An eigenvector of an operator is a vector such that the action of the operator is the same as multiplication by a constant, called the eigenvalue. The eigensystem of an operator is the set of all such eigenvectors and their associated eigenvalues.

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Computing modes with fourier beam propagation

For my final optics projects I have spent the last few weeks implementing the beam proportionate method with Fourier split steps. This now works really well. Now I am trying to compute the modes of ...
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27 views

Learning Lanczos eigenvalue iteration - what is wrong?

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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43 views

Using DSYEV in LAPACK

I am calculating eigenvalues of a symmetric matrix using DSYEV in LAPACK.The concerned matrix is in a lower triangular form in a .dat file. I was searching for a simple explanation of the parameters ...
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50 views

Fastest way to diagonalize square matrices with c++

I am currently writing software in c++ which solves the eigenvalue problem of sparse hermitian matrices. The size of the matrix depends on the user input, but as an estimation it will roughly be ...
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1answer
262 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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1answer
72 views

Numerical computation of Perron-Frobenius eigenvector

I would like to efficiently and (to the extent possible) reliably find the Perron-Frobenius eigenvector of non-negative matrices. These are not stochastic matrices, they are typically dense, and their ...
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36 views

Memory requirement to find eigenvalues and -vectors of large sparse matrix

How can I estimate how much memory will be needed to find eigenvalues and eigenvectors of a given large sparse matrix? I have a real symmetric matrix with roughly $5 \times 10^4$ rows and columns, ...
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36 views

Rotate complex eigenvectors

I'm computing the eigenvectors of a real non-symmetric Matrix. I know that complex eigenvectors would come in conjugate. However, does anybody know a algorithm to rotate such complex eigenvectors ...
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99 views

Eigenvalue-style optimization with quadratic constraints

Suppose $A\in\mathbb{R}^{n\times n}$ is symmetric and positive definite and that we have several symmetric matrices $B_i\in\mathbb{R}^{n\times n}$ that are low-rank and indefinite. I need an ...
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114 views

Does this partial eigen-expansion have a name?

This question is a follow-up to this one. Let $A\in \mathbb{R}^{n\times n}$ be large, sparse, symmetric and positive definite. Suppose for I already know $m<n$ eigenpairs of $A$, corresponding to ...
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1answer
87 views

Julia: ordschur command

I just started leaning Julia, and my command of it is still very preliminary. I am trying to reorder a Schur decomposition. In Matlab, I could use the ordqz command and just had to specify the ...
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75 views

What are some ideas to preprocess / precondition the following linear system?

Let $A\in \mathbb{R}^{n\times n}$ symmetric and positive semidefinite, and $\omega\in \mathbb{R}\setminus\{0\}$. I am interested in solving the following linear system for a range of values of ...
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What strategies / decompositions would be useful to solve the following linear system repeatedly if I only care about time to solution?

Let $A\in \mathbb{R}^{n\times n}$ be symmetric positive semidefinite, and $B\in \mathbb{R}^{n\times n}$ be symmetric positive definite. Suppose $B$ is block diagonal so it is easy to invert. (We ...
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1answer
65 views

Fast projection onto semidefinite cone

Lots of algorithms for semidefinite programming make use of the Frobenius projection onto the cone of semidefinite matrices: $$\mathcal{P}(A) = \min_{X\succeq0} \|A-X\|_{\mathrm{Fro}}^2.$$ Let's ...
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41 views

Nonlinear global optimization involving eigen-decomposition

I am trying to globally optimize a nonlinear (and nonconvex) objective function $\sum(\vec{Y}-g(\vec{\theta},\vec{X}))^2$ (regression) subject to some linear constrains ($\theta_{1,..,i} \ge 0$, ...
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143 views

Appropriate iterative linear solver for an eigenvalue problem

I'm trying to solve a generalized eigenvalue problem $$Ax = \lambda Bx, \quad A = A^\top > 0,\; B = B^\top > 0$$ with $\lambda \approx \sigma$ using Rayleigh Quotient Iteration (RQI) (RQI is ...
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1answer
54 views

Is there an efficient $O(n^2)$ way to get the eigen decomposition given a LDL factorization?

Let's say I have a LDL factorization of a matrix A. Is there an efficient $O(n^2)$ way to get the eigen decomposition of A given it's LDL factorization? Is there a more efficient way, in case L and ...
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165 views

Verification in Eigenvalue problems

Let us start with a problem of the form $$(\mathcal{L} + k^2) u=0$$ with a set of given boundary conditions (Dirichlet, Neumann, Robin, Periodic, Bloch-Periodic). This corresponds with finding the ...
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2answers
130 views

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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Eigenvalue problem of the symmetric real operator which corresponds to the symmetric positive definite matrix

I have a real symmetric function $C(x,y)$ defined on $x,y\in[0,\infty)$, i.e. $C(x,y)=C(y,x)$. I want to solve the eigenvalues problem, i.e. find eigen values and eigen functions: $$\lambda ...
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88 views

Integro-differential eigenvalue problem, COMSOL

In my research I encounter an eigenvalue integro-differential equation of the form: $$f_n(x,y)=\lambda_n\iint_D\frac{\nabla'\cdot\big\lbrace ...
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60 views

large eigenvalues with LAPACK

I have question about LAPACK. I calculate eigenvalues of a $16\times16$ Hermitian complex matrix with small entries by ZHEEV ...
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1answer
222 views

Armadillo eig_sym() for extracting eigenvalues. Is it parallel at all? [closed]

After wasting 3 days with scalapack, I gave up and moved to Armadillo, considering it uses lapack underneath its beatiful and easy interface. I would like to calculate the eigen values and eigen ...
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108 views

Iteratively finding both left and right eigenvectors for non-symmetric complex matrix

I have a complex, non-Hermitian matrix $\mathbf{A}$, for which I need to find a few eigenvalues and eigenvectors in the generalised eigenvalue problem: $$\mathbf{A}\cdot \mathbf{x} = \lambda ...
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97 views

How to choose (and many) threshold in Eigenface?

I edited my question trying to make it as short and precise. I am developing a prototype of a facial recognition system for my Graduation Project. I use Eigenface and my main source is the document ...
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104 views

most efficient way to calculate eigen states of a 2D or 3D potential (Matlab)

I know of several ways to calculated the eigen states of 1D potentials (i.e. DVR, Crank–Nicolson, etc). However I wonder what is the most efficient way to do the same for a N-Dimensional potential? ...
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198 views

How to impose boundary conditions on eigenfunction problems?

I am trying to solve for the eigenfunctions of a (1D) differential operator using finite differences: $$A \, f(x) = \lambda f(x)$$ Here is an example in Python where $A = \partial_x^4$: ...
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1answer
83 views

Parallel Monte Carlo simulation using PETSc

I am trying to do Monte Carlo simulation for a large problem which requires eigensolution of a matrix for each sample. The matrix itself is quite large so much so that I want the eigensolution itself ...
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69 views

Computing eigenpairs of singular matrix with ZGEEV?

I've never run into a singular matrix before, so bear with me. I have a complex non-symmetric matrix (about 1000 x 1000) that I know has a couple zero eigenvalues. It isn't guaranteed to be ...
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diagonalization of matrix - omitting small matrix elements

I was wondering whether there is some theorem that allows me to put an upper bound on the error introduced by omitting small matrix elements from a matrix before diagonalization. Let's assume we ...
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1answer
92 views

Numerical eigenbasis for a unitary matrix

Do you know what numerical software computes an eigenvector basis for a unitary matrix? Say I have a unitary matrix $U$. If its eigenvalues are simple (no multiplicities), then for instance Matlab ...
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218 views

calculating eigenvector components of a given vector

I have some vector $V$ which can be decomposed into the eigenspace of the hermitian sparse operator $M$: $V = \sum_i v_i \hat{m}_i$ Is there a way to find the $\hat{m}_i$ (the eigenvector itself) ...
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176 views

roots of polynomials of high degree: LinAlgError: Eigenvalues did not converge

I wrote a simple script to generate random polynoimals $\displaystyle f(z)= \sum_{k=0}^N a_k \frac{z^k}{\sqrt{k!}} $ of high degree and find their roots. For more discussion on random polyomials see ...
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1answer
53 views

LAPACK DGGEVX: BALANC option

I'm using DGGEVX routine from LAPACKE with BALANC option as shown below, but to my surprise changing BALANC option from 'N' to ...
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202 views

I am looking for a complex sparse matrix EigenVector solver for GPGPU; preferably CUDA

So far the closest I've found is ViennaCL, which has a Lanczos implementation for Eigenvalues. It is not clear that EigenVectors are produced by this library. Does anyone here know whether ViennaCL ...
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988 views

Eigenvectors: MATLAB vs LAPACK DGGEV or DGGEVX

If we call LAPACK DGGEV or DGGEVX routines for two badly-conditioned matrices in a C++ code, will we get the same eigen-values ...
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1answer
107 views

Fortran 2003 ARPACK wrapper

I wrote a Fortran 2003 wrapper for the ARPACK routine znaupd, basically translating the the example driver routine zndrv1 into modern Fortran 2003 language with automatic arrays. I initialize every ...
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807 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 ...
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1answer
494 views

Comparing Eigenvectors, Mathematica vs. Matlab

I am trying to create the same outputs in Mathematica and Matlab, however I am running into trouble aligning the eigenvectors with the eigenvalues, I think the Matlab is doing something slightly more ...
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315 views

Sparse smallest eigenvalue problem on a linear subspace?

I am interested in methods for solving the optimization problem $$ \begin{array}{rl} \arg\min_x & x^T A x \\ \mathrm{s.t.} & x^T x = 1 \\ & Bx = 0 \end{array} $$ where $A$ is symmetric ...
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1answer
103 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
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1answer
80 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 ...
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3answers
260 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 ...
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1answer
54 views

last column of SPD matrix given it's spectral decomposition

I'm working on this application where I get the spectral decomposition (O,D) of a matrix A for free (A is of full rank) but not A itself, and I need to only recover the last column of A. I was ...
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42 views

Dominant contributions of a quadratic form

Let $\Sigma$ be a covariance matrix (e.g. symmetric positive definite). For arbitrary vectors $\epsilon$, I need to compute $\chi^2 \equiv \epsilon^\top\Sigma^{-1}\epsilon$, which I do using a ...
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Locally conservative method for differential generalized eigenvalue problem

I have to approximate the smallest eigenvalue of the following generalized eigenvalue problem $$ - \nabla \cdot D(x) \nabla p(x) + \alpha(x) p(x) = \lambda \beta(x) p(x) $$ over a domain like ...
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183 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 $A=A^T, B_i=B_i^T(\forall i\in\mathbb{N}) \in \mathbb{R}^{n\times n}, \text{rank}(A)=n,$ ...
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2answers
528 views

Algorithm for directly finding the leading eigenvector of an irreducible matrix

According to the Perron-Frobenius theorem, a real matrix with only positive entries (or one with non-negative entries with a property called irreducibility) will have a unique eigenvector that ...
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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: $A=P\Lambda_1 P^T$, ...
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141 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 ...