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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163 views

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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89 views

solving a hyperbolic set of equations - upwind type method

I want to solve a set of hyperbolic equations (not the Euler equations) using an upwind type method. I am interested in using a first order upwind scheme and one that is not based on the method of ...
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151 views

what do zero real parts of eigenvalues mean? Any good references?

I am solving a 1D advection problem of the the form $$dQ/dt=[A]Q$$ where {Q} is the vector of unknowns and [A] is the matrix of coefficients of spatial discretisation. I have worked out the ...
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486 views

Accuracy of numerical methods in finding eigenvalues

I have a problem with assesing the accuracy of my numerical calculation. I have a 2nd order ODE. It is an eigenvalue problem of the form: $$ y'' + ay' + \lambda^2y = 0, $$ and the boundary condiations ...
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150 views

what do positive real parts of eigenvalues mean?

I am solving a 1D advection problem of the the form $$d{Q}/dt = [A]{Q}$$ where {Q} is the vector of unknowns and [A] is the matrix of coefficients of spatial discretisation. I have worked out the ...
2
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1answer
391 views

Improving my QZ-Algorithm (Include Shifts)

I Need to to solve an generalized Eigenvalue Problem and to compare two Methods (QR and QZ) concerning their convergence rate and execution time. I started with the Basic QR-Algorithm, implemented in ...
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405 views

linear stability analysis using spectral radius

I am analysing the stability of a series of 1D linear equations of the form \begin{equation} \frac{d}{dt} x = A x \end{equation} discretised using upwind and central finite volume methods, etc, with ...
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99 views

non convex, non linear optimization involving matrix differential equation solution

I'm trying to develop an inferential procedure for a multivariate dependent Markov process. Basically, the procedure could be considered as a non linear regression, with a known dependence structure ...
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454 views

Does the Lanczos starting vector have to be random?

In all descriptions of the Lanczos vector, it's said that the starting vector is random. But let's say I'm only interested in the eigenvector associated with the lowest eigenvalue (as is the case ...
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1answer
641 views

Sparse generalized eigensolver using OpenCL

I would like to solve a generalized eigenproblem of real sparse symmetric matrices. Is there an efficient library which utilizes OpenCL in order to find a limited amount of the smallest eigenvalues in ...
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2answers
285 views

How to correctly normalize modulus and phase of an eigenvector?

I am solving a linear stability problem using finite element discretization. Then, I have a generalised eigenvalue problem: $$ \lambda M x = J x.$$ I obtain complex eigenvalue and eigenvectors from ...
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178 views

Is it possible to use Eigtool for generalized problem pseudospectra?

I would like to use the Eigtool of professor Trefethen for pseudospectra but I have a generalized eigenvalue problem to solve: $$ \lambda M x = K x. $$ It seems that Eigtool takes only one matrix as ...
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How can I prove that two eigenvectors are orthogonal?

I obtained 6 eigenpairs of a matrix using eigs of Matlab. How can I demonstrate that these eigenvectors are orthogonal to each other? I am almost sure that I ...
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1answer
147 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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393 views

Arpack and Matlab give different values for eigenvalues

I am solving a generalized eigenvalues problem with inversed complex shift: $$(M-\sigma J)^{-1}J \boldsymbol{x} = \boldsymbol{x} \nu \enspace .$$ My matrices are obtained from a finite element ...
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4k views

Computational methods for finding the energy eigenvalues of the time-independent Schrodinger equation with arbitrary potential

I have seen in some papers that the energy levels in some arbitrary potential are specified. How can one find the energy levels in such arbitrary potentials. For example, $V(x)=\sin^2(x/2)$ with $x\in[...
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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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1answer
62 views

Locally evaluate nonlinear dynamic system's stability using eigenvalues

I'm working with Computational Neuroscience. I have a large Synaptic Matrix (x axis: presynaptic NeuronID, y axis: postsynaptic NeuronID) in a Modular network. This matrix is close to a random one and ...
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1k views

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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1answer
140 views

Correctly orthogonalizing and normalizing eigenvectors of a non-hermitian problem

I have some non-hermitian matrix $A$, that I have the left and right eigenvectors. (Calculated using SLEPc, by finding the eigenvectors of $A$ and $A^H$). I'm not sure how to orthogonalize them ...
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1answer
248 views

Compute eigenvectors of a matrix with known eigenvalue spectrum

If I have already accurately known the eigenvalue spectrum (i.e. all eigenvalues) of a matrix, is there any efficient numerical algorithm to compute all the eigenvectors corresponding to these ...
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1answer
567 views

Derivative of a generalized eigenvalue problem

I want to compute the derivative of a generalized eigenvalue $\lambda$ which is solution of $A u = \lambda Bu$ ($A,B,u,\lambda$ all depend on $t$; in my case $A,B$ are known explicitly, and the ...
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1answer
402 views

roots of polynomials with small coefficients

I would like to compute the roots of a polynomial with exponentially small coefficients. $$ \sum_{n=0}^N a_n \frac{z^n}{\sqrt{n!}} \tag{$\ast$}$$ where $a_n$ are Normal random variables with mean $0$...
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164 views

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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1answer
1k views

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
129 views

find the exact solution ref The finite element method using matlab by Kwon and Bang [closed]

The results found are as under how do we find the exact solution ref The finite element method using matlab by Kwon and Bang.Page no 280-281 Example no 8.10.1.
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177 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 ...
2
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1answer
513 views

Laplacian discretization for parametric curves

I know how to compute the discrete Laplacian of a graph and of a mesh (the Laplace-Beltrami operator). Is there an analogous definition for the computation of the Laplacian of a parametric curve ? ...
2
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1answer
704 views

eigs routine in octave [closed]

I am using octave and observed a problem with the eigs-routine for non symmetric matrices. Using GNU octave version 3.8.1 the code below gives significant difference of eigenvalues although same ...
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1answer
130 views

Parallel Monte Carlo simulation using PETSc

I am trying to do Monte Carlo simulation for a large problem which requires eigensolution of a matrix for each sample. The matrix itself is quite large so much so that I want the eigensolution itself ...
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2answers
163 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
199 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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307 views

ARPACK gives different answers from Matlab and NAG

I'm playing with ARPACK. I looked into the examples they provide, zndrv4.f illustrating the usage of the routine znaupd, in the directory of ARPACK/EXAMPLES/COMPLEX/. I also came cross NAG Fortran ...
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1answer
72 views

LAPACK DGGEVX: BALANC option

I'm using DGGEVX routine from LAPACKE with BALANC option as shown below, but to my surprise changing BALANC option from 'N' to ...
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431 views

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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2answers
3k views

Eigenvectors: MATLAB vs LAPACK DGGEV or DGGEVX

If we call LAPACK DGGEV or DGGEVX routines for two badly-conditioned matrices in a C++ code, will we get the same eigen-values &...
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Eigenvalues and Timestep restriction Follow up

This is a follow-up question to the previous questions I had on eigenvalues. Please let me know if I should edit the previous question itself for asking this. If the eigenvalues of a matrix ...
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1answer
721 views

Eigenvalues and Timestep restriction

For an equation of the kind $u_t=Au$, time step is usually defined by ensuring that the eigenvalues are within the stability region of the time-stepping scheme employed. If the eigenvalues are on ...
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108 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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2answers
985 views

Deflation for generalized eigenvalue problem

We know that principle component analysis (PCA) is a eigenvalue problem. Let $A$ be the covariance matrix of $X$, PCA aims to find the eigenvalue of $A$: $\max v'Av$, subject to $v'v=1$ Multiple ...
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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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2answers
868 views

Condition number from incomplete Cholesky factorization

I'm having difficulties patching together from what I read about obtaining the condition number of a real, symmetric, positive definite sparse matrix. In my code, I found that there is incomplete ...
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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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1answer
236 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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251 views

Method with low memory requirement for large-scale eigenvalue problem

I am working on the flow stability problem. In this work the main complication is solving generalized eigenvalue problem for a large scale Non-Hermitian matrix. I need only one eigenvalue (most left ...
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2answers
208 views

Azimuthal average in Fortran? Find indexes in Fortran?

I am working on an eigenvalue problem in fortran. I have used Lapack to solve the problem and get the eigenvalues and eigenvectors. This is done for $201\times101$ wavenumbers, only half the wavespace ...
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1answer
576 views

Applying Dirichlet b.c. to the Eigenvalue-Problem

If you use a FEM (on the variational formulation), you can discretize some continuous eigenvalue problem, $$L u = \lambda u \ \ \text{on} \ \Omega,$$ into some discrete, generalized eigenvalue problem,...
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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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1answer
2k views

Constructing matrix from eigenvalues, eigenvectors (Inconsistency with Matlab's eig())

Let $F_1$, $F_2$ be the foci points of an ellipse $\mathcal{E}\colon \mathbf{x}^TA\mathbf{x}=1$, $\mathbf{x}\in\mathbb{R}^2$, $A\in\mathbb{S}_{++}^{2}$. Let also $a$, $b$ be the semi-axes of $\mathcal{...
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1answer
1k views

Comparing Eigenvectors, Mathematica vs. Matlab

I am trying to create the same outputs in Mathematica and Matlab, however I am running into trouble aligning the eigenvectors with the eigenvalues, I think the Matlab is doing something slightly more ...