Skip to main content
OverflowAI is here! AI power for your Stack Overflow for Teams knowledge community. Learn more

Questions tagged [eigensystem]

An eigenvector of an operator is a vector such that the action of the operator is the same as multiplication by a constant, called the eigenvalue. The eigensystem of an operator is the set of all such eigenvectors and their associated eigenvalues.

Filter by
Sorted by
Tagged with
4 votes
0 answers
47 views

Robust Methods to Compute Left Eigenvector when Given Right Eigenvector and its Corresponding Eigenvalue

Given a corresponding right eigenpair, $(\lambda, v)$, of $A$, what robust methods exist for computing a left eigenvector, $w$ that satisfies $A^* w = \bar{\lambda}w$? My operator is has dimension on ...
vlovero's user avatar
  • 63
1 vote
1 answer
69 views

'eigs()' in Matlab gives inaccurate eigenvector when only several eigenvalues are calculated

I would like to report an issue which may be interesting in computational physics. Sometimes, to save time and memory, we use eigs() to calculate the first several ...
basic nutshell's user avatar
0 votes
0 answers
32 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
0 votes
0 answers
58 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
1 vote
0 answers
34 views

Can any discretization scheme reproduce the kane quasi-linear dispersion relation?

It is straightforward to solve the eigenproblem $$\hat{k}^2 \psi = E\psi$$using a finite-difference method. But what about$$\hat{k}^2 \psi = E(1+\alpha E)\psi$$I know it is possible to first solve the ...
DJames's user avatar
  • 417
1 vote
1 answer
166 views

Can Spectra find eigenvectors and eigenvalues of complex-valued matrices?

I am using Spectra to iteratively solve large-scale eigenvalue problems. I like it because it readily works on windows, and is header-only. I would like to know if it supports complex-valued matrices. ...
DJames's user avatar
  • 417
3 votes
1 answer
774 views

Time and memory required to diagonalize a 18000 by 18000 matrix using numpy in python

Can someone give an estimate of the Time and memory required to diagonalize a 20000 by 20000 complex hermitian matrix using numpy in python ?
Snpr_Physics's user avatar
1 vote
0 answers
120 views

A staggered grid for an eigenvalue problem (linear stability analysis)

I'm interested in extending the concept of a staggered grid (commonly used to solve the incompressible Navier-Stokes equations) to a linear stability analysis context. For example, we can consider ...
Samantha B.'s user avatar
1 vote
0 answers
136 views

Interface condition for 1D Helmholtz equation using finite element method

I want to implement a 1D Helmholtz equation with jump condition. The domain is $x=[0,1]$ and both ends have Dirichlet boundaries($p$=0). The 1D strong formulation is; $$c^2\nabla^2p + w^2p=0 \qquad \...
Ekrem Ekici's user avatar
1 vote
0 answers
43 views

Simulating Quantum Wave Function/Schrodinger Equation With A Time Varying Potential

I have solved the Time Independent Schrodinger Equation using the Numerov method and diagonalizing the Hamiltonian, in 1 - 3 dimensions. I suppose I could time-evolve it by multiplying every element ...
cgbsu's user avatar
  • 33
2 votes
0 answers
67 views

Sparse generalized symmetric eigensystem solver

Can anyone recommend a good software for solving generalized symmetric eigenvalue problems of the form, $$ A x = \lambda B x $$ where $A,B$ are symmetric and sparse, and $B$ is positive definite? I ...
vibe's user avatar
  • 1,058
4 votes
2 answers
159 views

Numerical estimation of eigenfunctions of Laplacian

Consider the Laplace equation, $$ \nabla^2 f(r,\theta,\phi) = 0 $$ in spherical coordinates. We know that the solution to this equation can be derived analytically, and is given by, $$ f(r,\theta,\phi)...
vibe's user avatar
  • 1,058
0 votes
2 answers
122 views

Eigenvectors of Laplacian

I am studying introduction to Multigrid methods. In all tutorials, authors write that eigenvectors of Laplacian (1D, finite difference) are given as $w_k(x_i) = \sin(k \pi x_i),$ where $x_i$ is a ...
student1's user avatar
2 votes
1 answer
177 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
5 votes
1 answer
274 views

Are Python/MATLAB/Mathematica numerical eigenvectors affected by eigenvalue degeneracies outside region of calculation?

I have a discretized 2D mesh over which I calculate eigenvalues and eigenvectors of some Hermitian 2 x 2 matrix at each point along a closed loop parameterized by parameter t. The eigenvectors are ...
TribalChief's user avatar
10 votes
1 answer
517 views

Generalization of eigendecomposition problem

Let $A\in \mathbb{R}^{n\times n}$ and $v \in \mathbb{R}^n$. We recognize $Av=\lambda v$ for some scalar $\lambda$ as an eigendecomposition problem. Suppose $\mu \in \mathbb{R}^n$, and let $\odot$ ...
waic's user avatar
  • 173
0 votes
1 answer
184 views

Computing eigenvalues of Schrodinger equation with spin

I want to solve a 2-dimensional particle in box problem with two electrons in the quantum well.I would like to take into account spin of electrons and Coulomb interactions to compute singlet and ...
celerion's user avatar
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
547 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
-2 votes
1 answer
98 views

Forward Euler Adaptive Step Size Stability

Given with a generalization using adaptive times-stepping as then is it still reasonable to assume that to ensure stability of the Euler’s forward method we need the growth factor for all n to be ...
Dognuts's user avatar
0 votes
0 answers
139 views

Large scale nonsymmetric eigenvalue problems in practical applications

I am trying to determine need for solving large scale nonsymmetric eigenvalue problems in both industry and academia. I am interested in any kind of problem where we cannot assume that matrices are ...
Carl Christian's user avatar
2 votes
0 answers
139 views

Why is LAPACK (seemingly) suboptimal for packed and banded eigenvalue problems?

Based on this LAPACK routines list, it looks like there is no relatively robust representation (RRR) driver routine for either packed or banded symmetric eigenvalue problems. According to the relevant ...
Grayscale's user avatar
  • 201
3 votes
1 answer
755 views

Boundary conditions in a finite element eigenvalue problem

I've been reading multiple papers and related posts for a while now, but I can't seem to find a specific answer to the issues I'm having so I hope someone can clarify things here. I'll provide some ...
n-claes's user avatar
  • 33
-1 votes
2 answers
522 views

2d Schrodinger Equation via matrix diagonalization in C

I am trying to solve the time-independent Schrodinger equation in two dimensions via discrete matrix diagonalization. I want the energy eigenvalues and the corresponding eigenfunctions for a given ...
user8384493's user avatar
4 votes
1 answer
3k views

Efficient ways to numerically evaluate matrix exponentials

What are some computationally efficient ways to solve matrix exponentials, i.e. functions of the form : $f(X)=e^{X}$, where $X$ is a square matrix? So far I have been able to diagonalise some ...
ScientificPythonNovice's user avatar
4 votes
3 answers
2k views

Fast and accurate eigenvalue computation for 3x3 posdef matrices

I am looking for a very fast and efficient algorithm for the computation of the eigenvalues of a $3\times 3$ symmetric positive definite matrix. The algorithm will be part of a massive computational ...
John Snow's user avatar
  • 139
8 votes
1 answer
188 views

Eigenvalue-like problem with coupled ODEs

I am looking at the following system of ODEs: \begin{array}{r}{\left[c_{2}(k)-\partial_{\tau}^{2}\right] \varphi_{2}\left(\tau \right)=f_{21}(\tau) \varphi_{1}\left(\tau \right)} \\ {\left[c_{1}(k)-...
KartMan's user avatar
  • 81
7 votes
0 answers
343 views

Implementation of Lanczos method that returns tridiagonal matrix

The Lanczos method can be used to obtain extremal eigenpairs of sparse symmetric or hermitian matrices. I know there are several implementations of the Lanczos method (as well as Arnoldi, Davidson, ...
delete000's user avatar
  • 171
0 votes
0 answers
48 views

A preconditioner for self-consistent iteration

I tried to derive a preconditioner for self-consistent iteration similar to section IX in arXiv:0804.2583. For simplicity, consider here only one orbital (one or two electrons) systems. Suppose that ...
tohoyn's user avatar
  • 331
3 votes
0 answers
117 views

Calculate the Bloch wave

The eigenvalue problem $$\frac{d^2u}{dx^2}+2i k\frac{du}{dx}-[k^2-6\sin(x)^2]u(x)=-\mu u(x)$$ gives the first five eigenvalues with $k=0$ or $k=1$ which are $2.06$, $2.26$, $5.16$, $6.81$, and $7....
yun shi's user avatar
  • 51
1 vote
0 answers
131 views

How can I maximise orthonormality between degenerate eigenvectors using ARPACK?

I am using ARPACK's zndrv1 to diagonalise a matrix (the context is quantum chemistry). While all vectors have a norm 1, as expected, vectors corresponding to degenerate eigenstates aren't always ...
DJames's user avatar
  • 417
5 votes
1 answer
829 views

Dirichlet boundary conditions in generalized eigenvalue problem

Let us consider a problem of the form $$(\mathcal{L} + k^2) u(\mathbf{x})=0\, ,\quad \forall \mathbf{x} \in \Omega$$ with Dirichlet boundary conditions $$u(\mathbf{x}) = 0, \quad \forall \mathbf{x} ...
nicoguaro's user avatar
  • 8,515
3 votes
1 answer
317 views

Eigenvectors associated to two quasi-degenerate eigenvalues

I need to find the smallest eigenvalue and the corresponding eigenvector of a sparse matrix $M$ whose dimension is $\approx 10^4$. Within Matlab enviroment, I use the command ...
AndreaPaco's user avatar
7 votes
0 answers
224 views

Solving a coupled eigen value problem

I have the following problem: $$\begin{bmatrix}A &B \\C& D\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}\lambda I_m & 0 \\ 0& \mu I_n\end{bmatrix}\begin{bmatrix}x \\y\...
xadu's user avatar
  • 171
1 vote
0 answers
52 views

The sign of Schrodinger equation

I have a question for the format of Schrodinger equation $$\psi(x,t) = \int_0^\infty c_n e^{-iE_nt/\hbar} \psi_n(x)$$ Why do we have $i$ instead of $-i$?
user4488's user avatar
2 votes
1 answer
257 views

Discrepancies between numerical and analytical solution for particle in a finite potential well?

Analytical Inside the box, the wavefunction is: \begin{equation} \frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} = E \psi(x) \iff \frac{d^2 \psi(x)}{dx^2} = k^2 \psi(x) \end{equation} where $k = \...
Ziezi's user avatar
  • 125
1 vote
0 answers
58 views

Methods to compute specific eigenvector components for a tridiagonal matrix

I have an application that is somewhat similar to the situation of computing Gaussian quadrature nodes and weights: simply put, I need to compute the eigenvalues and the last two (normalized) ...
篠原美菜子's user avatar
10 votes
2 answers
286 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 ...
Carl Christian's user avatar
2 votes
1 answer
287 views

Can the Power Method be used here?

Given a set of $n$ points on which a triangulation is performed, it is possible to construct coefficients $\lambda_{ij}>0$ such that each point $x_i$ is a convex combination of the points connected ...
kevin811's user avatar
5 votes
1 answer
172 views

Generate approximately semi-orthogonal tall matrix approximately satisfying constraints

I have a set of matrices $\{(A_i,D_i)\}$ for $i\in\{1,\ldots,n\}$, where: Each $D_j\in\mathbb{R}^{S\times S}$ is diagonal, and every entry on the main diagonal is non-negative. Each $A_j\in\mathbb{R}^...
user3658307's user avatar
4 votes
1 answer
407 views

Matrix exponential of hermitian matrix with eigenvectors from generalized eigenvalue problem

I want to calculate the following expression $$ \exp(-i\Delta t\mathbf{H}) $$ where $\mathbf{H}\in\mathbb{C}^{n\times n}$ is a hermitian matrix. Since I have a highly optimized eigensolver in the code ...
Lukk's user avatar
  • 83
6 votes
1 answer
170 views

Eigenvalue problems with extremely small gaps

I'm interested in numerically diagonalizing a class of structured, symmetric eigenvalue problems with potentially extremely small eigenvalue gaps. The question I have is how to design a numerically ...
deemaregee's user avatar
3 votes
0 answers
386 views

Left eigenvectors using ARPACK

I'm trying to find both the dominant $k$ left and right eigenvectors, that is, $$V_L\mathcal{A} = \Lambda V_L\\ \mathcal{A}V_R = V_R\Lambda\\ V_LV_R = I_{k\times k}$$ $V_L$ being the $k\times N$ ...
Nikko's user avatar
  • 31
4 votes
1 answer
758 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 ...
Justin Solomon's user avatar
0 votes
1 answer
457 views

How to do a Generalized Complex Schur (or QZ) Decomposition with Eigen C++? [closed]

I would like to do a Generalized Schur (or QZ) decomposition for a pair of complex matrices $A$ and $B$. I found the following class: ...
RangerBob's user avatar
  • 101
0 votes
1 answer
123 views

Non-linearities in modal analysis of flexible beam

I'm trying to analyse the behaviour of a wind turbine blade rigidly connected to a wall during fatigue testing, being excited in it's first mode in two orthogaonal directions simultaneously (the first ...
Peter Greaves's user avatar
1 vote
0 answers
63 views

Solving 2D Schrodinger Equation with ARPACK: Can I ensure all eigenvectors have the same phase?

I use arpack to solve the 2D Schrodinger, and eigenvalue problem of the form $$Hx = \epsilon x$$ on a uniform grid. All eigenvectors are real in my case. Arpack doesn't normalise the eigenvectors, ...
DJames's user avatar
  • 417
1 vote
0 answers
56 views

When numerically computing eigenstates during a coupled-mode-space NEGF calculation, do phases matter?

The coupled mode space NEGF method for computing transistor characteristics involves expanding the electronic wavefunction in a mode space basis $$\Psi(x,y,z) = \sum_n\phi_n(x)\xi_n(y,z;x)$$ where $\...
DJames's user avatar
  • 417
1 vote
0 answers
72 views

Eigenvalue ODE in Spherical Coordinates--Numerical

I wish to solve an eigenvalue problem: $$\nabla^{2}f=Ef $$ If I assume spherical symmetry $f(r,\theta,\phi)=f(r)$, I can reduce the problem to 1D: $$(\frac{2}{r}\frac{d}{dr}+\frac{d^{2}}{dr^{2}})f=...
Geoffrey Xiao's user avatar