Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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).

1
vote
1answer
63 views

Computing real normal modes from complex eigenvectors

I'm trying to get the normal modes of a system of springs and dasphots using the basic dynamic equations for a linear, damped elastic structure: $ M \ddot{u}(t) + C \dot{u}(t) + K u(t) = f(t) $ to ...
3
votes
0answers
90 views

Computing Small Eigenvalues with Sparse Symmetric Indefinite Mass Matrix

I want the eigenvalues of the following generalized eigenvalue problem: $$ Av = \lambda M v $$ where $A\in\mathbb{R}^{n\times n}$ is sparse, symmetric, and positive semi-definite $M\in\mathbb{R}^{n\...
6
votes
2answers
215 views

inertia count sparse matrix with dense low-rank perturbation

I would like to determine the number of negative eigenvalues (inertia count) of the $(N \times N)$ symmetric real matrix $K - \sigma M$, with $K$ a positive-definite sparse matrix and $M$ a positive-...
5
votes
1answer
49 views

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 ...
3
votes
1answer
64 views

Compute nearly degenerated eigen values and vectors

In a quantum physics calculation, I have to deal with a matrix that has a lot of eigenvalues really close to each other ($10^{-6}$ relative difference for example) and I can't manage to obtain ...
0
votes
1answer
71 views

Eigenvalue problem with periodic boundary conditions: Are my eigenvalues correct?

I am using a (central) finite difference scheme to solve the eigenvalue problem $$-\frac{d^2}{dx^2}u = \lambda u$$ with periodic boundary conditions on a unit interval. I use arpack's zndrv1 and ...
1
vote
1answer
57 views

Numerical solution for eigenvectors and eigenvalues of a Sturm-Liouville problem

I have to deal with the following problem in my research: $$\left[\frac{1}{D}F_{x}\right]_{x} + \frac{f D_{x}}{c D^{2}}F = 0$$ with boundary conditions $$F(0) = 0$$ $$F_{x}(L) = 0$$ where $f$ is ...
2
votes
0answers
54 views

Geometric interpretation of lemma

I am currently studying eigenvalue problems. I already worked through the Minimax-principles, seen why $\lambda_{h, m} \geq \lambda_m$, when comparing the eigenvalues of a discretization and the ...
6
votes
2answers
170 views

Nonlinear eigenvalue problem - MATLAB code

I'm trying to solve a nonlinear eigenvalue problem in MATLAB, still without success. It's a problem about graphene plasmonics. The nonlinear eigenvalue problem is given below: \begin{equation} \frac{...
4
votes
3answers
182 views

Efficient eigen-decomposition of covariance matrix

I am looking for an C/C++/Python algorithm implementation that calculates eigenvalues and eigenvectors of a symmetric, positive semidefinite covariance matrix. A general-purpose eigen-decomposition ...
2
votes
1answer
67 views

FEM-Laplace with Dirichlet in only a few points: Nonsingular operator?

Let's consider the FEM discretization of the Laplace operator without boundary conditions, i.e., $$ a(u,v) = \int_\Omega \nabla u \cdot \nabla v - \int_\Gamma (n\cdot \nabla u) v. $$ For one-...
3
votes
1answer
35 views

Partial diagonalisation of large symmetric positive-definite band-diagonal matrices

I want to partially diagonalise real sparse symmetric positive-definite matrices, that are of dimension $n = 10^5$ and I need on the order of $k = 500$ of the smallest eigenvalues and eigenvectors. ...
2
votes
0answers
51 views

Finite differenced eigenvalue prob. of inhomogeneous boundary conditions?

I am basically asking about eigenvalue problems of differential equations using some finite difference method (FDM). Usually the system is subject to some boundary conditions (BC), e.g., Dirichlet or ...
0
votes
0answers
36 views

Use Slepc from Matlab

Is there a direct way to use SLEPC from Matlab? I remember that in some old manuals there was some Matlab interface. However, in the last one, I cannot find any reference to this. For me it would be ...
5
votes
1answer
141 views

Nystrom approximation of SVD for asymmetric matrices

Suppose I have a symmetric matrix $K$. Subdivide $K$ into pieces as $$K=\begin{pmatrix} K_{11} & K_{12} \\ K_{21} & K_{22}\end{pmatrix},$$ where $K_{21}=K_{12}^\top$. Then, the Nystrom ...
0
votes
1answer
46 views

LAPACK zlapmt_ freezes code [closed]

I'm using LAPACK's zggev_ routine to solve some generalized eigenvalue problem. While it produces the correct results, I want the eigenvalues and according eigenvectors sorted by absolute value. For ...
0
votes
0answers
29 views

1D Helmholtz Equation Eigenmodes [duplicate]

I'm trying to simulate a 1D parallel plate waveguide. This is the eigenvalue problem: $$\left(\frac{d^2}{dx^2} + \omega^2\mu_0\epsilon_0\right)E = \beta^2E$$ As a matrix problem, if my 1D domain is ...
2
votes
1answer
78 views

Taylor expansion of error - Finite elements approximation

In some of my computations I calculate a scalar value $\lambda_h$ (in my case an eigenvalue) depending on a finite element discretization of the domain. Usually we can manage to find estimates of the ...
1
vote
1answer
40 views

LAPACK non-convergent eigenvalue algorithm for complex but not double matrix

I've encountered an odd issue with solving for the eigenvalues of the following matrix, in Matlab format: ...
2
votes
1answer
80 views

how to calculate the determinant of a projection of matrix to a subspace

I have a matrix $J$, and I know there are 3 existing zero eigenvalues and their eigenvectors. I want to detect if there is one extra eigenvalue to go cross zero (if it is zero, its eigenvalue will ...
2
votes
0answers
30 views

What is the complexity of calculating K-th largest real part eigenvalue of non-normal sparse matrix

I just need to calculate the largest real part of eigenvalues of a Jacobian which is highly non-normal and singular. Most of the eigenvalues are negative, and some of them are positive but near to ...
3
votes
0answers
73 views

What is Chebfun `eigs` doing

What is this doing? Looks like the original eigenvalue problem is converted into generalized eigenvalue problems with different dimensions of collocation points. Can someone explain more about this? ...
1
vote
0answers
80 views

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 ...
7
votes
0answers
160 views

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, ...
4
votes
0answers
71 views

Eigenvector with maximum overlap

Given a matrix $M$ and a vector $v$, is there an efficient method to find the normalized eigenvector of $M$ that is closest to $v$, in that it has maximal overlap. More explicitly, a vector $v$ can be ...
1
vote
0answers
42 views

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,...
6
votes
0answers
146 views

Multi-matrix orthogonal basis problem

Suppose we are given a set of symmetric, positive definite matrices $A_1,A_2,\ldots,A_k\in\mathbb{R}^{n\times n}$. Is there any numerical method or reduction to a known problem (e.g. eigenvalue ...
9
votes
2answers
187 views

Eigenvalues of Small Matrices

I'm writing a small numerical library for 2x2, 3x3, and 4x4 matrices (real, unsymmetric). A lot of numerical analysis texts highly recommend against computing the roots of the characteristic ...
0
votes
2answers
104 views

Computing Eigenvectors in MATLAB

I am assigned to compute eigenvalues and eigenvectors in MATLAB of a 2x2 matrix: $$ A = \left( \begin{matrix} 3 &0\\ 4 &5\\ \end{matrix} \right) $$ I know that the textbook's solution states ...
3
votes
2answers
267 views

Complex Eigenvalues using eig (Matlab)

I wanted to find and plot the eigenvalues of large matrices (around1000x1000). But discovered when using the eig function in matlab, it gives complex eigenvalues when it shouldn't. For example, in the ...
1
vote
0answers
123 views

How to make LAPACK eigenvectors orthogonal like Matlab?

I'm using LAPACK zgeev to calculate eigenvectors of a symmetric complex matrix of high dimensions ($n \approx 2000$). I need these eigenvectors to satisfy $$\sum_{...
1
vote
1answer
224 views

LAPACK sorting eigenvalues differently each time

I'm using LAPACK zgeev routine to get eigenvalues and eigenvectors of a symmetric matrix in C++. Problem is zgeev is being called in a loop but it sorts eigenvalues (and eigenvectors) differently ...
0
votes
2answers
180 views

Matlab, Mathematica & LAPACK returning 3 different eigenvectors

(I'm not sure which of math.se / stackoverflow / scicomp.se is the right place to ask this question) I have a C++ code which generates a complex matrix and then calculates its eigenvalues and ...
0
votes
1answer
52 views

Examples of finding eigenfunctions of coupled DEs

I am looking for examples of numerically finding eigenvalues/eigenfunctions of coupled DEs. If anyone is able to point me towards any examples, preferably with code included, it would be much ...
1
vote
0answers
65 views

Numerically solving generalized eigenproblem with Neumann conditions

I am interested in finding the eigenvalues/eigenfunctions of problems such as $$ \partial_{xx} u = \lambda \partial_{yy} u, $$ which can be solved as the generalised eigenvalue problem $$ \mathbf{A}...
6
votes
1answer
170 views

Correct eigenfunctions of Laplace operator by Finite Differences

I am trying to compute the eigenfunctions of the Laplace operator, i.e. finding $u$ in $$ -\nabla^2 u = \lambda u .$$ For now I am trying to do this in 1D, so $$ \nabla^2 = \partial_{xx} .$$ I am ...
0
votes
1answer
280 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} \...
1
vote
1answer
77 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 ...
2
votes
2answers
1k 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 ...
10
votes
2answers
344 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'...
1
vote
0answers
102 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. ...
1
vote
1answer
422 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 ...
1
vote
0answers
120 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: ...
4
votes
1answer
188 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 ...
1
vote
0answers
93 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 ...
1
vote
1answer
69 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 ...
1
vote
0answers
130 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 ...
4
votes
2answers
243 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. ...
1
vote
1answer
136 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 ...
1
vote
0answers
84 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 ...