# Tag Info

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### Numerical computation of Perron-Frobenius eigenvector

The algorithm used by eigs is called Arnoldi iteration (in shift-and-invert mode). You could think of it as a sophisticated way of doing an inverse power iteration....
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### Roots of a function for eigensystem

I suspect the main problem is the magnitude of the values. If you divide through by $\cosh$ to make all the numbers smaller, then ApproxFun doesn't seem to have a problem finding all the roots. ...

### Eigenvectors of Laplacian

They're on Wikipedia, for instance, in a page with the slightly unclear name of "Eigenvalues and eigenvectors of the second derivative".
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### Sorting eigenvalues by the dominant contribution

You'd like to sort eigenvalues/eigenvectors in a way that is continuous as you move through momentum. This highly constrains the sorting - for most k-points, you must sort the eigenvalues to be a ...
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### How to impose boundary conditions on eigenfunction problems?

Consider what happens when you approximate a function's derivative using finite differences near a boundary. If the boundary is at the point $x_0$, the point $x_1$ is just outside the boundary, then ...
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### Appropriate iterative linear solver for an eigenvalue problem

If your matrices are large, why not use a library like ARPACK? The shift-and-invert mode of ARPACK will help you calculate the eigenvalues close to $\sigma$. There are interfaces to ARPACK for most ...

### What are some ideas to preprocess / precondition the following linear system?

You have noticed that any eigenvector of $A$ is also an eigenvector of your composed matrix. When you compute eigenvectors of your matrix, you can use them in a deflation-type preconditioner, as ...
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### LAPACK sorting eigenvalues differently each time

You write, that you are computing the eigenvalues of a symmetric matrix. Does the matrix have real entries? In this case all eigenvalues are real, and you can use a symmetric eigenvalue solver, which ...
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### Nystrom approximation of SVD for asymmetric matrices

Nemtsov, Averbuchm, and Schclar's "Matrix compression using the Nyström method" (2016) seems relevant: The Nyström method is routinely used for out-of-sample extension of kernel matrices. We ...
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### Discrepancies between numerical and analytical solution for particle in a finite potential well?

I think that your potential for the numerical case is wrong. The potential should be a big positive number, so the solution tends to zero outside the well when the value of the potential increases. ...

### Solving the eigenvalue from a set of coupled second order differential equation numerically

This problem can be interpreted as a coupled convection-diffusion-reaction equation in two variables. You can use the Finite Element Method to solve it. Must be mentioned that you need to use some ...
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### Computing eigenvalues of Schrodinger equation with spin

When modelling spin in the Schrödinger equation, one has several alternatives which need to be chosen in advance. I'll copy an excerpt from a work of mine to give an overview (it considers atomic ...