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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13
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560 views

Fast Eigenvalue and SVD Solver for Structured Matrices

I am looking for a fast Eigenvalue and SVD solver for small dense structured matrices (Hankel and Toeplitz). I have searched for efficient implementations in libraries like MKL but I am not able to ...
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2answers
199 views

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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0answers
346 views

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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154 views

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\...
6
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335 views

Will penalty-augmented stiffness matrix cause numerical issues in eigenvalue analysis?

In the finite element method, we often construct the constraints of the system by adding penalty-function terms ( which often are many many magnitudes, up to $10^6$ order bigger than the largest ...
5
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76 views

Efficient way to find eigenvalues of complex symmetric matrix with real off-diagonal elements

My goal is to find all eigenvalues (and eigenvectors) in a given range of magnitudes of a complex symmetric matrix with real off-diagonal elements (only diagonal elements are complex). Currently I'm ...
5
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0answers
79 views

Improving convergence of Jacobi iteration to Schur form

I'm using SIMD processor arrays to compute the eigen-decomposition for large numbers of small (up to $32\times 32$) matrices. For assorted technical reasons, Jacobi iteration maps well to the SIMD ...
5
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0answers
154 views

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 ...
4
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189 views

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....
4
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0answers
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 ...
3
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58 views

Does shift-invert method has invertibility issue?

Please note that I have nearly zero background on numerical methods. I understand the shift invert method as described in SciPy Tutorial The main argument of the above link is as follows. Suppose we ...
3
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0answers
95 views

Why the MIRACLE of Lanczos/CG-like?

Lanczos/Arnoldi/Rietz/CG-like algorithm share the same core strategy... In each, a little miracle appears, most of the Gram-Schmidt inner products are zeroes! In others words, new direction need only ...
3
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0answers
257 views

Computing Small Eigenvalues with Sparse Symmetric Indefinite Mass Matrix

I want the eigenvalues of the following generalized eigenvalue problem: $$ Av = \lambda M v $$ where $A\in\mathbb{R}^{n\times n}$ is sparse, symmetric, and positive semi-definite $M\in\mathbb{R}^{n\...
3
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0answers
56 views

Preconditioning matrix with known spectrum

Assume I know all eigenvalues of a matrix $A$ fall into a certain set $\Omega \subset \mathbb{C}$. Is there any way I can exploit this knowledge to design a preconditioner for $A$? Some further ...
3
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0answers
327 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
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0answers
250 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 ...
3
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0answers
345 views

Solving PDE or eigenvalue problems without FEM

Do you know any methods/solvers for PDE or eigenvalue problems like $\begin{cases} \Delta u= 0\ (\text{ or }\lambda u) & \text{ in }\Omega \\ u =0 & \text{ on }\partial \Omega \end{cases}$ (...
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0answers
75 views

Why is LAPACK (seemingly) suboptimal for packed and banded eigenvalue problems?

Based on this LAPACK routines list, it looks like there is no relatively robust representation (RRR) driver routine for either packed or banded symmetric eigenvalue problems. According to the relevant ...
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58 views

Geometric interpretation of lemma

I am currently studying eigenvalue problems. I already worked through the Minimax-principles, seen why $\lambda_{h, m} \geq \lambda_m$, when comparing the eigenvalues of a discretization and the ...
2
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0answers
87 views

Finite differenced eigenvalue prob. of inhomogeneous boundary conditions?

I am basically asking about eigenvalue problems of differential equations using some finite difference method (FDM). Usually the system is subject to some boundary conditions (BC), e.g., Dirichlet or ...
2
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0answers
88 views

What is the complexity of calculating K-th largest real part eigenvalue of non-normal sparse matrix

I just need to calculate the largest real part of eigenvalues of a Jacobian which is highly non-normal and singular. Most of the eigenvalues are negative, and some of them are positive but near to ...
2
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136 views

What is Chebfun `eigs` doing

What is this doing? Looks like the original eigenvalue problem is converted into generalized eigenvalue problems with different dimensions of collocation points. Can someone explain more about this? ...
2
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157 views

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 ...
2
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355 views

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 ...
2
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0answers
101 views

Eigenvalue decomposition of the sum: $AA^T$ + diag($u$)

Suppose $A\in\mathbb{R}^{n\times c}$,$u\in\mathbb{R}^n$,$n\gg c$. The time complexity of eigenvalue decomposing directly for matrix $AA^T+\text{diag}(u)$ is $O(n^3)$. And it is easy to avoid $O(n^3)$ ...
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230 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 ...
2
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0answers
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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0answers
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 ...
2
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0answers
391 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
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0answers
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 ...
2
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0answers
48 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 ...
2
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0answers
59 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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0answers
75 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, ...
2
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1answer
388 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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0answers
48 views

Neural networks trained with subspace iteration algorithms for solving SCF eigenvalue problems

I am toying with the idea of using current libraries available with SCF (Self-Consistent Field) subspace iteration codes to train an ANN (Artificial Neural Network) to solve for SCF eigenvalue ...
1
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0answers
64 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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0answers
105 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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0answers
53 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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0answers
225 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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0answers
75 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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0answers
139 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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0answers
144 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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0answers
122 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 ...
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0answers
205 views

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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0answers
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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0answers
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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0answers
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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0answers
176 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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430 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 ...