# Questions tagged [eigenvalues]

Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144).

254 questions
Filter by
Sorted by
Tagged with
1answer
1k views

### Set of linear ordinary differential equations with a mass matrix

What methods are known for efficiently solving a large set of linear homogeneous ordinary differential equations of the following form? \begin{equation} \mathbf{B} \frac{d\mathbf{y}}{dt} = \mathbf{A} \...
1answer
129 views

### triple cross prouct of tensor

Im trying to compute a triple cross product of vectors a,b, and c in real space and integrate over the entire space. The result is a term in the hamiltonian for an electronic system so there are ...
2answers
4k views

### Eigen - Max and minimum eigenvalues of a sparse matrix

I am working with an Eigen::SparseMatrix matrix of type double. I would like to find the largest and the smallest eigenvalues. A solution of the problem is to convert it to dense and then find its ...
2answers
507 views

### Eigenvectors of a small norm adjustment

I have a dataset that is slowly changing, and I need to keep track of eigenvectors/eigenvalues of its covariance matrix. I've been using scipy.linalg.eigh, but it'...
0answers
139 views

### Eigenvalues using QR iteration

I'm trying find the eigenvalues of a matrix A using QR iteration with Householder. I used this code which I found from Cornell University that decomposes QR with Householder. ...
1answer
866 views

### LAPACK: ZHEEV and DSYEV give different eigenvalues for real symmetric matrix

exchangers, I have run into a bit of a puzzling problem. To solve an complex eigenvalue-problem, I make use of the LAPACK library function ZHEEV. To test the implementation I used a real symmetric ...
0answers
144 views

### Spectrum of the Laplace operator

I am studying the discretization of Poisson's equation in $1D$. In Matlab I created different discretization matrices (Laplace operator) according to different sizes of the mesh: ...
1answer
282 views

### SLEPc eigensolvers take long time to converge for large sparse symmetric matrices

I am leveraging the SLEPc library for solving the first $k$ (where $k = 3$ or $4$) eigenvalues and their corresponding vectors for a matrix of size 200,000. The matrix is sparse and symmetric. I ...
0answers
122 views

### Computing only the $k$ biggest eigenvalues and eigenvectors with Scalapack

Given that there are eigensolvers in Scalapack that use a divide and conquer method, is there any way we can use Scalapack functions to only compute the first $k$ dominant eigenvalues and ...
1answer
147 views

### Eigenvalue problem constrained with a penalty method

I am trying to constrain an eigenvalue problem. I am aware of the method utilizing the nullspace of the constraint vectors but I was wondering if it would be to use a penalty method for the same ...
0answers
205 views

### All eigenpairs of large sparse symmetric matrix

In advance I am sorry for my noobish question. I am a physics PHD student and basically I use python for my math/physics problems. But now I have a problem which requires more computing capacity and ...
2answers
312 views

### Using a subspace iteration method to obtain eigenvalues. Getting eigenvectors too but I don't understand why

I'm using an iterative subspace algorithm (dsrrit) to obtain the eigenvalues of an eigenvector equation $$-\nabla^2 \mathbf{x} = \lambda\mathbf{x}$$ where $\nabla^2$ is the usual Laplacian operator. ...
1answer
261 views

### Iteratively obtaining m eigenvectors using arpack: If I have a good initial guess, how do I use it?

I am trying out the arpack driver dsdrv1, which is used to iteratively obtain the first m eigenvectors from the eigenvalue problem. $$\hat{A}\mathbf{x} = \lambda\mathbf{x}$$ As it is an iterative ...
0answers
157 views

### Generalized eigenvalue with null space

Define $S\in\mathbb{R}^{n\times n}$ as $$S:=H+Q^\top V^{-1} Q.$$ $H,V$ are positive semidefinite. Here, $H$, $Q$, and $V$ are large, dense matrices but they are structured: I can write code for ...
1answer
143 views

2answers
4k views

### Quality of eigenvalue approximation in Lanczos method

I try to familiarize myself with iterative eigenvalue solvers such as Lanczos. So I tried rewrite it to python directly according to wiki. But it doesn't seem to work. The problem: it approximates ...
1answer
284 views

### Eigenvectors of slightly perturbed matrix

Assume I know eigen-pairs $(\epsilon_i,\vec\phi_i)$ for matrix operator $\hat H$ that $\hat H \vec \phi_i = \epsilon_i \vec \phi_i$ Now I have slightly perturbed $\hat H' = \hat H + \hat R$ were ...
1answer
1k views

### Power Iteration over Rayleigh Quotient Iteration?

It is a commonly known fact that the Rayleigh Quotient converges cubically (1), while the Power Iteration may converge slowly if the difference between the dominant and second-dominant eigenvalue is ...
1answer
2k views

### What is the state of the art algorithm for diagonalizing real symmetric matrices?

There are many methods for diagonalizing matrices, probably the most widely used is the combination of Householder transformations and the QR algorithm. Is there any superior method for diagonalizing ...
1answer
275 views

### Boundary conditions generalized eigenvalue problem

Consider the following eigenvalue problem \begin{equation} \mathcal {L} x(s) = \lambda x(s), \end{equation} where \begin{equation} \mathcal {L} = \alpha \partial^4_s + (s^2-1)\partial^2_s + s \...
0answers
327 views

### Computing eigenvectors from the QR algorithm

I've seen a few other posts on this topic but none have full answers. I'm trying to implement some eigen-decomposition algorithms. I've managed to get the Explicit QR algorithm and the Implicit (...
0answers
231 views

### scipy.linalg.sparse.eigsh does not work for generalised eigenvalues

I asked this question over at StackOverflow and someone told me that I'd get a better answer here. So here's my problem: I'm working on a machine learning project which involves doing a Principal ...
1answer
930 views

### Smallest eigenvalue without inverse

Suppose $A\in\mathbb{R}^{n\times n}$ is a symmetric, positive definite matrix. $A$ is big enough that it's expensive to solve $Ax=b$ directly. Is there an iterative algorithm for finding the ...
2answers
372 views

### Known issues with eigenvalue numerics?

Are there any known issues (such as precision issues) with $\mathsf{MATLAB}$ eig and charpoly functions for large enough $\{-1,0,+1\}$ matrices? Even if I change $1$ or $2$ entries between matrices ...
1answer
827 views

### Calculating left eigenvector when I know the right eigenvector

I'm using power iteration to find the dominant right eigenvector of some large-ish matrices ($1000\times 1000$ to $10000\times 10000$ or so, maybe I'll need to go bigger later) with non-negative ...