3,109 research outputs found
Fast and accurate con-eigenvalue algorithm for optimal rational approximations
The need to compute small con-eigenvalues and the associated con-eigenvectors
of positive-definite Cauchy matrices naturally arises when constructing
rational approximations with a (near) optimally small error.
Specifically, given a rational function with poles in the unit disk, a
rational approximation with poles in the unit disk may be obtained
from the th con-eigenvector of an Cauchy matrix, where the
associated con-eigenvalue gives the approximation error in the
norm. Unfortunately, standard algorithms do not accurately compute
small con-eigenvalues (and the associated con-eigenvectors) and, in particular,
yield few or no correct digits for con-eigenvalues smaller than the machine
roundoff. We develop a fast and accurate algorithm for computing
con-eigenvalues and con-eigenvectors of positive-definite Cauchy matrices,
yielding even the tiniest con-eigenvalues with high relative accuracy. The
algorithm computes the th con-eigenvalue in operations
and, since the con-eigenvalues of positive-definite Cauchy matrices decay
exponentially fast, we obtain (near) optimal rational approximations in
operations, where is the
approximation error in the norm. We derive error bounds
demonstrating high relative accuracy of the computed con-eigenvalues and the
high accuracy of the unit con-eigenvectors. We also provide examples of using
the algorithm to compute (near) optimal rational approximations of functions
with singularities and sharp transitions, where approximation errors close to
machine precision are obtained. Finally, we present numerical tests on random
(complex-valued) Cauchy matrices to show that the algorithm computes all the
con-eigenvalues and con-eigenvectors with nearly full precision
A modularity based spectral method for simultaneous community and anti-community detection
In a graph or complex network, communities and anti-communities are node sets
whose modularity attains extremely large values, positive and negative,
respectively. We consider the simultaneous detection of communities and
anti-communities, by looking at spectral methods based on various matrix-based
definitions of the modularity of a vertex set. Invariant subspaces associated
to extreme eigenvalues of these matrices provide indications on the presence of
both kinds of modular structure in the network. The localization of the
relevant invariant subspaces can be estimated by looking at particular matrix
angles based on Frobenius inner products
Spectra of large time-lagged correlation matrices from Random Matrix Theory
We analyze the spectral properties of large, time-lagged correlation matrices
using the tools of random matrix theory. We compare predictions of the
one-dimensional spectra, based on approaches already proposed in the
literature. Employing the methods of free random variables and diagrammatic
techniques, we solve a general random matrix problem, namely the spectrum of a
matrix , where is an Gaussian random
matrix and is \textit{any} , not necessarily symmetric
(Hermitian) matrix. As a particular application, we present the spectral
features of the large lagged correlation matrices as a function of the depth of
the time-lag. We also analyze the properties of left and right eigenvector
correlations for the time-lagged matrices. We positively verify our results by
the numerical simulations.Comment: 44 pages, 11 figures; v2 typos corrected, final versio
- …