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0
votes
1answer
48 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 ...
0
votes
0answers
9 views

How to determine if a homogeneous integral equation has non-trivial solutions? [migrated]

The equation is $f(p) = \int_{0}^{\Lambda} dk \; \mathcal{M}(k,p;E) \, f(k)$, where the kernel $\mathcal{M}$ is $\mathcal{M}(k,p;E) = \frac{2/ m\pi}{1 - \sqrt{E + 3p^2/4}} \frac{k}{p} \log\left( ...
3
votes
2answers
152 views

Determine numerical infinity for Schrodinger equation $−\psi''(x) + x^ 2 \psi(x) = E\psi(x)$

Consider the following Schrodinger equation for the harmonic oscillator with real $x$: $$ −ψ''(x) + x^ 2 ψ(x) = Eψ(x). $$ I solve the last equation using shooting method and implicit Runge-Kutta ...
3
votes
1answer
84 views

Stabilizing a 3x3 real symmetric matrix eigenvalue calculation

I have many 3x3 real symmetric matrices for which I need to determine the eigenvalues. Wikipedia gives a nice non-iterative algorithm for this case, which I have translated into C++: ...
0
votes
1answer
45 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 ...
0
votes
0answers
12 views

Computing modes with fourier beam propagation

For my final optics projects I have spent the last few weeks implementing the beam proportionate method with Fourier split steps. This now works really well. Now I am trying to compute the modes of ...
5
votes
1answer
77 views

Eigenvectors of slightly perturbed matrix

Assume I know eigen-pairs $(\epsilon_i,\vec\phi_i)$ for matrix operator $\hat H$ that $ \hat H \vec \phi_i = \epsilon_i \vec \phi_i $ Now I have slightly perturbed $\hat H' = \hat H + \hat R$ were ...
4
votes
1answer
100 views

Power Iteration over Rayleigh Quotient Iteration?

It is a commonly known fact that the Rayleigh Quotient converges cubically (1), while the Power Iteration may converge slowly if the difference between the dominant and second-dominant eigenvalue is ...
8
votes
1answer
139 views

What is the state of the art algorithm for diagonalizing real symmetric matrices?

There are many methods for diagonalizing matrices, probably the most widely used is the combination of Householder transformations and the QR algorithm. Is there any superior method for diagonalizing ...
0
votes
1answer
56 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 ...
0
votes
0answers
43 views

Using DSYEV in LAPACK

I am calculating eigenvalues of a symmetric matrix using DSYEV in LAPACK.The concerned matrix is in a lower triangular form in a .dat file. I was searching for a simple explanation of the parameters ...
1
vote
0answers
55 views

Computing eigenvectors from the QR algorithm

I've seen a few other posts on this topic but none have full answers. I'm trying to implement some eigen-decomposition algorithms. I've managed to get the Explicit QR algorithm and the Implicit ...
3
votes
0answers
49 views

scipy.linalg.sparse.eigsh does not work for generalised eigenvalues

I asked this question over at StackOverflow and someone told me that I'd get a better answer here. So here's my problem: I'm working on a machine learning project which involves doing a Principal ...
1
vote
0answers
44 views

Reorder eigenvalues in QZ factorization in Python-Scipy /Matlab [closed]

Python and Scipy seem capable of replicating the QZ factorization of Matlab when the option "complex" is used in the command scipy.scipy.linalg.qz Yet, it seems that is still not possible to obtain ...
10
votes
1answer
260 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 ...
4
votes
2answers
117 views

Known issues with eigenvalue numerics?

Are there any known issues (such as precision issues) with $\mathsf{MATLAB}$ eig and charpoly functions for large enough $\{-1,0,+1\}$ matrices? Even if I change $1$ or $2$ entries between matrices ...
3
votes
1answer
66 views

Calculating left eigenvector when I know the right eigenvector

I'm using power iteration to find the dominant right eigenvector of some large-ish matrices ($1000\times 1000$ to $10000\times 10000$ or so, maybe I'll need to go bigger later) with non-negative ...
3
votes
0answers
99 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 ...
1
vote
1answer
63 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 ...
3
votes
2answers
141 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 ...
0
votes
0answers
47 views

How to implement the eigendecomposition of a dense, non-hermitian matrix using cuSolver?

I am looking for a clear example on how to obtain the complete set of eigenvalues and eigenvectors of a matrix using cuSolver. I am also interested on the eigendecomposition of symmetric, tridiagonal ...
4
votes
2answers
208 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 ...
2
votes
0answers
106 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
votes
1answer
96 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 ...
3
votes
1answer
70 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 ...
2
votes
0answers
42 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 ...
5
votes
3answers
101 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 ...
1
vote
1answer
93 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 ...
0
votes
0answers
108 views

generalized eigenvalue problem, SLEPcEigenSolver no eigenvalues with negative reals

We are working on the model of a kinematic fluid dynamo in FEniCS. It tries to explore whether a magnetic field can be excited if a conducting fluid flows with a certain velocity field through some ...
0
votes
0answers
37 views

Eigenvalues from graph (modal)

I am a power engineer analyzing the dynamic response to system events. I am trying to re-create a modal analysis function from older software in python. In this context modal analysis is taking a ...
2
votes
2answers
130 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 ...
3
votes
2answers
66 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 ...
1
vote
2answers
186 views

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 ...
0
votes
1answer
62 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 ...
1
vote
0answers
167 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 ...
2
votes
1answer
176 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 ...
1
vote
0answers
33 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 ...
0
votes
0answers
33 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 ...
0
votes
1answer
222 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 ...
4
votes
1answer
71 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 ...
7
votes
1answer
128 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 ...
3
votes
1answer
108 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 ...
5
votes
1answer
159 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
votes
0answers
108 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 ...
0
votes
1answer
403 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 ...
2
votes
1answer
85 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.
0
votes
0answers
80 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
votes
1answer
223 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 ? ...
1
vote
0answers
249 views

eigs routine in octave

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 ...
1
vote
1answer
83 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 ...