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).

Filter by
Sorted by
Tagged with
0 votes
0 answers
64 views

Best transform of matrix to make it efficient for shift-then-invert?

I am using ARPACK to find the smallest eigenvalue of a matrix. I use the shift and invert method. That is, looking for the largest eigenvalue of $$ (A-\sigma I)^{-1}. $$ However, I do not know $\sigma$...
Ma Joad's user avatar
  • 161
4 votes
0 answers
180 views

Why for $A^T A$, it is faster to computer the eigenvalues of its inverse than itself?

I have written the following code in MATLAB. I also observe the same effect in Arnoldi iteration in ARPACK in C. ...
Ma Joad's user avatar
  • 161
1 vote
0 answers
68 views

Ways to speed up Lanczos algorithm when we have a very dense cluster of many eigenvalues?

Lanczos algorithm can be used to find the largest/smallest eigenvalues of matrices. I am trying to find a good library in C/C++/Rust for finding the smallest singular value (or eigenvalue). I have ...
Ma Joad's user avatar
  • 161
0 votes
1 answer
63 views

Eigenvalue problem and pseudoinverse of a product of sparse matrices

If I have some dense matrix that can be decomposed into a product of sparse matrices with known(but different) sparsity patterns. Can I somehow use this information to more efficiently compute its ...
HRI's user avatar
  • 113
0 votes
0 answers
22 views

using scipy.sparse.linalg.eigsh for degenerate states in Bose Hubbard model

I am currently writing a code for the Bose-Hubbard model, and I am calculating the ground states and single-particle density matrix for different values of U and J. As U=0, one would see how the ...
Lorenzo Carfora's user avatar
3 votes
0 answers
46 views

Preconditioned GMRES for nearly diagonalizable systems

I have been working with a matrix $A$ and preconditioner $P\approx A^{-1}$ that I've then applied GMRES to the (left) preconditioned linear system \begin{equation} P^{-1}Ax=P^{-1}b \end{equation} $P^{-...
Tucker's user avatar
  • 189
13 votes
8 answers
3k views

Real-world applications of eigendecomposition?

Cross-posted on Math.SE Are there real-world applications that call specifically for eigenvalues rather than singular values? I often see eigendecomposition used as "poor-man's SVD" For ...
Yaroslav Bulatov's user avatar
0 votes
0 answers
52 views

eigenvalues of inhomogeneous Helmholtz equation violate superposition using FEM

I am trying to solve the in-homogeneous Helmholtz equation with damping and forcing using finite element method (FEM) with FEniCSx. The equation is; $c^2\nabla^2p - c^2\omega i d \nabla^2p + \omega^2 ...
Ekrem Ekici's user avatar
0 votes
0 answers
18 views

How do you determine the Mott-Insulator to Superfluid transition in the Bose Hubbard System

I am doing some simulations on various systems expressed in 2nd quantization and one of the points of interest of mine was Phase transitions in the Bose-Hubbard model $$ H = \sum_{k} \{ t_k(b^\dagger_{...
Mephistopheles Faust's user avatar
12 votes
2 answers
2k views

Why are systems with clustered eigenvalues easy to solve?

I came across the following slide by Theo Diamandis & Zachary Frangella on what makes the linear system $Ax=b$ easy to solve using the conjugate gradient method. Transcription: CG converges ...
Yaroslav Bulatov's user avatar
3 votes
0 answers
132 views

Fast Fourier Transform on Meshes

I have a (closed, manifold, oriented) triangular mesh for which I build a matrix $L\in\mathbb{R}^{n\times n}$ discretising the negated Laplace-Beltrami operator. The matrix $L$ is symmetric positive ...
lightxbulb's user avatar
  • 1,994
1 vote
2 answers
123 views

Matlab eigs function with function handle

I am reading through the documentation of matlab function eigs specifically the function handle input version. Here it is: https://www.mathworks.com/help/matlab/ref/...
user8469759's user avatar
4 votes
1 answer
105 views

Numerical continuation of all eigenvalues of small, dense matrices

Consider a one-parameter family of matrices $A(q)\in \mathbb{R}^{n\times n}$, $q\in\mathbb{R}$. For my applications, $n$ is typically between $5$ and $50$ and $A(q)$ is generally dense, so direct ...
whpowell96's user avatar
  • 2,259
0 votes
0 answers
46 views

Different computational approaches to show that the conjecture is true

It seems that proving the statement given in the question https://math.stackexchange.com/questions/4601532/a-pen-and-paper-proof-for-a-matrix-implication is difficult analytically. I was thinking what ...
BAYMAX's user avatar
  • 229
4 votes
1 answer
1k views

Why are all eigen solvers iterative?

I have small dense square matrices for which I would like to compute the inverse by singular value decomposition, or equivalently solve the eigenvalue problem. While there are many direct methods for ...
Aurelius's user avatar
  • 2,365
2 votes
0 answers
54 views

Diagonalization of large sparse matrix, computational programme recommendation and methods

According to this link, All eigenpairs of large sparse symmetric matrix. The guy @Baranas seems to have given a very confident answer about solving the whole Eigen spectrum. May I know if anyone has ...
Lee Zhi Yan's user avatar
1 vote
1 answer
128 views

Conditioning and Stability of generalized eigenvalue problem

The (generalized) eigenvalue problems with a multiple eigenvalue are the ill-posed ones. I have two questions that should be simple for experts: (1) Is the eigenvalue problem much more sensitive to ...
alpx's user avatar
  • 113
1 vote
0 answers
84 views

Optimal Krylov subspace dimension and iteration limits for eigs

When using the eigs function in MATLAB, which is based off of ARPACK, one can manually modify the maximal dimension of the constructed Krylov subspaces, the maximum iteration counts, and the error ...
user45844's user avatar
3 votes
1 answer
226 views

Generalized eigenvalue problem for large, potentially ill-conditioned systems

Say that I have a generalized eigenvalue problem of the form $$Ax=\lambda Bx.$$ Using MATLAB, some naive ways that one may solve this is by either directly inverting $B$ then applying the ...
user45844's user avatar
1 vote
2 answers
373 views

How should I solve generalized eigenvalue problems in Python? (Orr-Sommerfeld equation)

I am trying to solve the Orr-Sommerfeld equation numerically, using the techniques given in this article. This leads to solving a generalized eigenvalue problem, that is, given two matrices $\mathbf A,...
K.defaoite's user avatar
1 vote
0 answers
74 views

Rayleigh-Ritz under LOBPCG : Nested iteration?

I am trying to understand the basic LOBPCG algorithm, as used in popular python libraries, primarily from the suggested resource on wiki. LOBPCG involves a subroutine to compute the coefficients of ...
Shark's user avatar
  • 11
3 votes
0 answers
71 views

Approximating eigenvalues of DPR1 matrix with special properties

In my application, I have a sum of diagonal $A$ and rank-1 $B$ $$T=\underbrace{\text{diag}(1-2\alpha h+2\alpha^2 h^2)}_A + \underbrace{\alpha^2 hh^T}_B$$ Where $h$ is a vector $\in \mathbb{R}^d$ with ...
Yaroslav Bulatov's user avatar
0 votes
0 answers
58 views

Eigenvalues of same operator expressed in two different orthonormal basis are coming out different

I have an operator $H$. I express $H$ as a matrix in the orthonormalized $\{ |e > \}$ basis. Then I diagonalize it to obtain eigenvalues, let's say for example $H$ is $6 \times 6$ and the ...
Snpr_Physics's user avatar
4 votes
1 answer
146 views

Eigenvalues of diagonal plus rank-one

I need to compute an eigendecomposition of an $n\times n$ matrix $$ D + c vv^\top = Q\Lambda Q^\top \tag{1} $$ in MATLAB, where $D$ is a real diagonal matrix, $c$ is a scalar, and $v$ is a real vector....
eepperly16's user avatar
2 votes
1 answer
139 views

qr-algorithm to find eigenvalues not returning expected values

I tried to compute eigen values with the QR-algorithm found here (there is also a wikipedia page also) ...
roi_saumon's user avatar
0 votes
2 answers
373 views

Solving a generalized eigenvalue problem with Chebyshev spectral method: How to impose boundary conditions into the matrices?

I'm solving a local instability problem for a pipe Poiseuille flow. The coordinate system is columnar, i.e., ($r,\theta,x$) (radial, tangential and axial). The basic flow is $\bar{u_r}=0, \bar{u_\...
Jack's user avatar
  • 1
2 votes
0 answers
43 views

Iterative methods for underestimate of smallest eigenvalue for large sparse matrices

I recently read the paper "EUCLIDEAN-NORM ERROR BOUNDS FOR SYMMLQ AND CG" by Estrin et al. and there they use an underestimate (i.e. something in $(0,\lambda_{min}]$) of the smallest ...
lightxbulb's user avatar
  • 1,994
2 votes
0 answers
173 views

Generalized Eigenvalue Problem using MATLAB

I'm trying to solve a generalized eigenvalue problem. I have two matrices $H$ and $S$ such that: $$ HX=λSX $$ I need to find the eigenvalues $\lambda$. The matrices $H$ and $S$ are real, asymmetric, ...
Beginner Noob's user avatar
2 votes
1 answer
113 views

Finding the correct order of eigenvectors of a parameter-dependent Hermitian matrix

so, I have a symmetric, analytic matrix $\mathbf{H}(x)$ ($x$ is real). Because $\mathbf{H}(x)$ is analytic and $x$ is real, it is possible to find analytic functions for the eigenvectors and the ...
cheetah's user avatar
  • 153
2 votes
1 answer
145 views

Numerical diagonalization of Hamiltonian

Framework I am trying to diagonalize the Bogoliubov-de Gennes Hamiltonian. The problem is that the Hamiltonian contains a Laplacian. This could be solved by using a discretized Laplacian. How I tried ...
Guest's user avatar
  • 21
1 vote
0 answers
104 views

Well-conditioned pseudospectral for computing eigenvalues to (partial) differential equations

I am working on writing a Chebyshev pseudospectral method (see for example "Chebsyhev and Fourier spectral methods" by John Boyd) to solve for the eigenvalues of differential equations of ...
physics_researcher's user avatar
1 vote
1 answer
1k views

How to implement Lax-Friedrich flux splitting with WENO scheme

I'm working on modeling a shock wave using the Euler equation with an advanced Equation of state and the fifth order WENO scheme. The equation are set up on the form: \begin{equation} \frac{\partial U}...
Twm1995's user avatar
  • 55
2 votes
1 answer
170 views

LOBPCG bad preconditioned performance for largest eigenpairs

The LOBPCG algorithm finds eigenpairs of the generalized eigenproblem $$ Ax = \lambda B x $$ where $B$ is symmetric and positive-definite, $A$ is symmetric. One of the features that makes LOBPCG so ...
Nico Schlömer's user avatar
0 votes
0 answers
75 views

Eigen-decomposition one eigenpair by one eigenpair?

Is it possible to conduct an Eigen-decomposition of a matrix one eigenpair by one eigenpair? And related to this question, what is the time complexity of truncated eigendecomposition? I am trying (...
Tan's user avatar
  • 109
7 votes
0 answers
136 views

Can we sparse solve a few eigenvalues specified by index range?

I need to solve a few eigenvalues of a large sparse matrix specified by their index range. These indices are according to the whole eigenspectrum sorted in algebraic (not absolute value) ascending ...
xiaohuamao's user avatar
1 vote
1 answer
1k views

Numerical Solution of the Schrödinger equation for hydrogen

I'm trying to solve the Schrödinger equation for the hydrogen atom in the following form numerically: $$\left[-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+V(r)+\frac{\hbar^2l(l+1)}{2mr^2}\right]R(r)=ER(r).$$ ...
mysterion123's user avatar
3 votes
1 answer
526 views

A method for finding the number of eigenvectors with a given, known eigenvalue

Is there a method for finding the number of eigenvectors with a given eigenvalue? I do not need the eigenvectors themselves, and to find the eigenvectors seems quite tough, given the comments on the ...
user196574's user avatar
4 votes
1 answer
99 views

Roundoff errors in FEM computations - generalized eigenvalues

This is a continuation of my previous question. I am trying to effectively compute a bound for the roundoff errors in some FEM computation (2d polygons, triangular meshes). Below I will write some of ...
Beni Bogosel's user avatar
  • 1,035
1 vote
1 answer
185 views

Accelerated Inverse Power Method with Rayleigh Quotient

I am considering implementing the accelerated inverse power (AIP) method with Rayleigh quotient to speed up eigendecomposition of a real square symmetric matrix. Halton (1996) gives an example ...
user avatar
3 votes
4 answers
499 views

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

I met a problem in solving a set of coupled differential equation, as shown below: $$A_1\psi_1(z)+A_2\frac{d^2\psi_1(z)}{dz^2}+A_3\frac{d\psi_2(z)}{dz}=\lambda\psi_1(z)$$ $$A_4\psi_2(z)+A_5\frac{d^2\...
JensenPang's user avatar
19 votes
3 answers
3k views

Why do we usually not want the eigenvalues of non-symmetric matrices?

I came across this line in a class note I am reading where it discusses finding eigenvalues of matrices. In reality we don't go all the way with Arnoldi. We stop at a decent value of 𝑘. Then the 𝑘 ...
CuriousMind's user avatar
3 votes
0 answers
205 views

How to obtain smallest eigenvalues with Arnoldi iteration

I understand that the Arnoldi iteration produces a basis which tends to include in its span the eigenvectors corresponding to eigenvalues of large magnitude (hence the analogy between the last vector ...
Pedro Secchi's user avatar
1 vote
1 answer
550 views

Numerical solution to the infinite well problem

I've used the following code to implement it ...
Photon's user avatar
  • 33
1 vote
0 answers
55 views

Reproducing a paper's result for Topological Insulators

For the past weeks I have been trying to reproduce Agarwala's results but I've been unsuccessful. From this paper I am trying to reproduce the first and last columns of Fig.2, by implementing eq.2; ...
Victor RM's user avatar
3 votes
1 answer
257 views

`eigsh` (Lanczos algorithm) slows down for degenerate eigenvalues

I have a complex Hermitian matrix of size about $70000\times 70000$. I want about 100 eigenvalues near 0. However, I know that every eigenvalues are two-fold degenerate. I found out that the running ...
Laplacian's user avatar
  • 171
1 vote
1 answer
624 views

Eigenvalue problem for ordinary differential equation

I am trying to compute the natural frequency of a cantilevered beam. The Euler-Bernoulli equation reduces to the following problem : $$ v''''=\lambda v, \text{with }, v(0)=0, v'(0)=0, v'''(1)=0,v''(1)...
GMV871's user avatar
  • 33
0 votes
1 answer
115 views

How are eigenvalues of a cell in the finite volume discretization calculated? (for euler fluid equations)

I am building a program to numerically solve the euler fluid equations based on finite volume discretization and am having trouble on a step where the eigenvalues of a cell need to be calculated ...
Frosty's user avatar
  • 27
3 votes
0 answers
136 views

Does shift-invert method has invertibility issue?

Please note that I have nearly zero background on numerical methods. I understand the shift invert method as described in SciPy Tutorial The main argument of the above link is as follows. Suppose we ...
Laplacian's user avatar
  • 171
2 votes
4 answers
957 views

Eigenvalue decomposition for a very huge matrix of medical images (such as the pixel physical coordinates of CT images)

I am trying to do eigenvalue decomposition for a huge matrix larger than 788000×788000 for medical image analysis. The matrix is not sparse and every element in the matrix has a real value. And, for ...
Michael Cheng's user avatar
0 votes
1 answer
326 views

Pseudospectrum of non square Matrix in Python

I have a rectangular matrix $A \in \mathbb{R}^{m \times n}$ ...
user avatar

1
2 3 4 5 6