# 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).

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### Fast Eigenvalue and SVD Solver for Structured Matrices

I am looking for a fast Eigenvalue and SVD solver for small dense structured matrices (Hankel and Toeplitz). I have searched for efficient implementations in libraries like MKL but I am not able to ...
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### Benchmark problems for eigenvalue reordering algorithms sought

Every real matrix $A$ can be reduce to real Schur form $T = U^T A U$ using an orthogonal similiary transform $U$. Here the matrix $T$ is quasi-triangular form with 1 by 1 or 2 by 2 blocks on the main ...
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### Eigenvalue with largest imaginary part

Iterative eigensolvers such as ARPACK, give the option to find a subset of the eigenvalues which have the largest imaginary part. My question is how do these algorithms work. As I understand it, ...
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### Improving convergence of Jacobi iteration to Schur form

I'm using SIMD processor arrays to compute the eigen-decomposition for large numbers of small (up to $32\times 32$) matrices. For assorted technical reasons, Jacobi iteration maps well to the SIMD ...
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### Eigenvalues of a Laplacian operator on an irregular mesh

I have the following setting: An irregularly-shaped domain, expressed as a mesh of points A Laplacian operator, together with boundary conditions I am looking for the eigenvalues of that operator, i....
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### Eigenvalue-style optimization with quadratic constraints

Suppose $A\in\mathbb{R}^{n\times n}$ is symmetric and positive definite and that we have several symmetric matrices $B_i\in\mathbb{R}^{n\times n}$ that are low-rank and indefinite. I need an ...
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### Why the MIRACLE of Lanczos/CG-like?

Lanczos/Arnoldi/Rietz/CG-like algorithm share the same core strategy... In each, a little miracle appears, most of the Gram-Schmidt inner products are zeroes! In others words, new direction need only ...
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### Estimating eigenvalues from time-dependent non-linear operator

I have a very sparse non-linear system $N(u) = 0$ that can be solved as a time-dependent ODE, $\frac{du}{dt} = N(u)$, and explicitly integrated until $\frac{du}{dt} = N(u) = 0$, e.g. by forward euler, ...
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### Improving my QZ-Algorithm (Include Shifts)

I Need to to solve an generalized Eigenvalue Problem and to compare two Methods (QR and QZ) concerning their convergence rate and execution time. I started with the Basic QR-Algorithm, implemented in ...
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### Eigenfaces Algorithm

This might be a silly quesntion but recently I've been trying to program the eigenface algorithm using PCA, so I arranged the face vectors vertically in a matrix X such as: X = [x1,x2,x3,...,xn]; In ...
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### Low memory algorithm for matrix diagonalisation

I'm trying to find the largest eigenvalues of very large $N \times N$ matrices ($N = 10^{10}$ and larger). The matrices are not sparse but the multiplication operation is fast. For now, I'm using ...
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### Appropriately handling boundary conditions in a PDE eigenvalue problem

Suppose I have a nonlinear second order Cauchy PDE $\dfrac{\partial p(x,t)}{\partial t}=N(p(x,t))$, where $N:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$, and a known fixed point $u(x)$. Mathematically,...
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### 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. ...
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### 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: ...
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### 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 ...
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### 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 ...
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### What's the optimal method to solve for the top eigenvectors of a very large, real, symmetric matrix of limited rank?

Consider a real symmetric matrix of dimension N~10^5 and rank m~2000. What is the most efficient algorithm for determining the top m eigenvectors? If the answer isn't obvious, are there existing ...
I have the following problem. I have two sequences of elements $A = [a_1,a_2,\cdots,a_n]$ and $B = [b_1,b_2,\cdots,b_m]$. I can build a matrix $D[n \times m]$ where $d_{ij} = d(a_i,b_j)$ My greedy ...