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Solve the 'Eigenvalue' problem efficiently [closed]

Usually, an eigenvalue of a matrix A is defined as |A-b*I|=0, where I is the identity matrix and |..| is for the determinant. Now my question becomes a little different, let's say A is a function of ...
4
votes
1answer
129 views

Numerically computing the eigenvalues of an infinite-dimensional tridiagonal matrix

I have one infinite dimensional tridiagonal matrix whose eigenvalues I have to compute. How can that be done numerically using Mathematica? Let me expose the concrete case I want to do it. I shall ...
3
votes
1answer
171 views

Eigenvalues of a non-Hermitian complex periodic potential

I have an eigenvalue problem: $$-\frac{d^2}{dx^2} \psi(x) +V(x)\psi(x) = E \psi(x)$$ where $V(x)$ is a complex periodic potential: $$V(x) = 4[\cos^2(x) + i 0.3 \sin(2x)]$$ It has been claimed that ...
3
votes
1answer
66 views

ParallelDo gives different solution to Eigensystem

I am trying to calculate the eigensystem of a large matrix (e.g. 256x256). I have found that when I do this within a ParallelDo (because I am actually calculating many of these eigensystems), the ...
2
votes
1answer
62 views

Taking a derivative of an eigenvector

I'm trying to calculate the derivative of an eigenvector that I obtain by ...
10
votes
1answer
569 views

Nonlinear ODE eigenvalue problem

How does one find eigenvalues $\lambda$ of the following problem? $$ \frac{\mathrm{d}^2 u}{\mathrm{d}x^2} = \lambda \left( -u + u^2 \right),$$ $$ u(0) = u(1) = 0. $$ Can this be tackled by ...
3
votes
2answers
309 views

Why ODE's naive finite difference matrix works well for different boundary conditions

We know finite difference method (FDM) can replace $y''(x)$ as $\frac{1}{h^2}[y(x+h)+y(x-h)-2y(x)]$ or so. The naive way to write down the matrix of the differential operator is like the following, ...
0
votes
0answers
66 views

Testing a (numerical) matrix for positivity

I’ve been testing certain randomly-generated $6 \times 6$ symmetric (and also Hermitian) matrices ($H)$ for positive definiteness, using the command ($n$, of course, being a count variable), ...
1
vote
1answer
164 views

NDEigensystem in a complicated case

Apologies for a boring question. I am trying to modify the standard Mathematica example for my needs. The only differences in my case are: A more complicated potential (double-well, grows rapidly). ...
3
votes
1answer
182 views

Seemingly wrong eigenvectors for numerical matrix whose elements differ in scale by orders of magnitude

I stumbled upon an issue with computing eigenvalues of a matrix which I discovered while debugging some code which uses reconstructing the matrix from its eigenvalues/eigenvectors. First of all, here ...
5
votes
1answer
256 views

Eigenvalues of large symmetric matrices

When I try to compute the eigenvalues of the adjacency matrix of a very large graph I get, what can be charitably described as, garbage. In particular, since the graph is four-regular, the eigenvalues ...