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4
votes
1answer
64 views

Applying Dirichlet b.c. to the Eigenvalue-Problem

If you use a FEM (on the variational formulation), you can discretize some continuous eigenvalue problem, $$L u = \lambda u \ \ \text{on} \ \Omega,$$ into some discrete, generalized eigenvalue ...
0
votes
2answers
134 views

Most efficient library to diagonalize exactly large hermitian or unitary matrices

I am working on a physics problem which requires obtaining the exact eigenvalues and eigenvectors of Hermitian and Unitary matrices numerically. Naturally I would like to ask the experts what are the ...
2
votes
1answer
66 views

Constructing matrix from eigenvalues, eigenvectors (Inconsistency with Matlab's eig())

Let $F_1$, $F_2$ be the foci points of an ellipse $\mathcal{E}\colon \mathbf{x}^TA\mathbf{x}=1$, $\mathbf{x}\in\mathbb{R}^2$, $A\in\mathbb{S}_{++}^{2}$. Let also $a$, $b$ be the semi-axes of ...
2
votes
1answer
135 views

Comparing Eigenvectors, Mathematica vs. Matlab

I am trying to create the same out puts in Mathmatica and Matlab, however I am running into trouble aligning the eigenvectors with the eigenvalues, I think the Matlab is doing something slighly more ...
1
vote
1answer
49 views

Fast way to compute all eigenvalues of a dense Hermitian matrix

I am finding the eigenvalues of dense NxN Hermitian matrix which is calculated from a density operator in quantum physics. All the eigenvalues are needed as I need to calculate the sum of the absolute ...
1
vote
1answer
34 views

eigenvalue of small symmetric matrices

If I am to solve a symmetric eigenvalue system $A=QDQ^T$, where $A\in\mathcal{R}^{n\times n}$ and $n$ is small (in the range 4 - 64); I want all the eigenvectors and eigenvalues; There are two major ...
1
vote
1answer
45 views

How can QR iteration with complex matrices produce complex diagonal entries?

In Lapack (zhseqr) and matlab, the eigenvalues of a complex matrix are computed successfully. I notice that QR iteration or algorithm is involved with that process. QR iteration repeats to call QR ...
5
votes
1answer
154 views

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 ...
0
votes
1answer
81 views

Flop counts for LAPACK symmetric eigenvalue routines DSYEV, DSYEVD, DSYEVX and DSYEVR

LAPACK has following 4 routines for calculating eigenvalues of a real symmetric matrix; namely DSYEV, DSYEVD, DSYEVX and DSYEVR (DSYEVR being the recommended one). If I were to calculate both ...
2
votes
3answers
239 views

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 ...
1
vote
1answer
52 views

Error propagation on GSL eigenvalues computation

The problem comes from the need to estimate the error propagation of the spectral norm computation of a square matrix $A$ of which I know the components' $A_{ij}$ absolute error. The fundamental step ...
1
vote
0answers
56 views

Lanczos algorithm with thick restart on a dynamic matrix

According to a recommendation, this is a re-post of that. currently, I'm working on a way to compute the 2 biggest eigenvalues of a real, symmetric, huge and sparse matrix that changes a few entries ...
2
votes
1answer
64 views

Finding Interior eigenvalues using Davidson algorithm

Is it possible to find interior eigenvalues closer to some lambda using Davidson method. I was searching online but found that most people use Jacobi-Davidson method for that. Thanks
5
votes
1answer
70 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 ...
3
votes
2answers
194 views

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 ...
1
vote
3answers
133 views

How to find the smallest positive eigenvalue of a large general system if they are all in +/- pairs of real eigenvalues

I used MKL Lapack SBGVX because it can solve for "selected" eigenvalues/modes (both positive and negative), thinking it would be efficient, but it is extremely slow when compared with Bathe's subspace ...
5
votes
1answer
227 views

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 ...
1
vote
0answers
49 views

Shift-Invert in Anasazi/Belos using Tpetra Sparse Matrices

I am currently trying to port my code over to Trilinos because the problems that I am working on are too big for LAPACK/ARPACK. Specifically I am computing the generalized eigenvalues/eigenvectors: ...
1
vote
0answers
87 views

Algorithms to compute largest gap between smallest nonzero eigenvalues of sparse symmetric matrix

I am looking mainly for c/c++ implementations but also for theoretical algorithms to compute gaps between smallest positive eigenvalues of symmetric, singular matrix or real numbers. To be precise, I ...
4
votes
0answers
78 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 ...
1
vote
0answers
77 views

Is it possible to construct such a symmetric matrix with desired eigenvalues?

Suppose a real, dense and asymmetric square matrix $A\in\mathbb{R}^{n\times n}$, all its eigenvalues $\lambda_i \in \mathbb R$ Is it possible to construct a symmetric matrix $B\in\mathbb{R}^{n\times ...
1
vote
0answers
54 views

Efficient way to do congruent transformation using matrix inverse?

I know a square self-adjoint matrix $S_{vv}$ and I want to find: $S_{rr} = HS_{vv}H^{\dagger}$ where $\dagger$ denotes conjugate transpose. I do not know $H$ but I do know $H^{-1}$. What is the ...
5
votes
2answers
159 views

Is there guaranteed global solver for such an eigenvalue problem?

The original nonlinear optimization problem I have is as follows: For constant symmetric matrices $A=A^T, B_i=B_i^T(\forall i\in\mathbb{N}) \in \mathbb{R}^{n\times n}, \text{rank}(A)=n,$ ...
2
votes
0answers
43 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: $A=P\Lambda_1 P^T$, ...
3
votes
1answer
87 views

What is the most efficient way to obtain the max eigenvalue of a specific symmetric matrix via Eigen C++

Suppose I have a symmetric matrix $A_{1000\times 1000}$, which can be represented by: $A = J G J^T$ where $J$ in 1000x3 is full column rank dense matrix; $G$ in 3x3 is a nonsingular dense matrix. ...
0
votes
0answers
59 views

Sorting eigenvalues by the dominant contribution

[Edited to simplify the question] I am trying to associate the eigenvalues $E$ of a matrix $H$ to the original rows of the matrix. Moreover, it would be trivial to sort the eigenvalues in ascending ...
2
votes
1answer
114 views

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) ...
3
votes
1answer
162 views

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 $N$ ...
8
votes
0answers
193 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 ...
5
votes
1answer
208 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 ...
1
vote
1answer
106 views

Is there congruent transform implementation for dense symmetric matrix in Eigen(C++)?

I need to determine whether a real dense symmetric matrix is positive definite or not. One possible way is to obtain all the eigen values and check the sign of the minimum eigen value but requires ...
1
vote
3answers
99 views

Real eigenvalues finding

I have a question about matrix diagonalization. I don't know if this is the right forum... Is there a way to compute the smallest real eigenvalue (and eigenvector if possible) of a general real nxn ...
2
votes
1answer
125 views

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 ...
5
votes
1answer
397 views

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 ...
0
votes
0answers
23 views

Size reduction of matrices in dispersion curve calculation

I have an energy dispersion curve resulted from eigen values of E(k) = eig(T exp(ik) + T' exp(-ik) + H0). Where H0 and T are NxN square matrices and T' is transpose of T and k is wave-vector. So there ...
1
vote
3answers
287 views

Eigenvectors: Mathematica vs. LAPACK dgeev

I've been using LAPACK dgeev in FORTRAN in the last months spending hours to diagonalize ~4000*4000 matrices. It takes about 2'75 hours to find eigenvalues and ...
5
votes
2answers
216 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 ...
5
votes
2answers
372 views

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 ...
2
votes
0answers
54 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
votes
1answer
201 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 ...
5
votes
2answers
223 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 ...
4
votes
3answers
2k 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
votes
1answer
170 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
votes
1answer
87 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
votes
1answer
129 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 ...
9
votes
1answer
552 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
votes
2answers
400 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 ...
6
votes
2answers
153 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 ...
2
votes
1answer
224 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
votes
1answer
274 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 ...