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
-2
votes
1answer
65 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 ...
0
votes
0answers
84 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 ...
2
votes
0answers
75 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 ...
2
votes
1answer
195 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 ...
-1
votes
2answers
149 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 ...
3
votes
1answer
2k 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 ...
3
votes
2answers
614 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 ...
8
votes
1answer
141 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)-...
7
votes
0answers
152 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, ...
0
votes
0answers
38 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 ...
3
votes
0answers
72 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....
1
vote
0answers
64 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 ...
6
votes
1answer
327 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} ...
2
votes
1answer
190 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 ...
7
votes
0answers
152 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\...
1
vote
0answers
47 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 $-i$?
2
votes
1answer
104 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 = \...
1
vote
0answers
53 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
votes
2answers
189 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 ...
2
votes
1answer
179 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 ...
5
votes
1answer
140 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}^...
4
votes
1answer
289 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 ...
6
votes
1answer
128 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 ...
2
votes
0answers
179 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 $k\times N$ ...
4
votes
1answer
442 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
263 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: ...
0
votes
1answer
113 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 ...
1
vote
0answers
56 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, ...
1
vote
0answers
45 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 $\...
1
vote
0answers
53 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=...
8
votes
0answers
340 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, ...
6
votes
1answer
347 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 ...
5
votes
0answers
154 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
vote
1answer
151 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(-...
5
votes
0answers
83 views

How to construct a eigensolver targeting a specific type of matrix

I need to diagonalize such kind of matrix during research: it's n-by-n, with it's upper-left (n-1)-by-(n-1) corner be diagonal while the nth row & column are dense. It's observed that the self ...
1
vote
0answers
218 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
522 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 ...
2
votes
0answers
57 views

Scaling for a nonsymmetric eigenvalue problem

I have an eigenvalue problem emerging from the internal vibro-acoustic coupling. The eigenvalue problem is nonsymmetric but it was proven in literature that it results in real eigenvalues and ...
7
votes
1answer
592 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
117 views

Stability of dark solitons in a harmonic trap

This question is based upon a research article which I am trying to reproduce. One of the main result of this paper is the condition on transverse confinement of the Bose-Einstein Condensate(BEC) to ...
2
votes
1answer
380 views

Preconditioning ARPACK eigenvalue solver

I am working on a generalized eigenvalue problem of the form $$ \boldsymbol{A}\cdot\boldsymbol{x}=\lambda\boldsymbol{B}\cdot\boldsymbol{x} $$ where $\boldsymbol{B}$ is not symmetric positive. ...
1
vote
0answers
204 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 ...
0
votes
2answers
147 views

Efficient Eigen Solver

I am working on a flow physics problem (2D), which ultimately boils down to solving Eigen value problem. Even for a grid size of 60 x 60, the matrix in the Eigen value problem end up with 1260 million ...
2
votes
0answers
153 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 ...
2
votes
0answers
352 views

Numerical solution of Dirac equation (eigenvalue problem)

Suppose we have equation of the form: $$H \Psi = E \Psi $$ where $H$ is Dirac Hamiltonian (also my question can be answered by people who are not familiar with Dirac Hamiltonian but familiar with ...
4
votes
0answers
185 views

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....
4
votes
0answers
70 views

Differential eigenproblem with eigenvalue in boundary condition

Statement of the problem I need to (numerically) solve an eigenproblem of the type $$-\omega^2\mathcal{D}_1\vec{x}=\mathcal{D}_2\vec{x}$$ on the interval $[-1,1]$, where $\mathcal{D}_1$ and $\...
6
votes
2answers
1k views

Numerical Solution to Schrödinger Equation--Multiple Wells

I am trying to solve for the allowed wavefunctions and energies for a 1D quartic potential well. To do this I am using the patching method (https://engineering.dartmouth.edu/microeng/otherweb/...
2
votes
1answer
362 views

finite difference frequency domain Eigenvalue matrix to get eigenmodes

I am using the FDFD method to calculate eigenmodes for an empty wave guide. It's a 1x1meter structure with a PEC boundary. (Here I have 6x6 points to make it simple). Sounds simple, but I can't get it ...
6
votes
1answer
168 views

Roots of a function for eigensystem

I want to find the roots for $\kappa$ for the equation $$\sqrt{\alpha - 1} \cos{\left (\frac{\sqrt{2} \sqrt{\alpha - 1}}{2 \sqrt{\epsilon}} \right )} \cosh{\left (\frac{\sqrt{2} \sqrt{\alpha + 1}}{2 \...