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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.

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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\...
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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 $-i$?
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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 = \...
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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) ...
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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 ...
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118 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 ...
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119 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}^...
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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 ...
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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 ...
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55 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 $k\times N$ ...
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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 ...
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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 ...
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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: ...
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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 ...
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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, ...
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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 $\...
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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=...
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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, ...
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103 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 ...
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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 ...
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1answer
107 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(-...
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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 ...
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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_{...
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260 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 ...
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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 ...
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1answer
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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 ...
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1answer
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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 ...
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1answer
213 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. ...
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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 ...
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129 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 ...
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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 ...
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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 ...
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Compute all eigenvalues of a large sparse matrix [duplicate]

I am a physicist and I am trying to calculate the vibrational spectrum of a very large systems. I need whole spectrum of vibrational modes, that leads to a symmetric system. Basically, I need to solve ...
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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....
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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 $\...
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2answers
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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/...
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1answer
223 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 ...
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1answer
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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 \...
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Existing solver for large scale load dependent Ritz vectors problem

I have a large scale system with the following equations of motions $M\ddot z(t)+K z(t)=F(t)$ $M$ is the mass matrix $K$ is the stiffness matrix And I want to solve the equation for displacement $...
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Coupled system of differential equations with eigenvalue

I have the following system of coupled differential equations (the Bogoliubov-de Gennes equations for a certain geometry): $$ \begin{cases} -\frac{1}{2}u''(r) + \big[\frac{1}{2r^2} + f_1(r)\big]u(r) +...
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1answer
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Quality of eigenvalue approximation in Lanczos method

I try to familiarize myself with iterative eigenvalue solvers such as Lanczos. So I tried rewrite it to python directly according to wiki. But it doesn't seem to work. The problem: it approximates ...
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1answer
346 views

Fastest way to diagonalize square matrices with c++

I am currently writing software in c++ which solves the eigenvalue problem of sparse hermitian matrices. The size of the matrix depends on the user input, but as an estimation it will roughly be ...
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1answer
574 views

Smallest eigenvalue without inverse

Suppose $A\in\mathbb{R}^{n\times n}$ is a symmetric, positive definite matrix. $A$ is big enough that it's expensive to solve $Ax=b$ directly. Is there an iterative algorithm for finding the ...
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1answer
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Numerical computation of Perron-Frobenius eigenvector

I would like to efficiently and (to the extent possible) reliably find the Perron-Frobenius eigenvector of non-negative matrices. These are not stochastic matrices, they are typically dense, and their ...
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Memory requirement to find eigenvalues and -vectors of large sparse matrix

How can I estimate how much memory will be needed to find eigenvalues and eigenvectors of a given large sparse matrix? I have a real symmetric matrix with roughly $5 \times 10^4$ rows and columns, ...
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Eigenvalue-style optimization with quadratic constraints

Suppose $A\in\mathbb{R}^{n\times n}$ is symmetric and positive definite and that we have several symmetric matrices $B_i\in\mathbb{R}^{n\times n}$ that are low-rank and indefinite. I need an ...
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2answers
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Does this partial eigen-expansion have a name?

This question is a follow-up to this one. Let $A\in \mathbb{R}^{n\times n}$ be large, sparse, symmetric and positive definite. Suppose for I already know $m<n$ eigenpairs of $A$, corresponding to ...
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1answer
313 views

Julia: ordschur command

I just started leaning Julia, and my command of it is still very preliminary. I am trying to reorder a Schur decomposition. In Matlab, I could use the ordqz command and just had to specify the ...
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1answer
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What are some ideas to preprocess / precondition the following linear system?

Let $A\in \mathbb{R}^{n\times n}$ symmetric and positive semidefinite, and $\omega\in \mathbb{R}\setminus\{0\}$. I am interested in solving the following linear system for a range of values of $\...
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What strategies / decompositions would be useful to solve the following linear system repeatedly if I only care about time to solution?

Let $A\in \mathbb{R}^{n\times n}$ be symmetric positive semidefinite, and $B\in \mathbb{R}^{n\times n}$ be symmetric positive definite. Suppose $B$ is block diagonal so it is easy to invert. (We ...