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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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0answers
73 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
796 views

Estimating the spectral radius when the dominant eigenvalues are complex conjugates

I want to determine the spectral radius of a large non-symmetric matrix $A$ whose dominant eigenvalues are a pair of complex conjugates. My first instinct was to use a power iteration with a starting ...
6
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2answers
876 views

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 ...
5
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3answers
5k views

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 ...
4
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1answer
515 views

Solving Coupled ODE eigenvalue problem

I've been trying to find some resources that would help me figure out how to numerically solve a coupled system of ODEs which is also an eigenvalue problem. The system is something like: $ \tag{1} ...
2
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1answer
149 views

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 \...
3
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1answer
305 views

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&\...
14
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1answer
3k views

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 ...
4
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2answers
1k views

2D Schrödinger time-independent finite difference and eigenvalues

I'm learning about numerical methods to obtain the eigenvalues of a system. I have to find the eigenvalues for the time-independent Schrödinger equation but I'm having some difficulties understanding ...
7
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2answers
188 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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1answer
377 views

Singular matrix - but SVD works - what does the eigenvalues mean? Find I the “dependent” lines?

please can I ask a bit stupid question? I have a complex matrix A as a set of equations. I wanted to find the solution of Ax=b where b is vector of right-hand side. So I have called zgetrf on A (does ...
5
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1answer
765 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 ...
6
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1answer
3k views

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 ...
3
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0answers
330 views

Solving PDE or eigenvalue problems without FEM

Do you know any methods/solvers for PDE or eigenvalue problems like $\begin{cases} \Delta u= 0\ (\text{ or }\lambda u) & \text{ in }\Omega \\ u =0 & \text{ on }\partial \Omega \end{cases}$ (...
6
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1answer
566 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 ...
19
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1answer
605 views

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 ...
6
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1answer
261 views

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 ...
3
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1answer
144 views

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 ...
7
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2answers
847 views

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?