107,086 research outputs found
A fast solver for linear systems with displacement structure
We describe a fast solver for linear systems with reconstructable Cauchy-like
structure, which requires O(rn^2) floating point operations and O(rn) memory
locations, where n is the size of the matrix and r its displacement rank. The
solver is based on the application of the generalized Schur algorithm to a
suitable augmented matrix, under some assumptions on the knots of the
Cauchy-like matrix. It includes various pivoting strategies, already discussed
in the literature, and a new algorithm, which only requires reconstructability.
We have developed a software package, written in Matlab and C-MEX, which
provides a robust implementation of the above method. Our package also includes
solvers for Toeplitz(+Hankel)-like and Vandermonde-like linear systems, as
these structures can be reduced to Cauchy-like by fast and stable transforms.
Numerical experiments demonstrate the effectiveness of the software.Comment: 27 pages, 6 figure
Solving Sparse Integer Linear Systems
We propose a new algorithm to solve sparse linear systems of equations over
the integers. This algorithm is based on a -adic lifting technique combined
with the use of block matrices with structured blocks. It achieves a sub-cubic
complexity in terms of machine operations subject to a conjecture on the
effectiveness of certain sparse projections. A LinBox-based implementation of
this algorithm is demonstrated, and emphasizes the practical benefits of this
new method over the previous state of the art
Curriculum Guidelines for Undergraduate Programs in Data Science
The Park City Math Institute (PCMI) 2016 Summer Undergraduate Faculty Program
met for the purpose of composing guidelines for undergraduate programs in Data
Science. The group consisted of 25 undergraduate faculty from a variety of
institutions in the U.S., primarily from the disciplines of mathematics,
statistics and computer science. These guidelines are meant to provide some
structure for institutions planning for or revising a major in Data Science
A distributed-memory package for dense Hierarchically Semi-Separable matrix computations using randomization
We present a distributed-memory library for computations with dense
structured matrices. A matrix is considered structured if its off-diagonal
blocks can be approximated by a rank-deficient matrix with low numerical rank.
Here, we use Hierarchically Semi-Separable representations (HSS). Such matrices
appear in many applications, e.g., finite element methods, boundary element
methods, etc. Exploiting this structure allows for fast solution of linear
systems and/or fast computation of matrix-vector products, which are the two
main building blocks of matrix computations. The compression algorithm that we
use, that computes the HSS form of an input dense matrix, relies on randomized
sampling with a novel adaptive sampling mechanism. We discuss the
parallelization of this algorithm and also present the parallelization of
structured matrix-vector product, structured factorization and solution
routines. The efficiency of the approach is demonstrated on large problems from
different academic and industrial applications, on up to 8,000 cores.
This work is part of a more global effort, the STRUMPACK (STRUctured Matrices
PACKage) software package for computations with sparse and dense structured
matrices. Hence, although useful on their own right, the routines also
represent a step in the direction of a distributed-memory sparse solver
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