Faster Geometric Algorithms via Dynamic Determinant Computation

The computation of determinants or their signs is the core procedure in many important geometric algorithms, such as convex hull, volume and point location. As the dimension of the computation space grows, a higher percentage of the total computation time is consumed by these computations. In this paper we study the sequences of determinants that appear in geometric algorithms. The computation of a single determinant is accelerated by using the information from the previous computations in that sequence. We propose two dynamic determinant algorithms with quadratic arithmetic complexity when employed in convex hull and volume computations, and with linear arithmetic complexity when used in point location problems. We implement the proposed algorithms and perform an extensive experimental analysis. On one hand, our analysis serves as a performance study of state-of-the-art determinant algorithms and implementations. On the other hand, we demonstrate the supremacy of our methods over state-of-the-art implementations of determinant and geometric algorithms. Our experimental results include a 20 and 78 times speed-up in volume and point location computations in dimension 6 and 11 respectively.Comment: 29 pages, 8 figures, 3 table

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

Solving large linear algebraic systems in the context of integrable non-abelian Laurent ODEs

Abstract The paper reports on a computer algebra program LSSS (Linear Selective Systems Solver) for solving linear algebraic systems with rational coefficients. The program is especially efficient for very large sparse systems that have a solution in which many variables take the value zero. The program is applied to the symmetry investigation of a non-abelian Laurent ODE introduced recently by M. Kontsevich. The computed symmetries confirmed that a Lax pair found for this system earlier generates all first integrals of degree at least up to 14

Interoperability in the OpenDreamKit Project: The Math-in-the-Middle Approach

OpenDreamKit --- "Open Digital Research Environment Toolkit for the Advancement of Mathematics" --- is an H2020 EU Research Infrastructure project that aims at supporting, over the period 2015--2019, the ecosystem of open-source mathematical software systems. From that, OpenDreamKit will deliver a flexible toolkit enabling research groups to set up Virtual Research Environments, customised to meet the varied needs of research projects in pure mathematics and applications. An important step in the OpenDreamKit endeavor is to foster the interoperability between a variety of systems, ranging from computer algebra systems over mathematical databases to front-ends. This is the mission of the integration work package (WP6). We report on experiments and future plans with the \emph{Math-in-the-Middle} approach. This information architecture consists in a central mathematical ontology that documents the domain and fixes a joint vocabulary, combined with specifications of the functionalities of the various systems. Interaction between systems can then be enriched by pivoting off this information architecture.Comment: 15 pages, 7 figure

Computational techniques in graph homology of the moduli space of curves

The object of this thesis is the automated computation of the rational (co)homology of the moduli spaces of smooth marked Riemann surfaces Mg;n. This is achieved by using a computer to generate a chain complex, known in advance to have the same homology as Mg;n, and explicitly spell out the boundary operators in matrix form. As an application, we compute the Betti numbers of some moduli spaces Mg;n. Our original contribution is twofold. In Chapter 3, we develop algorithms for the enumeration of fatgraphs and their automorphisms, and the computation of the homology of the chain complex formed by fatgraphs of a given genus g and number of boundary components n. In Chapter 4, we describe a new practical parallel algorithm for performing Gaussian elimination on arbitrary matrices with exact computations: projections indicate that the size of the matrices involved in the Betti number computation can easily exceed the computational power of a single computer, so it is necessary to distribute the work over several processing units. Experimental results prove that our algorithm is in practice faster than freely available exact linear algebra codes. An effective implementation of the fatgraph algorithms presented here is available at http://code.google.com/p/fatghol. It has so far been used to compute the Betti numbers of Mg;n for (2g + n) 6 6. The Gaussian elimination code is likewise publicly available as open-source software from http://code.google.com/p/rheinfall