19,616 research outputs found
Towards an exact adaptive algorithm for the determinant of a rational matrix
In this paper we propose several strategies for the exact computation of the
determinant of a rational matrix. First, we use the Chinese Remaindering
Theorem and the rational reconstruction to recover the rational determinant
from its modular images. Then we show a preconditioning for the determinant
which allows us to skip the rational reconstruction process and reconstruct an
integer result. We compare those approaches with matrix preconditioning which
allow us to treat integer instead of rational matrices. This allows us to
introduce integer determinant algorithms to the rational determinant problem.
In particular, we discuss the applicability of the adaptive determinant
algorithm of [9] and compare it with the integer Chinese Remaindering scheme.
We present an analysis of the complexity of the strategies and evaluate their
experimental performance on numerous examples. This experience allows us to
develop an adaptive strategy which would choose the best solution at the run
time, depending on matrix properties. All strategies have been implemented in
LinBox linear algebra library
Reconstructing Rational Functions with
We present the open-source library for the
reconstruction of multivariate rational functions over finite fields. We
discuss the involved algorithms and their implementation. As an application, we
use in the context of integration-by-parts reductions and
compare runtime and memory consumption to a fully algebraic approach with the
program .Comment: 46 pages, 3 figures, 6 tables; v2: matches published versio
Generic design of Chinese remaindering schemes
We propose a generic design for Chinese remainder algorithms. A Chinese
remainder computation consists in reconstructing an integer value from its
residues modulo non coprime integers. We also propose an efficient linear data
structure, a radix ladder, for the intermediate storage and computations. Our
design is structured into three main modules: a black box residue computation
in charge of computing each residue; a Chinese remaindering controller in
charge of launching the computation and of the termination decision; an integer
builder in charge of the reconstruction computation. We then show that this
design enables many different forms of Chinese remaindering (e.g.
deterministic, early terminated, distributed, etc.), easy comparisons between
these forms and e.g. user-transparent parallelism at different parallel grains
Progressive growing of self-organized hierarchical representations for exploration
Designing agent that can autonomously discover and learn a diversity of
structures and skills in unknown changing environments is key for lifelong
machine learning. A central challenge is how to learn incrementally
representations in order to progressively build a map of the discovered
structures and re-use it to further explore. To address this challenge, we
identify and target several key functionalities. First, we aim to build lasting
representations and avoid catastrophic forgetting throughout the exploration
process. Secondly we aim to learn a diversity of representations allowing to
discover a "diversity of diversity" of structures (and associated skills) in
complex high-dimensional environments. Thirdly, we target representations that
can structure the agent discoveries in a coarse-to-fine manner. Finally, we
target the reuse of such representations to drive exploration toward an
"interesting" type of diversity, for instance leveraging human guidance.
Current approaches in state representation learning rely generally on
monolithic architectures which do not enable all these functionalities.
Therefore, we present a novel technique to progressively construct a Hierarchy
of Observation Latent Models for Exploration Stratification, called HOLMES.
This technique couples the use of a dynamic modular model architecture for
representation learning with intrinsically-motivated goal exploration processes
(IMGEPs). The paper shows results in the domain of automated discovery of
diverse self-organized patterns, considering as testbed the experimental
framework from Reinke et al. (2019)
Extended Rate, more GFUN
We present a software package that guesses formulae for sequences of, for
example, rational numbers or rational functions, given the first few terms. We
implement an algorithm due to Bernhard Beckermann and George Labahn, together
with some enhancements to render our package efficient. Thus we extend and
complement Christian Krattenthaler's program Rate, the parts concerned with
guessing of Bruno Salvy and Paul Zimmermann's GFUN, the univariate case of
Manuel Kauers' Guess.m and Manuel Kauers' and Christoph Koutschan's
qGeneratingFunctions.m.Comment: 26 page
Development of symbolic algorithms for certain algebraic processes
This study investigates the problem of computing the exact greatest common divisor of two polynomials relative to an orthogonal basis, defined over the rational number field. The main objective of the study is to design and implement an effective and efficient symbolic algorithm for the general class of dense polynomials, given the rational number defining terms of their basis. From a general algorithm using the comrade matrix approach, the nonmodular and modular techniques are prescribed. If the coefficients of the generalized polynomials are multiprecision integers, multiprecision arithmetic will be required in the construction of the comrade matrix and the corresponding systems coefficient matrix. In addition, the application of the nonmodular elimination technique on this coefficient matrix extensively applies multiprecision rational number operations. The modular technique is employed to minimize the complexity involved in such computations. A divisor test algorithm that enables the detection of an unlucky reduction is a crucial device for an effective implementation of the modular technique. With the bound of the true solution not known a priori, the test is devised and carefully incorporated into the modular algorithm. The results illustrate that the modular algorithm illustrate its best performance for the class of relatively prime polynomials. The empirical computing time results show that the modular algorithm is markedly superior to the nonmodular algorithms in the case of sufficiently dense Legendre basis polynomials with a small GCD solution. In the case of dense Legendre basis polynomials with a big GCD solution, the modular algorithm is significantly superior to the nonmodular algorithms in higher degree polynomials. For more definitive conclusions, the computing time functions of the algorithms that are presented in this report have been worked out. Further investigations have also been suggested
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