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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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 ...
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92 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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358 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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302 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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46 views

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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58 views

Are the eigenvalues of the product matrix of two real symmetric square matrices also real values?

Suppose $A,B \in \mathbb{R}^{n\times n}; A=A^T, B=B^T$, let $C = AB, D =BA$, If we have all the real eigenvalues of $A$ and $B$, e.g. the eigenvalue decomposition of them: $A=P\Lambda_1 P^T$, $B=Q\...
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74 views

Estimating eigenvalues from time-dependent non-linear operator

I have a very sparse non-linear system $N(u) = 0$ that can be solved as a time-dependent ODE, $\frac{du}{dt} = N(u)$, and explicitly integrated until $\frac{du}{dt} = N(u) = 0$, e.g. by forward euler, ...
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1answer
308 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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3answers
1k views

Eigenvectors: Mathematica vs. LAPACK dgeev

I've been using LAPACK dgeev in FORTRAN in the last months spending hours to diagonalize ~4000*4000 matrices. It takes about 2'75 hours to find eigenvalues and ...
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1answer
196 views

How to compute all the eigenvalues of a large sparse matrix using matlab?

In matlab, there are 2 commands named "eig" for full matrices and "eigs" for sparse matrices to compute eigenvalues of a matrix. And eig(A) computes all the eigenvalues of a full matrix and eigs(A) ...
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1answer
161 views

Is LAPACK behind the cutting edge of dense linear algebra?

I have been digging into some numerical linear algebra lately, and reading in particular about how LAPACK solves symmetric eigenvalue problems. I noticed that the ...
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3answers
479 views

Real eigenvalues finding

I have a question about matrix diagonalization. I don't know if this is the right forum... Is there a way to compute the smallest real eigenvalue (and eigenvector if possible) of a general real nxn ...
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1answer
470 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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2answers
1k 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 ...
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2answers
834 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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1answer
73 views

LAPACK non-convergent eigenvalue algorithm for complex but not double matrix

I've encountered an odd issue with solving for the eigenvalues of the following matrix, in Matlab format: ...
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1answer
821 views

Set of linear ordinary differential equations with a mass matrix

What methods are known for efficiently solving a large set of linear homogeneous ordinary differential equations of the following form? \begin{equation} \mathbf{B} \frac{d\mathbf{y}}{dt} = \mathbf{A} \...
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1answer
58 views

Appropriate algorithm for smallest several eigen(values|vectors) of a ~300 DoF system?

I'm no expert on the different types of algorithms to compute eigenvalues and vectors for a real, symmetric matrix (coming from linear mass and stiffness matrices for a frame FEA model). I am looking ...
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3answers
597 views

How to find the smallest positive eigenvalue of a large general system if they are all in +/- pairs of real eigenvalues

I used MKL Lapack SBGVX because it can solve for "selected" eigenvalues/modes (both positive and negative), thinking it would be efficient, but it is extremely slow when compared with Bathe's subspace ...
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1answer
706 views

Sorting eigenvalues by the dominant contribution

[Edited to simplify the question] I am trying to associate the eigenvalues $E$ of a matrix $H$ to the original rows of the matrix. Moreover, it would be trivial to sort the eigenvalues in ascending ...
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1answer
51 views

Newman algorithm yielding different result to what is given in his paper

Summary I am trying to implement Newman's algorithm for community detection, outlined in this paper. I am testing my implementation against one of the datasets used in that paper to benchmark the ...
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1answer
127 views

How can I evaluate more accurate energy eigenvalues from Schrodinger equation using shooting method?

I am trying to use the "shooting method" for solving Schrodinger's equation for a reasonably arbitrary potential in 1D. But the eigenvalues so evaluated in the case of potentials that do not have hard ...
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1answer
99 views

Reference for QR algorithm for complex matrix

I am trying to find out if the known QR algorithm to find the eigenvalues of a real matrix, which can be found in the book Fundamentals of Matrix Computations, can also be used for complex matrices ...
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1answer
197 views

FEM or FD eigenvalue equation to get wave number instead of cutoff frequency

To get cutoff frequencies and eigenmode field distributions for a waveguide, one uses following equation: $$1/\epsilon ∇ \times 1/\mu ∇ \times E = \omega^2 E$$ With $ \omega^2 $ as eigenvalues. This ...
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1answer
128 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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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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2answers
202 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
62 views

eigenvalue of small symmetric matrices

If I am to solve a symmetric eigenvalue system $A=QDQ^T$, where $A\in\mathcal{R}^{n\times n}$ and $n$ is small (in the range 4 - 64); I want all the eigenvectors and eigenvalues; There are two major ...
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1answer
293 views

Error propagation on GSL eigenvalues computation

The problem comes from the need to estimate the error propagation of the spectral norm computation of a square matrix $A$ of which I know the components' $A_{ij}$ absolute error. The fundamental step ...
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1answer
79 views

Jacobi iterative method

I'm using Jacobi iterative method for finding eigenvalue and eigenvector for hermitian or symmetric matrix. Eigenvectors corresponding to eigenvalues are not exact. The third eigenvector is totally ...
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1answer
83 views

Computing real normal modes from complex eigenvectors

I'm trying to get the normal modes of a system of springs and dasphots using the basic dynamic equations for a linear, damped elastic structure: $ M \ddot{u}(t) + C \dot{u}(t) + K u(t) = f(t) $ to ...
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1answer
115 views

Numerical solution for eigenvectors and eigenvalues of a Sturm-Liouville problem

I have to deal with the following problem in my research: $$\left[\frac{1}{D}F_{x}\right]_{x} + \frac{f D_{x}}{c D^{2}}F = 0$$ with boundary conditions $$F(0) = 0$$ $$F_{x}(L) = 0$$ where $f$ is ...
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1answer
198 views

Use SLEPc from Matlab

Is there a direct way to use SLEPc from Matlab? I remember that in some old manuals there was some Matlab interface. However, in the last one, I cannot find any reference to this. For me, it would be ...
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1answer
748 views

LAPACK: ZHEEV and DSYEV give different eigenvalues for real symmetric matrix

exchangers, I have run into a bit of a puzzling problem. To solve an complex eigenvalue-problem, I make use of the LAPACK library function ZHEEV. To test the implementation I used a real symmetric ...
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1answer
135 views

Eigenvalue problem constrained with a penalty method

I am trying to constrain an eigenvalue problem. I am aware of the method utilizing the nullspace of the constraint vectors but I was wondering if it would be to use a penalty method for the same ...
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1answer
113 views

Finding the lowest $n$ eigenvalues of a band-diagonal Matrix

I have a real sparse matrix of the form $$ \left( \begin{array}{ccc} h_{11} & h_{12} & 0 & h_{14} & & & \\ h_{21} & h_{22} & h_{23} & 0 & h_{25} & & ...
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1answer
88 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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1answer
61 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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1answer
132 views

Lanczos algorithm with thick restart on a dynamic matrix

According to a recommendation, this is a re-post of that. currently, I'm working on a way to compute the 2 biggest eigenvalues of a real, symmetric, huge and sparse matrix that changes a few entries ...
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1answer
304 views

Is there congruent transform implementation for dense symmetric matrix in Eigen(C++)?

I need to determine whether a real dense symmetric matrix is positive definite or not. One possible way is to obtain all the eigen values and check the sign of the minimum eigen value but requires ...
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57 views

When does reduced integration lead to artificial zero energy modes in stiffness matrix?

This question relates to the topic of locking free finite element development. In the case of application of reduced integration to global stiffness matrix for the Timoshenko beam element with ...
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0answers
85 views

Eigenfaces Algorithm

This might be a silly quesntion but recently I've been trying to program the eigenface algorithm using PCA, so I arranged the face vectors vertically in a matrix X such as: X = [x1,x2,x3,...,xn]; In ...
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100 views

Low memory algorithm for matrix diagonalisation

I'm trying to find the largest eigenvalues of very large $N \times N$ matrices ($N = 10^{10}$ and larger). The matrices are not sparse but the multiplication operation is fast. For now, I'm using ...
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48 views

Appropriately handling boundary conditions in a PDE eigenvalue problem

Suppose I have a nonlinear second order Cauchy PDE $\dfrac{\partial p(x,t)}{\partial t}=N(p(x,t))$, where $N:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$, and a known fixed point $u(x)$. Mathematically,...
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203 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_{...
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73 views

Numerically solving generalized eigenproblem with Neumann conditions

I am interested in finding the eigenvalues/eigenfunctions of problems such as $$ \partial_{xx} u = \lambda \partial_{yy} u, $$ which can be solved as the generalised eigenvalue problem $$ \mathbf{A}...
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1answer
117 views

triple cross prouct of tensor

Im trying to compute a triple cross product of vectors a,b, and c in real space and integrate over the entire space. The result is a term in the hamiltonian for an electronic system so there are ...
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136 views

Eigenvalues using QR iteration

I'm trying find the eigenvalues of a matrix A using QR iteration with Householder. I used this code which I found from Cornell University that decomposes QR with Householder. ...
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138 views

Spectrum of the Laplace operator

I am studying the discretization of Poisson's equation in $1D$. In Matlab I created different discretization matrices (Laplace operator) according to different sizes of the mesh: ...
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115 views

Computing only the $k$ biggest eigenvalues and eigenvectors with Scalapack

Given that there are eigensolvers in Scalapack that use a divide and conquer method, is there any way we can use Scalapack functions to only compute the first $k$ dominant eigenvalues and ...