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).

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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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65 views

What's the optimal method to solve for the top eigenvectors of a very large, real, symmetric matrix of limited rank?

Consider a real symmetric matrix of dimension N~10^5 and rank m~2000. What is the most efficient algorithm for determining the top m eigenvectors? If the answer isn't obvious, are there existing ...
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40 views

Minimum effort merging of two sets

I have the following problem. I have two sequences of elements $A = [a_1,a_2,\cdots,a_n]$ and $B = [b_1,b_2,\cdots,b_m]$. I can build a matrix $D[n \times m]$ where $d_{ij} = d(a_i,b_j)$ My greedy ...
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Estimating the second largest eigenvalue

I am currently dealing with the following problem. I am given a matrix $A$ of order $n \times n$ where $n \leq 20.$ The principal $(n-1) \times (n-1)$ matrix of $A$ is symmetric and contains only $\{...
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165 views

Compute eigenvalues with Arpack

I am using Arpack to compute the eigenvalues of the problem $\lambda Mx = Ax$ with reverse shift method with complex shift. $A$ and $M$ are real, $M$ is symmetric. Then, I use znaupd e zneupd. I use ...
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I am looking for a complex sparse matrix EigenVector solver for GPGPU; preferably CUDA

So far the closest I've found is ViennaCL, which has a Lanczos implementation for Eigenvalues. It is not clear that EigenVectors are produced by this library. Does anyone here know whether ViennaCL ...
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105 views

smallest eigenvalues for linear elasticity

I want to compute a few tens of the smallest eigenvalues of a linear system which is a discretization of a linear elasticity. In the presence of additional constraints like Dirichlet boundary ...
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89 views

Eigenvalue analysis of preconditioned partial differential operator

today, I encountered a confused problem by accident, but I have no ideas to deal with it fully. The question can be described as follows: for example, when we need to use FDM/FEM to discrete the ...
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79 views

Time-stepper approach to eigenvalue problem

For a linear system $$ M \dot{u} = Au \qquad \textrm{or} \qquad \dot{u} = L u $$ The generalized eigenvalue problem is $$ A e = \lambda M e $$ We can use the time-stepper approach which essentially ...
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187 views

Shift-Invert in Anasazi/Belos using Tpetra Sparse Matrices

I am currently trying to port my code over to Trilinos because the problems that I am working on are too big for LAPACK/ARPACK. Specifically I am computing the generalized eigenvalues/eigenvectors: $$...
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95 views

Is it possible to construct such a symmetric matrix with desired eigenvalues?

Suppose a real, dense and asymmetric square matrix $A\in\mathbb{R}^{n\times n}$, all its eigenvalues $\lambda_i \in \mathbb R$ Is it possible to construct a symmetric matrix $B\in\mathbb{R}^{n\times ...
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138 views

Efficient way to do congruent transformation using matrix inverse?

I know a square self-adjoint matrix $S_{vv}$ and I want to find: $S_{rr} = HS_{vv}H^{\dagger}$ where $\dagger$ denotes conjugate transpose. I do not know $H$ but I do know $H^{-1}$. What is the ...
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51 views

Size reduction of matrices in dispersion curve calculation

I have an energy dispersion curve obtained from the eigenvalues of $$E(k) = \text{eig}(T e^{ik} + T^H e^{-ik} + H_0),$$ where $H_0$ and $T$ are $N\times N$ square matrices, $T^H$ is the Hermitian ...
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381 views

Eigenvalue problem with periodic boundary conditions: Are my eigenvalues correct?

I am using a (central) finite difference scheme to solve the eigenvalue problem $$-\frac{d^2}{dx^2}u = \lambda u$$ with periodic boundary conditions on a unit interval. I use arpack's zndrv1 and ...
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301 views

Matlab, Mathematica & LAPACK returning 3 different eigenvectors

(I'm not sure which of math.se / stackoverflow / scicomp.se is the right place to ask this question) I have a C++ code which generates a complex matrix and then calculates its eigenvalues and ...
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Order of eigenvalue problem using c++ Eigen library

I have the following 6x6 matrix (taken from Google Books p. 129): For background info: All the entries depend on the momentum $k$. Getting the eigenvalues of this matrix for each $k$ corresponds to ...
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1answer
182 views

Solve eigenvalue problem using finite differences without vectorization

I am interested in solving the problem $-A u = \lambda u$ on a finite differences grid on a square. In my case, the operator $A$ is of the type $-\Delta + \mu I$, where $\mu$ is there to impose some ...
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Flop counts for LAPACK symmetric eigenvalue routines DSYEV, DSYEVD, DSYEVX and DSYEVR

LAPACK has following 4 routines for calculating eigenvalues of a real symmetric matrix; namely DSYEV, DSYEVD, DSYEVX and DSYEVR (DSYEVR being the recommended one). If I were to calculate both ...
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1answer
135 views

How does the animation work in eigenvalue problem of FEM

I have used free vibration analysis in FEM. After analysis, we can usually use animation to see the motion of each eigenmode (In Abaqus or Comsol, I would choose either half harmonic or full harmonic)...
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631 views

Computing Eigenvectors in MATLAB

I am assigned to compute eigenvalues and eigenvectors in MATLAB of a 2x2 matrix: $$ A = \left( \begin{matrix} 3 &0\\ 4 &5\\ \end{matrix} \right) $$ I know that the textbook's solution states ...
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1answer
79 views

Most efficient way to compute eigenvectors / values of this matrix?

I have a symmetric $ 3 \times 3 $ matrix $A$ and I need to compute the eigenvectors and eigenvalues of this. I know that I can use something like Lapack, but I also know that this can be computed ...
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1answer
95 views

Construct tridiagonal matrix from eigenvalues

I have a sort of reverse problem, and I'm not sure if there is a simple solution. I have a tridiagonal Hermitian matrix: $$ A = \begin{bmatrix} 0 & a_1 & 0 & 0 & 0 \\ a_1 & 0 &...
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157 views

Computing eigenpairs of singular matrix with ZGEEV?

I've never run into a singular matrix before, so bear with me. I have a complex non-symmetric matrix (about 1000 x 1000) that I know has a couple zero eigenvalues. It isn't guaranteed to be ...
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1answer
62 views

Analytic formula for $\arg\max_{\|z\|_\infty \le 1}z^T A z$, where $A=uu^T+vv^T$

Let $u$ and $v$ be column vectors of size $n \gg 1$ (not both zero), and consider the matrix $A:=uu^T+vv^T$ Question What is an analytic formula for $\arg\max_{\|z\|_\infty \le 1}z^TAz=\arg\max_{\|z\...
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77 views

Numerical Stability of a Generalized Spatial Discretization Scheme

After reading the matrix stability chapter (10) of Hirsch [1], I decided to dive in the reference list of the chapter. One of the papers [2], which is cited as reference shows an very interesting ...
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52 views

An Upper Bound of $\left<H,X\right>_F$ with Constrainted Rank

We have $X=[\mathbf{x}_1,\mathbf{x}_2...,\mathbf{x}_n]\in\mathbb{R}^{d\times n}$, $H=[\mathbf{h}_1,\mathbf{h}_2...,\mathbf{h}_n] \in\mathbb{R}^{d\times n}$, and $d<n$. $H$ has rank $r\leq d$ and $X$...
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1answer
58 views

LAPACK zlapmt_ freezes code [closed]

I'm using LAPACK's zggev_ routine to solve some generalized eigenvalue problem. While it produces the correct results, I want the eigenvalues and according eigenvectors sorted by absolute value. For ...
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1answer
62 views

Examples of finding eigenfunctions of coupled DEs

I am looking for examples of numerically finding eigenvalues/eigenfunctions of coupled DEs. If anyone is able to point me towards any examples, preferably with code included, it would be much ...
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1answer
251 views

Boundary conditions generalized eigenvalue problem

Consider the following eigenvalue problem \begin{equation} \mathcal {L} x(s) = \lambda x(s), \end{equation} where \begin{equation} \mathcal {L} = \alpha \partial^4_s + (s^2-1)\partial^2_s + s \...
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224 views

Diagonalize a circulant-plus-rank-one matrix

Suppose $A=uv^T$ where $u$ and $v$ are non-zero column vectors in $\mathbb{R}^n, n\geq 3$. $\lambda=0$ is an eigenvalue of $A$ since $A$ is not of full rank. $\lambda=v^Tu$ is also an eigenvalue of $A$...
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2answers
3k views

Most efficient library to diagonalize exactly large hermitian or unitary matrices

I am working on a physics problem which requires obtaining the exact eigenvalues and eigenvectors of Hermitian and Unitary matrices numerically. Naturally I would like to ask the experts what are the ...
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42 views

Are there any constraints on eigenvalues that are used in inverse iteration?

What is the result of the method for multiple eigenvalues? Is there any case for which this method will not work altogether?
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69 views

Numerical scheme to calculate the normal mode of a set of hyperbolic PDEs?

I would like to solve the linearised, ideal, MHD equations, where the gas pressure is zero. $$\frac{\partial u_x}{\partial t}=v_A^2(x,z)\left[\nabla_{||}b_x - \frac{\partial b_{||}}{\partial x}\right],...
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43 views

Fast algorithm for computing lower mode shapes and natural frequencies in MATLAB using sparse stiffness and mass matrices

I am looking for a fast algorithm for computing eigenvalues and eigenvectors from sparse stiffness and mass matrices in MATLAB. The eig(K, M) doesn't work with ...
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24 views

Simulating Anderson model, have problem with momentum representation (MATLAB)

I want to change from real-space representation to momentum-space representation I have a Hamilton-operator (Anderson-model), and I calculated some kind of entropy of its eigenstates (this is working, ...
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114 views

Finding second excited state of Schrödinger equation with secant Runge Kutta method

In our assignment, we are required to find the energies of the ground state and the first two excited states of the Schrödinger equation in a harmonic potential: $$V = \frac{50 x^2}{(10^{-11})^2}\, .$...
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Armadillo eig_sym() for extracting eigenvalues. Is it parallel at all? [closed]

After wasting 3 days with scalapack, I gave up and moved to Armadillo, considering it uses lapack underneath its beatiful and easy interface. I would like to calculate the eigen values and eigen ...
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132 views

Algorithms to compute largest gap between smallest nonzero eigenvalues of sparse symmetric matrix

I am looking mainly for c/c++ implementations but also for theoretical algorithms to compute gaps between smallest positive eigenvalues of symmetric, singular matrix or real numbers. To be precise, I ...
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2answers
72 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 ...

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