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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Parallel vs Serial computing using Python Qutip. Why is serial faster?
I am trying to learn parallel computing using qutip's parallel_map function. I've tried to write a basic and simple code to understand the differences between parallel and serial calculation of ...
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'eigs()' in Matlab gives inaccurate eigenvector when only several eigenvalues are calculated
I would like to report an issue which may be interesting in computational physics. Sometimes, to save time and memory, we use eigs() to calculate the first several ...
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Best transform of matrix to make it efficient for shift-then-invert?
I am using ARPACK to find the smallest eigenvalue of a matrix. I use the shift and invert method. That is, looking for the largest eigenvalue of
$$
(A-\sigma I)^{-1}.
$$
However, I do not know $\sigma$...
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Why for $A^T A$, it is faster to computer the eigenvalues of its inverse than itself?
I have written the following code in MATLAB. I also observe the same effect in Arnoldi iteration in ARPACK in C.
...
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Ways to speed up Lanczos algorithm when we have a very dense cluster of many eigenvalues?
Lanczos algorithm can be used to find the largest/smallest eigenvalues of matrices. I am trying to find a good library in C/C++/Rust for finding the smallest singular value (or eigenvalue). I have ...
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Eigenvalue problem and pseudoinverse of a product of sparse matrices
If I have some dense matrix that can be decomposed into a product of sparse matrices with known(but different) sparsity patterns. Can I somehow use this information to more efficiently compute its ...
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using scipy.sparse.linalg.eigsh for degenerate states in Bose Hubbard model
I am currently writing a code for the Bose-Hubbard model, and I am calculating the ground states and single-particle density matrix for different values of U and J. As U=0, one would see how the ...
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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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Real-world applications of eigendecomposition?
Cross-posted on Math.SE
Are there real-world applications that call specifically for eigenvalues rather than singular values?
I often see eigendecomposition used as "poor-man's SVD"
For ...
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eigenvalues of inhomogeneous Helmholtz equation violate superposition using FEM
I am trying to solve the in-homogeneous Helmholtz equation with damping and forcing using finite element method (FEM) with FEniCSx. The equation is;
$c^2\nabla^2p - c^2\omega i d \nabla^2p + \omega^2 ...
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How do you determine the Mott-Insulator to Superfluid transition in the Bose Hubbard System
I am doing some simulations on various systems expressed in 2nd quantization and one of the points of interest of mine was Phase transitions in the Bose-Hubbard model
$$
H = \sum_{k} \{ t_k(b^\dagger_{...
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Why are systems with clustered eigenvalues easy to solve?
I came across the following slide by Theo Diamandis & Zachary Frangella on what makes the linear system $Ax=b$ easy to solve using the conjugate gradient method.
Transcription:
CG converges ...
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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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Matlab eigs function with function handle
I am reading through the documentation of matlab function eigs specifically the function handle input version.
Here it is: https://www.mathworks.com/help/matlab/ref/...
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Numerical continuation of all eigenvalues of small, dense matrices
Consider a one-parameter family of matrices $A(q)\in \mathbb{R}^{n\times n}$, $q\in\mathbb{R}$. For my applications, $n$ is typically between $5$ and $50$ and $A(q)$ is generally dense, so direct ...
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Different computational approaches to show that the conjecture is true
It seems that proving the statement given in the question https://math.stackexchange.com/questions/4601532/a-pen-and-paper-proof-for-a-matrix-implication is difficult analytically. I was thinking what ...
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Why are all eigen solvers iterative?
I have small dense square matrices for which I would like to compute the inverse by singular value decomposition, or equivalently solve the eigenvalue problem.
While there are many direct methods for ...
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Diagonalization of large sparse matrix, computational programme recommendation and methods
According to this link, All eigenpairs of large sparse symmetric matrix. The guy @Baranas seems to have given a very confident answer about solving the whole Eigen spectrum. May I know if anyone has ...
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Conditioning and Stability of generalized eigenvalue problem
The (generalized) eigenvalue problems with a multiple eigenvalue are the ill-posed ones.
I have two questions that should be simple for experts:
(1) Is the eigenvalue problem much more sensitive to ...
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Optimal Krylov subspace dimension and iteration limits for eigs
When using the eigs function in MATLAB, which is based off of ARPACK, one can manually modify the maximal dimension of the constructed Krylov subspaces, the maximum iteration counts, and the error ...
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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.$$ Using MATLAB, some naive ways that one may solve this is by either
directly inverting $B$ then applying the ...
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How should I solve generalized eigenvalue problems in Python? (Orr-Sommerfeld equation)
I am trying to solve the Orr-Sommerfeld equation numerically, using the techniques given in this article. This leads to solving a generalized eigenvalue problem, that is, given two matrices $\mathbf A,...
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Rayleigh-Ritz under LOBPCG : Nested iteration?
I am trying to understand the basic LOBPCG algorithm, as used in popular python libraries,
primarily from the suggested resource on wiki.
LOBPCG involves a subroutine to compute the coefficients of ...
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Approximating eigenvalues of DPR1 matrix with special properties
In my application, I have a sum of diagonal $A$ and rank-1 $B$
$$T=\underbrace{\text{diag}(1-2\alpha h+2\alpha^2 h^2)}_A + \underbrace{\alpha^2 hh^T}_B$$
Where $h$ is a vector $\in \mathbb{R}^d$ with ...
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Eigenvalues of same operator expressed in two different orthonormal basis are coming out different
I have an operator $H$. I express $H$ as a matrix in the orthonormalized $\{ |e > \}$ basis. Then I diagonalize it to obtain eigenvalues, let's say for example $H$ is $6 \times 6$ and the ...
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Eigenvalues of diagonal plus rank-one
I need to compute an eigendecomposition of an $n\times n$ matrix
$$
D + c vv^\top = Q\Lambda Q^\top \tag{1}
$$
in MATLAB, where $D$ is a real diagonal matrix, $c$ is a scalar, and $v$ is a real vector....
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qr-algorithm to find eigenvalues not returning expected values
I tried to compute eigen values with the QR-algorithm found here (there is also a wikipedia page also)
...
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Solving a generalized eigenvalue problem with Chebyshev spectral method: How to impose boundary conditions into the matrices?
I'm solving a local instability problem for a pipe Poiseuille flow.
The coordinate system is columnar, i.e., ($r,\theta,x$) (radial, tangential and axial).
The basic flow is $\bar{u_r}=0, \bar{u_\...
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Iterative methods for underestimate of smallest eigenvalue for large sparse matrices
I recently read the paper "EUCLIDEAN-NORM ERROR BOUNDS FOR SYMMLQ AND CG" by Estrin et al. and there they use an underestimate (i.e. something in $(0,\lambda_{min}]$) of the smallest ...
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Generalized Eigenvalue Problem using MATLAB
I'm trying to solve a generalized eigenvalue problem. I have two matrices $H$ and $S$ such that:
$$
HX=λSX
$$
I need to find the eigenvalues $\lambda$. The matrices $H$ and $S$ are real, asymmetric, ...
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Finding the correct order of eigenvectors of a parameter-dependent Hermitian matrix
so, I have a symmetric, analytic matrix $\mathbf{H}(x)$ ($x$ is real). Because $\mathbf{H}(x)$ is analytic and $x$ is real, it is possible to find analytic functions for the eigenvectors and the ...
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Numerical diagonalization of Hamiltonian
Framework
I am trying to diagonalize the Bogoliubov-de Gennes Hamiltonian. The problem is that the Hamiltonian contains a Laplacian. This could be solved by using a discretized Laplacian.
How I tried ...
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Well-conditioned pseudospectral for computing eigenvalues to (partial) differential equations
I am working on writing a Chebyshev pseudospectral method (see for example "Chebsyhev and Fourier spectral methods" by John Boyd) to solve for the eigenvalues of differential equations of ...
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How to implement Lax-Friedrich flux splitting with WENO scheme
I'm working on modeling a shock wave using the Euler equation with an advanced Equation of state and the fifth order WENO scheme. The equation are set up on the form:
\begin{equation}
\frac{\partial U}...
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LOBPCG bad preconditioned performance for largest eigenpairs
The LOBPCG algorithm finds eigenpairs of the generalized eigenproblem
$$
Ax = \lambda B x
$$
where $B$ is symmetric and positive-definite, $A$ is symmetric. One of the features that makes LOBPCG so ...
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Eigen-decomposition one eigenpair by one eigenpair?
Is it possible to conduct an Eigen-decomposition of a matrix one eigenpair by one eigenpair?
And related to this question, what is the time complexity of truncated eigendecomposition?
I am trying (...
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Can we sparse solve a few eigenvalues specified by index range?
I need to solve a few eigenvalues of a large sparse matrix specified by their index range. These indices are according to the whole eigenspectrum sorted in algebraic (not absolute value) ascending ...
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Numerical Solution of the Schrödinger equation for hydrogen
I'm trying to solve the Schrödinger equation for the hydrogen atom in the following form numerically:
$$\left[-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+V(r)+\frac{\hbar^2l(l+1)}{2mr^2}\right]R(r)=ER(r).$$
...
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A method for finding the number of eigenvectors with a given, known eigenvalue
Is there a method for finding the number of eigenvectors with a given eigenvalue? I do not need the eigenvectors themselves, and to find the eigenvectors seems quite tough, given the comments on the ...
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Roundoff errors in FEM computations - generalized eigenvalues
This is a continuation of my previous question. I am trying to effectively compute a bound for the roundoff errors in some FEM computation (2d polygons, triangular meshes). Below I will write some of ...
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Accelerated Inverse Power Method with Rayleigh Quotient
I am considering implementing the accelerated inverse power (AIP) method with Rayleigh quotient to speed up eigendecomposition of a real square symmetric matrix. Halton (1996) gives an example ...
user39981
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Solving the eigenvalue from a set of coupled second order differential equation numerically
I met a problem in solving a set of coupled differential equation, as shown below:
$$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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Why do we usually not want the eigenvalues of non-symmetric matrices?
I came across this line in a class note I am reading where it discusses finding eigenvalues of matrices.
In reality we don't go all the way with Arnoldi. We stop at a decent value of 𝑘. Then the 𝑘 ...
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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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Reproducing a paper's result for Topological Insulators
For the past weeks I have been trying to reproduce Agarwala's results but I've been unsuccessful. From this paper I am trying to reproduce the first and last columns of Fig.2, by implementing eq.2; ...
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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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Eigenvalue problem for ordinary differential equation
I am trying to compute the natural frequency of a cantilevered beam.
The Euler-Bernoulli equation reduces to the following problem :
$$
v''''=\lambda v, \text{with }, v(0)=0, v'(0)=0, v'''(1)=0,v''(1)...
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How are eigenvalues of a cell in the finite volume discretization calculated? (for euler fluid equations)
I am building a program to numerically solve the euler fluid equations based on finite volume discretization and am having trouble on a step where the eigenvalues of a cell need to be calculated ...
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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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