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.
181
questions
1
vote
0
answers
32
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 ...
- 403
1
vote
1
answer
96
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. ...
- 403
3
votes
1
answer
335
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 ?
- 31
1
vote
0
answers
93
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 ...
- 11
1
vote
0
answers
91
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;
$$c^2\nabla^2p + w^2p=0 \qquad \...
- 11
1
vote
0
answers
41
views
Simulating Quantum Wave Function/Schrodinger Equation With A Time Varying Potential
I have solved the Time Independent Schrodinger Equation using the Numerov method and diagonalizing the Hamiltonian, in 1 - 3 dimensions. I suppose I could time-evolve it by multiplying every element ...
- 33
2
votes
0
answers
63
views
Sparse generalized symmetric eigensystem solver
Can anyone recommend a good software for solving generalized symmetric eigenvalue problems of the form,
$$
A x = \lambda B x
$$
where $A,B$ are symmetric and sparse, and $B$ is positive definite?
I ...
- 1,038
4
votes
2
answers
150
views
Numerical estimation of eigenfunctions of Laplacian
Consider the Laplace equation,
$$
\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)...
- 1,038
0
votes
2
answers
118
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 ...
- 13
2
votes
1
answer
147
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 ...
- 3,108
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 (...
- 109
5
votes
1
answer
222
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 ...
- 151
10
votes
1
answer
499
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 $\odot$ ...
- 173
0
votes
1
answer
160
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 ...
- 21
1
vote
1
answer
171
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 ...
user39981
3
votes
4
answers
357
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\...
- 131
-2
votes
1
answer
92
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 ...
- 1
0
votes
0
answers
118
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 ...
- 1,381
2
votes
0
answers
117
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 ...
- 201
3
votes
1
answer
595
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 ...
- 33
-1
votes
2
answers
458
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 ...
- 95
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 ...
4
votes
3
answers
1k
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 ...
- 139
8
votes
1
answer
155
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)-...
- 81
7
votes
0
answers
329
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, ...
- 171
0
votes
0
answers
47
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 ...
- 331
3
votes
0
answers
111
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)$$
gives the first five eigenvalues with $k=0$ or $k=1$ which are $2.06$, $2.26$, $5.16$, $6.81$, and $7....
- 51
1
vote
0
answers
116
views
How can I maximise orthonormality between degenerate eigenvectors using ARPACK?
I am using ARPACK's zndrv1 to diagonalise a matrix (the context is quantum chemistry). While all vectors have a norm 1, as expected, vectors corresponding to degenerate eigenstates aren't always ...
- 403
5
votes
1
answer
700
views
Dirichlet boundary conditions in generalized eigenvalue problem
Let us consider a problem of the form
$$(\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} ...
- 8,312
3
votes
1
answer
293
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
...
- 203
7
votes
0
answers
218
views
Solving a coupled eigen value problem
I have the following problem:
$$\begin{bmatrix}A &B \\C& D\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}\lambda I_m & 0 \\ 0& \mu I_n\end{bmatrix}\begin{bmatrix}x \\y\...
- 171
1
vote
0
answers
51
views
The sign of Schrodinger equation
I have a question for the format of Schrodinger equation
$$\psi(x,t) = \int_0^\infty c_n e^{-iE_nt/\hbar} \psi_n(x)$$
Why do we have $i$ instead of $-i$?
- 11
2
votes
1
answer
239
views
Discrepancies between numerical and analytical solution for particle in a finite potential well?
Analytical
Inside the box, the wavefunction is:
\begin{equation}
\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} = E \psi(x) \iff
\frac{d^2 \psi(x)}{dx^2} = k^2 \psi(x)
\end{equation}
where $k = \...
- 125
1
vote
0
answers
58
views
Methods to compute specific eigenvector components for a tridiagonal matrix
I have an application that is somewhat similar to the situation of computing Gaussian quadrature nodes and weights: simply put, I need to compute the eigenvalues and the last two (normalized) ...
- 11
10
votes
2
answers
271
views
Benchmark problems for eigenvalue reordering algorithms sought
Every real matrix $A$ can be reduce to real Schur form $T = U^T A U$ using an orthogonal similiary transform $U$. Here the matrix $T$ is quasi-triangular form with 1 by 1 or 2 by 2 blocks on the main ...
- 1,381
2
votes
1
answer
261
views
Can the Power Method be used here?
Given a set of $n$ points on which a triangulation is performed, it is possible to construct coefficients $\lambda_{ij}>0$ such that each point $x_i$ is a convex combination of the points connected ...
- 41
5
votes
1
answer
166
views
Generate approximately semi-orthogonal tall matrix approximately satisfying constraints
I have a set of matrices $\{(A_i,D_i)\}$ for $i\in\{1,\ldots,n\}$, where:
Each $D_j\in\mathbb{R}^{S\times S}$ is diagonal, and every entry on the main diagonal is non-negative.
Each $A_j\in\mathbb{R}^...
- 183
4
votes
1
answer
387
views
Matrix exponential of hermitian matrix with eigenvectors from generalized eigenvalue problem
I want to calculate the following expression
$$
\exp(-i\Delta t\mathbf{H})
$$
where $\mathbf{H}\in\mathbb{C}^{n\times n}$ is a hermitian matrix. Since I have a highly optimized eigensolver in the code ...
- 83
6
votes
1
answer
164
views
Eigenvalue problems with extremely small gaps
I'm interested in numerically diagonalizing a class of structured, symmetric eigenvalue problems with potentially extremely small eigenvalue gaps. The question I have is how to design a numerically ...
- 303
3
votes
0
answers
348
views
Left eigenvectors using ARPACK
I'm trying to find both the dominant $k$ left and right eigenvectors, that is,
$$V_L\mathcal{A} = \Lambda V_L\\
\mathcal{A}V_R = V_R\Lambda\\
V_LV_R = I_{k\times k}$$
$V_L$ being the $k\times N$ ...
- 31
4
votes
1
answer
717
views
Nystrom approximation of SVD for asymmetric matrices
Suppose I have a symmetric matrix $K$. Subdivide $K$ into pieces as
$$K=\begin{pmatrix} K_{11} & K_{12} \\ K_{21} & K_{22}\end{pmatrix},$$
where $K_{21}=K_{12}^\top$. Then, the Nystrom ...
- 1,341
0
votes
1
answer
415
views
How to do a Generalized Complex Schur (or QZ) Decomposition with Eigen C++? [closed]
I would like to do a Generalized Schur (or QZ) decomposition for a pair of complex matrices $A$ and $B$.
I found the following class:
...
- 101
0
votes
1
answer
121
views
Non-linearities in modal analysis of flexible beam
I'm trying to analyse the behaviour of a wind turbine blade rigidly connected to a wall during fatigue testing, being excited in it's first mode in two orthogaonal directions simultaneously (the first ...
- 143
1
vote
0
answers
63
views
Solving 2D Schrodinger Equation with ARPACK: Can I ensure all eigenvectors have the same phase?
I use arpack to solve the 2D Schrodinger, and eigenvalue problem of the form
$$Hx = \epsilon x$$
on a uniform grid. All eigenvectors are real in my case.
Arpack doesn't normalise the eigenvectors, ...
- 403
1
vote
0
answers
52
views
When numerically computing eigenstates during a coupled-mode-space NEGF calculation, do phases matter?
The coupled mode space NEGF method for computing transistor characteristics involves expanding the electronic wavefunction in a mode space basis
$$\Psi(x,y,z) = \sum_n\phi_n(x)\xi_n(y,z;x)$$
where $\...
- 403
1
vote
0
answers
68
views
Eigenvalue ODE in Spherical Coordinates--Numerical
I wish to solve an eigenvalue problem:
$$\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=...
- 105
8
votes
0
answers
421
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, ...
- 243
6
votes
1
answer
622
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 ...
- 243
5
votes
0
answers
158
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 ...
- 1,341
1
vote
1
answer
216
views
Matrix exponential by eigenvectors - implementation issues
I posted a similar question yesterday but I deleted it since I think that I had to reformulate it after some insights.
I want to calculate
$$
\exp(-i\Delta t\,\mathcal{H}) = V\,\mathrm{diag}(\{\exp(-...
- 83