108 research outputs found
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
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
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
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
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
Microscopic derivation of multi-channel Hubbard models for ultracold nonreactive molecules in an optical lattice
Recent experimental advances in the cooling and manipulation of bialkali
dimer molecules have enabled the production of gases of ultracold molecules
that are not chemically reactive. It has been presumed in the literature that
in the absence of an electric field the low-energy scattering of such
nonreactive molecules (NRMs) will be similar to atoms, in which a single
-wave scattering length governs the collisional physics. However, in Ref.
[1], it was argued that the short-range collisional physics of NRMs is much
more complex than for atoms, and that this leads to a many-body description in
terms of a multi-channel Hubbard model. In this work, we show that this
multi-channel Hubbard model description of NRMs in an optical lattice is robust
against the approximations employed in Ref. [1] to estimate its parameters. We
do so via an exact, albeit formal, derivation of a multi-channel resonance
model for two NRMs from an ab initio description of the molecules in terms of
their constituent atoms. We discuss the regularization of this two-body
multi-channel resonance model in the presence of a harmonic trap, and how its
solutions form the basis for the many-body model of Ref. [1]. We also
generalize the derivation of the effective lattice model to include multiple
internal states (e.g., rotational or hyperfine). We end with an outlook to
future research.Comment: 19 pages, 4 figure
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})
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