335 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
An efficient multi-core implementation of a novel HSS-structured multifrontal solver using randomized sampling
We present a sparse linear system solver that is based on a multifrontal
variant of Gaussian elimination, and exploits low-rank approximation of the
resulting dense frontal matrices. We use hierarchically semiseparable (HSS)
matrices, which have low-rank off-diagonal blocks, to approximate the frontal
matrices. For HSS matrix construction, a randomized sampling algorithm is used
together with interpolative decompositions. The combination of the randomized
compression with a fast ULV HSS factorization leads to a solver with lower
computational complexity than the standard multifrontal method for many
applications, resulting in speedups up to 7 fold for problems in our test
suite. The implementation targets many-core systems by using task parallelism
with dynamic runtime scheduling. Numerical experiments show performance
improvements over state-of-the-art sparse direct solvers. The implementation
achieves high performance and good scalability on a range of modern shared
memory parallel systems, including the Intel Xeon Phi (MIC). The code is part
of a software package called STRUMPACK -- STRUctured Matrices PACKage, which
also has a distributed memory component for dense rank-structured matrices
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