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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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++:
...
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Implementation of a $O(n \log(n))$ method to compute eigenvalues of real symmetric tridiagonal matrices
I just came upon this paper, which details the implementation of a fast method to get eigenvalues of tridiagonal symmetric matrices :
Coakley, Ed S.; Rokhlin, Vladimir, A fast divide-and-conquer ...
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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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Solving the eigenvalue from a set of coupled second order differential equation numerically
I met a problem in solving a set of coupled differential equation, as shown below:
$$A_1\psi_1(z)+A_2\frac{d^2\psi_1(z)}{dz^2}+A_3\frac{d\psi_2(z)}{dz}=\lambda\psi_1(z)$$
$$A_4\psi_2(z)+A_5\frac{d^2\...
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A method for finding the number of eigenvectors with a given, known eigenvalue
Is there a method for finding the number of eigenvectors with a given eigenvalue? I do not need the eigenvectors themselves, and to find the eigenvectors seems quite tough, given the comments on the ...
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Lanczos algorithm for finding top eigenvalues of a matrix sum
I am trying to find the top k leading eigenvalues of a NumPy matrix (using python dot product notation) L@L + a*[email protected], where $L$ ...
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Efficient solution to a structured symmetric linear system with condition number estimation
I have a real-valued linear system $Hx = b$ where $H$ is symmetric matrix** (not necessarily positive/negative definite) with a very particular structure:
$$
H = \begin{bmatrix} D && B \\ B^T &...
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Using a subspace iteration method to obtain eigenvalues. Getting eigenvectors too but I don't understand why
I'm using an iterative subspace algorithm (dsrrit) to obtain the eigenvalues of an eigenvector equation
$$
-\nabla^2 \mathbf{x} = \lambda\mathbf{x}
$$
where $\nabla^2$ is the usual Laplacian operator. ...
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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 ...
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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 ...
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Jacobi Iteration diverges?
I'm working through a problem in a textbook as follows:
"Consider the $d \times d$ Toeplitz matrix
$$ A = \left[ \begin{array}{ccccc} 2 & 1 & 0 & \cdots & 0 \\
-1 &2 &1&\...
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Generalized eigenvalue problem for large, potentially ill-conditioned systems
Say that I have a generalized eigenvalue problem of the form $$Ax=\lambda Bx.$$ Using MATLAB, some naive ways that one may solve this is by either
directly inverting $B$ then applying the ...
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Known issues with eigenvalue numerics?
Are there any known issues (such as precision issues) with $\mathsf{MATLAB}$ eig and charpoly functions for large enough $\{-1,0,+1\}$ matrices?
Even if I change $1$ or $2$ entries between matrices ...
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Eigenvalues and Timestep restriction
For an equation of the kind $u_t=Au$, time step is usually defined by ensuring that the eigenvalues are within the stability region of the time-stepping scheme employed.
If the eigenvalues are on ...
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Generalized eigenvalue problem using ARPACK
Is it possible to solve the eigenvalue problem:
$$Ax = \lambda Mx$$
using ARPACK when $A$ and $M$ are both non-symmetric complex matrices? According to this documentation, the function ...
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What is the worst case complexity of the symmetric tridiagonal QR eigenvalue algorithm?
Ignoring eigenvectors, the shifted QR algorithm for computing eigenvalues in the symmetric tridiagional case costs $O(n)$ per iteration, converges globally, and converges cubically near the end. What ...
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Efficient computation of leading eigenvector of a matrix product of the form $ADA^T$, where $D$ is diagonal
Let $A=[A_1|\ldots|A_m] \in \mathbb R^{n \times m}$ with $n \gg m \gg 1$ and $D=\text{diag}(d_1,\ldots,d_m)$ where $d_1,\ldots,d_m > 0$, and consider the $n\times n$ positive-definite matrix $X=\...
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Iteratively obtaining m eigenvectors using arpack: If I have a good initial guess, how do I use it?
I am trying out the arpack driver dsdrv1, which is used to iteratively obtain the first m eigenvectors from the eigenvalue problem.
$$
\hat{A}\mathbf{x} = \lambda\mathbf{x}
$$
As it is an iterative ...
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Finding eigenvalues of a complex symmetric tridiagonal matrix
I am trying to find specific eigenvalues and -vectors of a large complex symmetric tridiagonal matrix (at least 10000x10000, and ideally larger). I know roughly which eigenvalues I am looking for, so ...
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`eigsh` (Lanczos algorithm) slows down for degenerate eigenvalues
I have a complex Hermitian matrix of size about $70000\times 70000$. I want about 100 eigenvalues near 0. However, I know that every eigenvalues are two-fold degenerate. I found out that the running ...
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Quick way to find a common basis of eigenvectors between 2 matrices : valid or not?
Following the advise of @Federico Polonion a previous post, one suggested, to find a basis of common eigen vectors between 2 matrices, to simply do :
Generate 2 ...
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Eigenvectors associated to two quasi-degenerate eigenvalues
I need to find the smallest eigenvalue and the corresponding eigenvector of a sparse matrix $M$ whose dimension is $\approx 10^4$.
Within Matlab enviroment, I use the command
...
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How to correctly normalize modulus and phase of an eigenvector?
I am solving a linear stability problem using finite element discretization. Then, I have a generalised eigenvalue problem:
$$ \lambda M x = J x.$$
I obtain complex eigenvalue and eigenvectors from ...
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How to use eigenvalue information to efficiently diagonalize matrices?
I apologize if this question in a more general form has been asked before. I have a tridiagonal Toeplitz matrix $K$, whose eigenvalues and eigenvectors are known analytically for any dimension $N$ [1]....
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Solving the elliptic eigenproblem with periodic boundary conditions
Given an energy level $\mu$, I'm looking to calculate the eigenvectors corresponding to the time-independent Schrodinger operator on the torus (that is, periodic boundary conditions)-- $H = -h^2 \...
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SLEPc eigensolvers take long time to converge for large sparse symmetric matrices
I am leveraging the SLEPc library for solving the first $k$ (where $k = 3$ or $4$) eigenvalues and their corresponding vectors for a matrix of size 200,000. The matrix is sparse and symmetric. I ...
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Preconditioned GMRES for nearly diagonalizable systems
I have been working with a matrix $A$ and preconditioner $P\approx A^{-1}$ that I've then applied GMRES to the (left) preconditioned linear system
\begin{equation}
P^{-1}Ax=P^{-1}b
\end{equation}
$P^{-...
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Fast Fourier Transform on Meshes
I have a (closed, manifold, oriented) triangular mesh for which I build a matrix $L\in\mathbb{R}^{n\times n}$ discretising the negated Laplace-Beltrami operator. The matrix $L$ is symmetric positive ...
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Approximating eigenvalues of DPR1 matrix with special properties
In my application, I have a sum of diagonal $A$ and rank-1 $B$
$$T=\underbrace{\text{diag}(1-2\alpha h+2\alpha^2 h^2)}_A + \underbrace{\alpha^2 hh^T}_B$$
Where $h$ is a vector $\in \mathbb{R}^d$ with ...
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How to obtain smallest eigenvalues with Arnoldi iteration
I understand that the Arnoldi iteration produces a basis which tends to include in its span the eigenvectors corresponding to eigenvalues of large magnitude (hence the analogy between the last vector ...
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Does shift-invert method has invertibility issue?
Please note that I have nearly zero background on numerical methods.
I understand the shift invert method as described in SciPy Tutorial
The main argument of the above link is as follows. Suppose we ...
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Why the MIRACLE of Lanczos/CG-like?
Lanczos/Arnoldi/Rietz/CG-like algorithm share the same core strategy...
In each, a little miracle appears, most of the Gram-Schmidt inner products are zeroes! In others words, new direction need only ...
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Computing Small Eigenvalues with Sparse Symmetric Indefinite Mass Matrix
I want the eigenvalues of the following generalized eigenvalue problem:
$$ Av = \lambda M v $$
where
$A\in\mathbb{R}^{n\times n}$ is sparse, symmetric, and positive semi-definite
$M\in\mathbb{R}^{n\...
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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 ...
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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 ...
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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 (...
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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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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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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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Eigenvalues of a sparse banded nonsymmetric matrix from an elliptic operator
I have a sparse matrix coming from the discretization of a 3D elliptic PDE. The matrix is banded with seven non-zeros diagonals. The sparsity pattern of the matrix looks like this (the actual matrix ...
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Python scipy eigh(Arpack) giving wrong eigenvalues for generalized eigenvalue problem
I am trying to solve a generalized eigenvalue problem using Arpack, right now the code is using LAPACK but that's too slow, we only need a few eigenvalues and the matrices are sparse so using Arpack ...
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Can ARPACK exploit hermiticity when diagonalising a complex matrix?
I have noticed arpack comes with a driver dsdrv1 that exploits symmetry of a real-valued matrix.
Is there a way to analogously exploit a Hermitian matrix in some way via z--- drivers?
The manual ...
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Eigenvalue decomposition for a very huge matrix of medical images (such as the pixel physical coordinates of CT images)
I am trying to do eigenvalue decomposition for a huge matrix larger than 788000×788000 for medical image analysis. The matrix is not sparse and every element in the matrix has a real value. And, for ...
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Zero Eigenvalues in Lanczos Algorithm
I need to find the smallest few eigenvalues of a Hamiltonian (exact diagonalization) I use Python, and SciPy's built-in sparse eigenvalue solver. I notice, however, that for my small system (only a ...
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Algorithm for Eigenvalue Problem of a Real Symmetric nxn Matrix
I have a nxn covariance matrix (so, real, symmetric, dense, nxn... I mean positive semi-definite). 'n' may be very very very big! I'd like to solve partial (3 largest) eigenvalue (+eigenvectors) ...
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Faster eigenvector routine for non-symmetric matrices with real eigensystem?
I have non-symmetric real-valued matrices with real-valued eigensystems. How to compute eigenvectors efficiently?
Using scipy.linalg.eig (which calls ...
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Eigenvectors of Black-box matrix
$\DeclareMathOperator{\diag}{diag}$
Consider the generalized eigenproblem $A\mathbf{x}=\lambda B\mathbf{x}$. When solving PDEs numerically (specifically, I am interested on finding the Dirichlet ...
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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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qr-algorithm to find eigenvalues not returning expected values
I tried to compute eigen values with the QR-algorithm found here (there is also a wikipedia page also)
...
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Numerical diagonalization of Hamiltonian
Framework
I am trying to diagonalize the Bogoliubov-de Gennes Hamiltonian. The problem is that the Hamiltonian contains a Laplacian. This could be solved by using a discretized Laplacian.
How I tried ...