263 research outputs found
A Purely Functional Computer Algebra System Embedded in Haskell
We demonstrate how methods in Functional Programming can be used to implement
a computer algebra system. As a proof-of-concept, we present the
computational-algebra package. It is a computer algebra system implemented as
an embedded domain-specific language in Haskell, a purely functional
programming language. Utilising methods in functional programming and prominent
features of Haskell, this library achieves safety, composability, and
correctness at the same time. To demonstrate the advantages of our approach, we
have implemented advanced Gr\"{o}bner basis algorithms, such as Faug\`{e}re's
and , in a composable way.Comment: 16 pages, Accepted to CASC 201
Partial Gröbner bases for multiobjective integer linear optimization
This paper presents a new methodology for solving multiobjective integer linear programs (MOILP) using tools from algebraic geometry. We introduce the concept of partial Gr¨obner basis for a family of multiobjective programs where the right-hand side varies. This new structure extends the notion of Gr¨obner basis for the single objective case to the case of multiple objectives, i.e., when there is a partial ordering instead of a total ordering over the feasible vectors. The main property of these bases is that the partial reduction of the integer elements in the kernel of the
constraint matrix by the different blocks of the basis is zero. This property allows us to prove that this new construction is a test family for a family of multiobjective programs. An algorithm “´a la Buchberger” is developed to compute partial Gr¨obner bases, and two different approaches are
derived, using this methodology, for computing the entire set of Pareto-optimal solutions of any MOILP problem. Some examples illustrate the application of the algorithm, and computational experiments are reported on several families of problems.Ministerio de Educación y Cienci
Parallel and distributed Gr\"obner bases computation in JAS
This paper considers parallel Gr\"obner bases algorithms on distributed
memory parallel computers with multi-core compute nodes. We summarize three
different Gr\"obner bases implementations: shared memory parallel, pure
distributed memory parallel and distributed memory combined with shared memory
parallelism. The last algorithm, called distributed hybrid, uses only one
control communication channel between the master node and the worker nodes and
keeps polynomials in shared memory on a node. The polynomials are transported
asynchronous to the control-flow of the algorithm in a separate distributed
data structure. The implementation is generic and works for all implemented
(exact) fields. We present new performance measurements and discuss the
performance of the algorithms.Comment: 14 pages, 8 tables, 13 figure
- …