16,277 research outputs found
The Anderson model of localization: a challenge for modern eigenvalue methods
We present a comparative study of the application of modern eigenvalue
algorithms to an eigenvalue problem arising in quantum physics, namely, the
computation of a few interior eigenvalues and their associated eigenvectors for
the large, sparse, real, symmetric, and indefinite matrices of the Anderson
model of localization. We compare the Lanczos algorithm in the 1987
implementation of Cullum and Willoughby with the implicitly restarted Arnoldi
method coupled with polynomial and several shift-and-invert convergence
accelerators as well as with a sparse hybrid tridiagonalization method. We
demonstrate that for our problem the Lanczos implementation is faster and more
memory efficient than the other approaches. This seemingly innocuous problem
presents a major challenge for all modern eigenvalue algorithms.Comment: 16 LaTeX pages with 3 figures include
Spectral Theory of Sparse Non-Hermitian Random Matrices
Sparse non-Hermitian random matrices arise in the study of disordered
physical systems with asymmetric local interactions, and have applications
ranging from neural networks to ecosystem dynamics. The spectral
characteristics of these matrices provide crucial information on system
stability and susceptibility, however, their study is greatly complicated by
the twin challenges of a lack of symmetry and a sparse interaction structure.
In this review we provide a concise and systematic introduction to the main
tools and results in this field. We show how the spectra of sparse
non-Hermitian matrices can be computed via an analogy with infinite dimensional
operators obeying certain recursion relations. With reference to three
illustrative examples --- adjacency matrices of regular oriented graphs,
adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency
matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the
use of these methods to obtain both analytic and numerical results for the
spectrum, the spectral distribution, the location of outlier eigenvalues, and
the statistical properties of eigenvectors.Comment: 60 pages, 10 figure
Second order accurate distributed eigenvector computation for extremely large matrices
We propose a second-order accurate method to estimate the eigenvectors of
extremely large matrices thereby addressing a problem of relevance to
statisticians working in the analysis of very large datasets. More
specifically, we show that averaging eigenvectors of randomly subsampled
matrices efficiently approximates the true eigenvectors of the original matrix
under certain conditions on the incoherence of the spectral decomposition. This
incoherence assumption is typically milder than those made in matrix completion
and allows eigenvectors to be sparse. We discuss applications to spectral
methods in dimensionality reduction and information retrieval.Comment: Complete proofs are included on averaging performanc
Sensitivity of Markov chains for wireless protocols
Network communication protocols such as the IEEE 802.11 wireless protocol are currently best modelled as Markov chains. In these situations we have some protocol parameters , and a transition matrix from which we can compute the steady state (equilibrium) distribution and hence final desired quantities , which might be for example the throughput of the protocol. Typically the chain will have thousands of states, and a particular example of interest is the Bianchi chain defined later. Generally we want to optimise , perhaps subject to some constraints that also depend on the Markov chain. To do this efficiently we need the gradient of with respect to , and therefore need the gradient of and other properties of the chain with respect to . The matrix formulas available for this involve the so-called fundamental matrix, but are there approximate gradients available which are faster and still sufficiently accurate? In some cases BT would like to do the whole calculation in computer algebra, and get a series expansion of the equilibrium with respect to a parameter in . In addition to the steady state , the same questions arise for the mixing time and the mean hitting times. Two qualitative features that were brought to the Study Group’s attention were:
* the transition matrix is large, but sparse.
* the systems of linear equations to be solved are generally singular and need some additional normalisation condition, such as is provided by using the fundamental matrix.
We also note a third highly important property regarding applications of numerical linear algebra:
* the transition matrix is asymmetric.
A realistic dimension for the matrix in the Bianchi model described below is 8064×8064, but on average there are only a few nonzero entries per column. Merely storing such a large matrix in dense form would require nearly 0.5GBytes using 64-bit floating point numbers, and computing its LU factorisation takes around 80 seconds on a modern microprocessor. It is thus highly desirable to employ specialised algorithms for sparse matrices. These algorithms are generally divided between those only applicable to symmetric matrices, the most prominent being the conjugate-gradient (CG) algorithm for solving linear equations, and those applicable to general matrices. A similar division is present in the literature on numerical eigenvalue problems
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