutator with it, i.e. \delta A =[J,A] for an operator A where the change is due to the Lorentz transformation U = \exp{\epsilon J} = 1 + \epsilon J...

Write a recursive function to compute the Jacobi symbol J(a, n), is defined for relatively prime integers a and n, a> o, \mbox{ and } n > 0 by the formula J(a, n) = 1 \mbox{ if a = 1, } = J(a/2, n)*(-1)^\frac{n^2-1}{8} \mbox{ if a is even, } =J(n \% a, a)*(-1)^\frac{(a-1)*(n-1)}{4} \mbox{...

Homework Statement Let D be the set of points (x,y) in R^2 for which 0 is ≤ x ≤ 1 and 0 ≤ y ≤ 1. Find a function g: R^2 --> R for which: ∫_0^1 ∫_0^1 h(x,y)dxdy = ∫_0^1∫_0^1 h(y^5, x^3) * g(x,y)dxdy is true for all functions h: D--> R integrable over D In the question before this I...

what if Jacobi Method's condition did not meet? Homework Statement solve by Jacobi Method upto four decimal places 8x+y-z= 8 2x+y+9z= 12 x-8y+12z = 35 Homework Equations The Attempt at a Solution since the condition of convergence of jacobi method is |A1| > |B1|+|C1|...

Homework Statement The transformation f is defined by: R^2 --> R^2 and is defined by: f(x,y) = (y^5, x^3) Find the jacobi matrix and its determinant Homework Equations f(x,y) = (y^5, x^3) The Attempt at a Solution I would start by differentiating y^5 with respect to x and then y, then...

Homework Statement (Ax = B) A: 3.1410 -2.7180 1.4140 -1.7321 9.8690 2.7180 -7.3890 0.4280 2.2360 -2.4490 1.0000 -1.4140 31.0060 7.3890 -2.6450 0.1110 B: 3.316 0 3.141 1.414 The question in my Numerical Methods assignment asks to use the Jacobi Iterative method to solve the system...

Couple of days ago I downloaded a book on numerical optimization, hoping to get clearer picture on some techniques. But, I'm surprised that some of the concepts were not precisely separated from one another. Namely, in the part of "coordinate descent methods" (cdm), I found that, in the...

Homework Statement Prove the chain rule for Jacobi determinants \frac{d(f,g)}{d(u,v)} * \frac{d(u,v)}{d(x,y)}=\frac{d(f,g)}{d(x,y)} Homework Equations Definition of Jacobi determinant \frac{d(f,g)}{d(u,v)} = \frac{d(f,g)}{d(u,v)} = det \begin{bmatrix} \frac{df}{du}&\frac{df}{dv} \\...

Homework Statement Evaluate the number of iterations that are needed to have 10^-9 precision with the Jacobi, Gauss-Seidel, and SOR ( with ω=1.5) methods. Compare these 3 methods for different values of n - for instance 3≤n≤20. Plot the convergence curves for the 3 methods for each n. Homework...

How we prove that rate of convergence of gauss-Seidel method is approximately twice that of Jacobi iterative method without doing an example itself ? What's the general proof of this statement ? I didn't fin in any book ? Can anyone please help me ?

hey Folks, please have a look at the attached Ex from MTW. does somebody know what is the meaning of the parallel bars in the first levi civita symbol ? Is there a typo in this EX perhaps? I would have expected that on the right hand side one would see the product which is shown in the first...

Homework Statement Reduce the equation \partial_\mu {*} F^{\mu \nu} = 0 into the following form of the Jacobi Identity: \partial_\lambda F_{\mu \nu} + \partial_\mu F_{\lambda \nu} + \partial_\nu F_{\lambda \mu} = 0 The Attempt at a Solution I can't figure out what the '*' is supposed to...

Could anyone give me a simple explanation as to why the Fermat/Hamilton principle would be called more general than the Jacobi least time principle? I am trying to understand what differences would result from using the one principle vs. the other; eg: where/in what way would the Jacobi least...

Hello, I'm unfamiliar with the notation used in this problem with the commas. I understand matricies, identities, etc. but not sure about the commas.. Question 3.2.9: Verify the Jacobi Identity: [A,[B,C]] = [B,[A,C]] - [C,[A,B]] I see the BAC CAB rule here, but not sure how to show it...

I was reading a book on the zeta function and came across this attributed to Jacobi. I have no idea where to find a source about this so maybe someone can give me some direction. Let \psi(x) = {\sum}^{\infty}}_{n=1}e^{-n^2 \pi x}. How do you show that \frac{1+2\psi(x)}{1+2\psi(1/x)} =...