4,187 research outputs found
A Householder-based algorithm for Hessenberg-triangular reduction
The QZ algorithm for computing eigenvalues and eigenvectors of a matrix
pencil requires that the matrices first be reduced to
Hessenberg-triangular (HT) form. The current method of choice for HT reduction
relies entirely on Givens rotations regrouped and accumulated into small dense
matrices which are subsequently applied using matrix multiplication routines. A
non-vanishing fraction of the total flop count must nevertheless still be
performed as sequences of overlapping Givens rotations alternately applied from
the left and from the right. The many data dependencies associated with this
computational pattern leads to inefficient use of the processor and poor
scalability.
In this paper, we therefore introduce a fundamentally different approach that
relies entirely on (large) Householder reflectors partially accumulated into
block reflectors, by using (compact) WY representations. Even though the new
algorithm requires more floating point operations than the state of the art
algorithm, extensive experiments on both real and synthetic data indicate that
it is still competitive, even in a sequential setting. The new algorithm is
conjectured to have better parallel scalability, an idea which is partially
supported by early small-scale experiments using multi-threaded BLAS. The
design and evaluation of a parallel formulation is future work
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
An Incremental Algorithm for Computing Cylindrical Algebraic Decompositions
In this paper, we propose an incremental algorithm for computing cylindrical
algebraic decompositions. The algorithm consists of two parts: computing a
complex cylindrical tree and refining this complex tree into a cylindrical tree
in real space. The incrementality comes from the first part of the algorithm,
where a complex cylindrical tree is constructed by refining a previous complex
cylindrical tree with a polynomial constraint. We have implemented our
algorithm in Maple. The experimentation shows that the proposed algorithm
outperforms existing ones for many examples taken from the literature
Using the distribution of cells by dimension in a cylindrical algebraic decomposition
We investigate the distribution of cells by dimension in cylindrical
algebraic decompositions (CADs). We find that they follow a standard
distribution which seems largely independent of the underlying problem or CAD
algorithm used. Rather, the distribution is inherent to the cylindrical
structure and determined mostly by the number of variables.
This insight is then combined with an algorithm that produces only
full-dimensional cells to give an accurate method of predicting the number of
cells in a complete CAD. Since constructing only full-dimensional cells is
relatively inexpensive (involving no costly algebraic number calculations) this
leads to heuristics for helping with various questions of problem formulation
for CAD, such as choosing an optimal variable ordering. Our experiments
demonstrate that this approach can be highly effective.Comment: 8 page
Fast Parallel Randomized QR with Column Pivoting Algorithms for Reliable Low-rank Matrix Approximations
Factorizing large matrices by QR with column pivoting (QRCP) is substantially
more expensive than QR without pivoting, owing to communication costs required
for pivoting decisions. In contrast, randomized QRCP (RQRCP) algorithms have
proven themselves empirically to be highly competitive with high-performance
implementations of QR in processing time, on uniprocessor and shared memory
machines, and as reliable as QRCP in pivot quality.
We show that RQRCP algorithms can be as reliable as QRCP with failure
probabilities exponentially decaying in oversampling size. We also analyze
efficiency differences among different RQRCP algorithms. More importantly, we
develop distributed memory implementations of RQRCP that are significantly
better than QRCP implementations in ScaLAPACK.
As a further development, we introduce the concept of and develop algorithms
for computing spectrum-revealing QR factorizations for low-rank matrix
approximations, and demonstrate their effectiveness against leading low-rank
approximation methods in both theoretical and numerical reliability and
efficiency.Comment: 11 pages, 14 figures, accepted by 2017 IEEE 24th International
Conference on High Performance Computing (HiPC), awarded the best paper priz
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