1,621 research outputs found
The use of Lanczos's method to solve the large generalized symmetric definite eigenvalue problem
The generalized eigenvalue problem, Kx = Lambda Mx, is of significant practical importance, especially in structural enginering where it arises as the vibration and buckling problem. A new algorithm, LANZ, based on Lanczos's method is developed. LANZ uses a technique called dynamic shifting to improve the efficiency and reliability of the Lanczos algorithm. A new algorithm for solving the tridiagonal matrices that arise when using Lanczos's method is described. A modification of Parlett and Scott's selective orthogonalization algorithm is proposed. Results from an implementation of LANZ on a Convex C-220 show it to be superior to a subspace iteration code
Network Density of States
Spectral analysis connects graph structure to the eigenvalues and
eigenvectors of associated matrices. Much of spectral graph theory descends
directly from spectral geometry, the study of differentiable manifolds through
the spectra of associated differential operators. But the translation from
spectral geometry to spectral graph theory has largely focused on results
involving only a few extreme eigenvalues and their associated eigenvalues.
Unlike in geometry, the study of graphs through the overall distribution of
eigenvalues - the spectral density - is largely limited to simple random graph
models. The interior of the spectrum of real-world graphs remains largely
unexplored, difficult to compute and to interpret.
In this paper, we delve into the heart of spectral densities of real-world
graphs. We borrow tools developed in condensed matter physics, and add novel
adaptations to handle the spectral signatures of common graph motifs. The
resulting methods are highly efficient, as we illustrate by computing spectral
densities for graphs with over a billion edges on a single compute node. Beyond
providing visually compelling fingerprints of graphs, we show how the
estimation of spectral densities facilitates the computation of many common
centrality measures, and use spectral densities to estimate meaningful
information about graph structure that cannot be inferred from the extremal
eigenpairs alone.Comment: 10 pages, 7 figure
Parallel Matrix-free polynomial preconditioners with application to flow simulations in discrete fracture networks
We develop a robust matrix-free, communication avoiding parallel, high-degree
polynomial preconditioner for the Conjugate Gradient method for large and
sparse symmetric positive definite linear systems. We discuss the selection of
a scaling parameter aimed at avoiding unwanted clustering of eigenvalues of the
preconditioned matrices at the extrema of the spectrum. We use this
preconditioned framework to solve a block system arising in the
simulation of fluid flow in large-size discrete fractured networks. We apply
our polynomial preconditioner to a suitable Schur complement related with this
system, which can not be explicitly computed because of its size and density.
Numerical results confirm the excellent properties of the proposed
preconditioner up to very high polynomial degrees. The parallel implementation
achieves satisfactory scalability by taking advantage from the reduced number
of scalar products and hence of global communications
A parallel Block Lanczos algorithm and its implementation for the evaluation of some eigenvalues of large sparse symmetric matrices on multicomputers
In the present work we describe HPEC (High Performance Eigenvalues Computation), a parallel software package for the
evaluation of some eigenvalues of a large sparse symmetric matrix. It implements an efficient and portable Block Lanczos
algorithm for distributed memory multicomputers. HPEC is based on basic linear algebra operations for sparse and dense
matrices, some of which have been derived by ScaLAPACK library modules. Numerical experiments have been carried out
to evaluate HPEC performance on a cluster of workstations with test matrices from Matrix Market and Higham’s collections.
A comparison with a PARPACKroutine is also detailed. Finally, parallel performance is evaluated on random matrices, using
standard parameters
Spectrum of the Dirac Operator and Multigrid Algorithm with Dynamical Staggered Fermions
Complete spectra of the staggered Dirac operator \Dirac are determined in
quenched four-dimensional gauge fields, and also in the presence of
dynamical fermions.
Periodic as well as antiperiodic boundary conditions are used.
An attempt is made to relate the performance of multigrid (MG) and conjugate
gradient (CG) algorithms for propagators with the distribution of the
eigenvalues of~\Dirac.
The convergence of the CG algorithm is determined only by the condition
number~ and by the lattice size.
Since~'s do not vary significantly when quarks become dynamic,
CG convergence in unquenched fields can be predicted from quenched
simulations.
On the other hand, MG convergence is not affected by~ but depends on
the spectrum in a more subtle way.Comment: 19 pages, 8 figures, HUB-IEP-94/12 and KL-TH 19/94; comes as a
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