91 research outputs found
Random matrices over a DVR and LU factorization
23 pagesLet R be a discrete valuation ring (DVR) and K be its fraction field. If M is a matrix over R admitting a LU decomposition, it could happen that the entries of the factors L and U do not lie in R, but just in K. Having a good control on the valuations of these entries is very important for algorithmic applications. In the paper, we prove that in average these valuations are not too large and explain how one can apply this result to provide an efficient algorithm computing a basis of a coherent sheaf over A^1 from the knowledge of its stalks
Random matrix over a DVR and LU factorization
Let R be a discrete valuation ring (DVR) and K be its fraction field. If M is
a matrix over R admitting a LU decomposition, it could happen that the entries
of the factors L and U do not lie in R, but just in K. Having a good control on
the valuations of these entries is very important for algorithmic applications.
In the paper, we prove that in average these valuations are not too large and
explain how one can apply this result to provide an efficient algorithm
computing a basis of a coherent sheaf over A^1 from the knowledge of its
stalks.Comment: 23 page
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
Resultants and subresultants of p-adic polynomials
We address the problem of the stability of the computations of resultants and
subresultants of polynomials defined over complete discrete valuation rings
(e.g. Zp or k[[t]] where k is a field). We prove that Euclide-like algorithms
are highly unstable on average and we explain, in many cases, how one can
stabilize them without sacrifying the complexity. On the way, we completely
determine the distribution of the valuation of the principal subresultants of
two random monic p-adic polynomials having the same degree
Tracking p-adic precision
We present a new method to propagate -adic precision in computations,
which also applies to other ultrametric fields. We illustrate it with many
examples and give a toy application to the stable computation of the SOMOS 4
sequence
Three real-space discretization techniques in electronic structure calculations
A characteristic feature of the state-of-the-art of real-space methods in
electronic structure calculations is the diversity of the techniques used in
the discretization of the relevant partial differential equations. In this
context, the main approaches include finite-difference methods, various types
of finite-elements and wavelets. This paper reports on the results of several
code development projects that approach problems related to the electronic
structure using these three different discretization methods. We review the
ideas behind these methods, give examples of their applications, and discuss
their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status
solidi (b) - basic solid state physics" devoted to the CECAM workshop "State
of the art developments and perspectives of real-space electronic structure
techniques in condensed matter and molecular physics". v2: Minor stylistic
and typographical changes, partly inspired by referee comment
A fast algorithm for computing the characteristic polynomial of the p-curvature
International audienceWe discuss theoretical and algorithmic questions related to the -curvature of differential operators in characteristic . Given such an operator~, and denoting by the characteristic polynomial of its -curvature, we first prove a new, alternative, description of . This description turns out to be particularly well suited to the fast computation of when is large: based on it, we design a new algorithm for computing , whose cost with respect to~ is \softO(p^{0.5}) operations in the ground field. This is remarkable since, prior to this work, the fastest algorithms for this task, and even for the subtask of deciding nilpotency of the -curvature, had merely slightly subquadratic complexity \softO(p^{1.79})
Point Counting On Genus 2 Curves
For cryptographic purposes, counting points on the jacobian variety of a given hyperelliptic curve is of great importance. There has been several approaches to obtain the cardinality of such a group, specially for hyperelliptic curves of genus 2. The best known algorithm for counting points on genus 2 curves over prime fields of large characteristic is a variant of Schoof’s genus 1 algorithm. Following a recent work of Gaudry and Schost, we show how to speed up the current state of the art genus 2 point counting algorithm by proposing various computational improvements to its basic arithmetical ingredients
- …