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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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Difficulty with Spectral Method using Chebyshev Polynomials

I am having a bit of difficulty in trying to understand a paper. The paper uses spectral method to solve for an eigenvalue that comes from a system of coupled ODEs. I will write out only one equation ...
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Why does SciPy eigsh() produce erroneous eigenvalues in case of harmonic oscillator?

I'm developing some larger code to perform eigenvalue computations of huge sparse matrices, in the context of computational physics. I test my routines against the simple harmonic oscillator in one ...
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Testing if a matrix is positive semi-definite

I have a list ${\cal L}$ of symmetric matrices that I need to check for positive semi-definiteness (i.e their eigenvalues are non-negative.) The comment above implies that one could do it by ...
11
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598 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 ...
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Compute all eigenvalues of a very big and very sparse adjacency matrix

I have two graphs with nearly n~100000 nodes each. In both graphs, each node is connected to exactly 3 other nodes so the adjacency matrix is symmetric and very sparse. The hard part is I need all ...
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368 views

Eigenvectors of a small norm adjustment

I have a dataset that is slowly changing, and I need to keep track of eigenvectors/eigenvalues of its covariance matrix. I've been using scipy.linalg.eigh, but it'...
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428 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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Eigenvalues of Small Matrices

I'm writing a small numerical library for 2x2, 3x3, and 4x4 matrices (real, unsymmetric). A lot of numerical analysis texts highly recommend against computing the roots of the characteristic ...
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585 views

Implementation of Jacobi-Davidson method for cubic eigenvalue problem

I have a large cubic eigenvalue problem: $$\left(\mathbf{A}_0 + \lambda\mathbf{A}_1 + \lambda^2\mathbf{A}_2 + \lambda^3\mathbf{A}_3\right)\mathbf{x} = 0.$$ I could solve this by converting to a ...
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How to use Lanczos method to compute eigenvalues and eigenvectors

I have a sparse and symmetric matrix A(n x n). The method Lanczos tranforms matrix A into tridiagonal and symmetric matrix T and the Lanczos vectors in matrix V. From there how do I compute k ...
8
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1answer
775 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 ...
8
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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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How does the QR algorithm applied to a real matrix returns complex eigenvalues?

I'm a noob into eigenvalues algorithms, but something call my attention. QR algorithm works with real/complex matrices producing real/complex eigenvalues. However, it can not produce complex ...
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Can all eigenvalues of a Hermitian Toeplitz matrix be computed in $\tilde{O}(n)$ time?

I know there are "superfast" $O(n \log^p n)$ algorithms for solving Toeplitz linear systems. Is it possible to compute all eigenvalues of such a matrix with the same complexity?
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1answer
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Eigenvalues of $ab^T$

In deriving a Newton scheme, I end up with a Jacobian matrix of the form $J=I+ab^T$ where $a,b$ are vectors. For practical reasons, I want to approximate it by a symmetric positive definite matrix. ...
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Solving a generalised eigenvalue problem

I have a generalized eigenvalue problem in the standard form $\lambda \mathbf{B} \mathbf{x} = \mathbf{A} \mathbf{x} $, resulting from a finite difference discretization of a coupled system of two ...
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186 views

Left and right eigenspaces of the product of Gramians

I solve the Lyapunov equations : $$ A W_C E^T + E W_C A^T + B B^T = 0 $$ $$ A^T W_O E^T + E W_O A + C^T C = 0 $$ to obtain $ W_C $ and $W_O$. My aim is to get the left and right eigenspaces of $W_C ...
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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 ...
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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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3answers
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Nonlinear eigenvalue problem - MATLAB code

I'm trying to solve a nonlinear eigenvalue problem in MATLAB, still without success. It's a problem about graphene plasmonics. The nonlinear eigenvalue problem is given below: \begin{equation} \frac{...
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3answers
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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 ...
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1answer
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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 $x\in[...
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Largest negative eigenvalue

Is there an efficient way to find the largest negative eigenvalue of a matrix? The matrix in question is a Markov matrix. Computing the full spectrum of the matrix by decomposing it is not an ...
6
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1answer
268 views

Correct eigenfunctions of Laplace operator by Finite Differences

I am trying to compute the eigenfunctions of the Laplace operator, i.e. finding $u$ in $$ -\nabla^2 u = \lambda u .$$ For now I am trying to do this in 1D, so $$ \nabla^2 = \partial_{xx} .$$ I am ...
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Is there an eigenvalue estimation method more accurate than Gershgorin's, which uses no multiplication?

Suppose I have a real symmetric matrix. I would like to tell wether it has at least $k$ strictly positive eigenvalues, but using only additions (no multiplications). Is there a method that I could use?...
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matlab eigs: wrong eigenvalues for tridiagonal matrix

I try to compute eigenvalues of the tridiagonal matrix coming from finite difference scheme. For small mesh size, eigs works well. But for large size it fails. Here is an example where eigs fails. Is ...
6
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1answer
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Simple Lanczos algorithm code to obtain eigenvalues and eigenvectors of a symmetric matrix

I would like to write a simple program (in C) using Lanczos algorithm. I came across a Matlab example which helped me to understand a bit further the algorithm, however from this piece of code I can't ...
6
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2answers
236 views

inertia count sparse matrix with dense low-rank perturbation

I would like to determine the number of negative eigenvalues (inertia count) of the $(N \times N)$ symmetric real matrix $K - \sigma M$, with $K$ a positive-definite sparse matrix and $M$ a positive-...
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1answer
189 views

Eigenvector with maximum overlap

Given a matrix $M$ and a vector $v$, is there an efficient method to find the normalized eigenvector of $M$ that is closest to $v$, in that it has maximal overlap. More explicitly, a vector $v$ can be ...
6
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1answer
297 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 $0$...
6
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2answers
396 views

What algorithms are known for computing exact eigenvalues for rational matrices?

Let $M$ be a matrix which has the following properties: 1) $M$ is Hermitian 2) $M$ has only rational entries 3) $M$ is known to have rational eigenvalues What algorithms are there for exactly ...
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Computing smallest eigenvectors of a sparse matrix, having its inverse

I'm a bit confused by the vast amount of literature on solving eigenvalue problems. I have a sparse (large) matrix which I have already factored (by Cholesky or LDU). I would like to compute few ...
6
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1answer
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Dense generalized hermitian indefinite eigenvalue problem

Lapack contains a driver routine to solve dense generalized Hermitian positive definite eigenvalue problems of the form $Ax=\lambda Bx$, where $A$ and $B$ are both Hermitian, and $B$ is positive ...
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2answers
586 views

Numerical Solution to Schrödinger Equation--Multiple Wells

I am trying to solve for the allowed wavefunctions and energies for a 1D quartic potential well. To do this I am using the patching method (https://engineering.dartmouth.edu/microeng/otherweb/...
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1answer
519 views

Estimate extreme eigenvalues with CG

CG may be used to estimate the extremal eigenvalues of a SPD matrix (by computing eigenvalues of tridiagonal matrix associated with the Lanczos algorithm). After a few iterations the largest ...
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148 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 ...
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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 ...
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Is there guaranteed global solver for such an eigenvalue problem?

The original nonlinear optimization problem I have is as follows: For constant symmetric matrices $A=A^T, B_i=B_i^T(\forall i\in\mathbb{N}) \in \mathbb{R}^{n\times n}, \text{rank}(A)=n,$ $$\arg\min\...
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1answer
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Nystrom approximation of SVD for asymmetric matrices

Suppose I have a symmetric matrix $K$. Subdivide $K$ into pieces as $$K=\begin{pmatrix} K_{11} & K_{12} \\ K_{21} & K_{22}\end{pmatrix},$$ where $K_{21}=K_{12}^\top$. Then, the Nystrom ...
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1answer
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Numerical solution of two coupled nonlinear eigenvalue problems

I would like to numerically solve the following system of coupled nonlinear differential equations: $$ -\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a + \left( g_a |...
5
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1answer
172 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 ...
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1answer
657 views

Inverse iteration to find the null singular vector of a rank-deficient matrix

I have an $n \times n$ unsymmetric matrix $A$ that results from the discretization of an ill-posed Poisson problem, and thus is rank-deficient with null space of dimension one. I want to compute just ...
5
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1answer
131 views

Computing eigendecomposition of a Hermitian matrix that is almost unitary

I have a dense Hermitian matrix that is approximately unitary, so it has eigenvalues that are $\sim \pm1$. I would like to compute all the eigenvectors corresponding to the $+1$ eigenvalue (not ...
5
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1answer
705 views

Eigenvalue Decomposition of Hermitian Matrix in Scala

I'm working on helping my friend create code to perform the Overlap Dirac Operator and have come across one part that I'm not sure what to do. I need to compute the Eigenvalues and corresponding ...
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139 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\...
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1answer
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Quality of eigenvalue approximation in Lanczos method

I try to familiarize myself with iterative eigenvalue solvers such as Lanczos. So I tried rewrite it to python directly according to wiki. But it doesn't seem to work. The problem: it approximates ...
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3answers
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Algorithm for Principal Eigenvector of a Real Symmetric 3x3 Matrix

I have a 3x3 covariance matrix (so, real, symmetric, dense, 3x3), I would like it's principal eigenvector, and speed is a concern. Is there a fast algorithm for this specific problem? I've seen ...
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993 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++: ...
4
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3answers
165 views

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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214 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 ...