115 research outputs found
New Structured Matrix Methods for Real and Complex Polynomial Root-finding
We combine the known methods for univariate polynomial root-finding and for
computations in the Frobenius matrix algebra with our novel techniques to
advance numerical solution of a univariate polynomial equation, and in
particular numerical approximation of the real roots of a polynomial. Our
analysis and experiments show efficiency of the resulting algorithms.Comment: 18 page
Solving parametric systems of polynomial equations over the reals through Hermite matrices
We design a new algorithm for solving parametric systems having finitely many
complex solutions for generic values of the parameters. More precisely, let with and
, be the algebraic set
defined by and be the projection . Under the
assumptions that admits finitely many complex roots for generic values of
and that the ideal generated by is radical, we solve the following
problem. On input , we compute semi-algebraic formulas defining
semi-algebraic subsets of the -space such that
is dense in and the number of real points in
is invariant when varies over each .
This algorithm exploits properties of some well chosen monomial bases in the
algebra where is the ideal generated by in
and the specialization property of the so-called Hermite
matrices. This allows us to obtain compact representations of the sets by
means of semi-algebraic formulas encoding the signature of a symmetric matrix.
When satisfies extra genericity assumptions, we derive complexity bounds on
the number of arithmetic operations in and the degree of the
output polynomials. Let be the maximal degree of the 's and , we prove that, on a generic , one can compute
those semi-algebraic formulas with operations in and that the polynomials involved
have degree bounded by .
We report on practical experiments which illustrate the efficiency of our
algorithm on generic systems and systems from applications. It allows us to
solve problems which are out of reach of the state-of-the-art
HPC-GAP: engineering a 21st-century high-performance computer algebra system
Symbolic computation has underpinned a number of key advances in Mathematics and Computer Science. Applications are typically large and potentially highly parallel, making them good candidates for parallel execution at a variety of scales from multi-core to high-performance computing systems. However, much existing work on parallel computing is based around numeric rather than symbolic computations. In particular, symbolic computing presents particular problems in terms of varying granularity and irregular task sizes thatdo not match conventional approaches to parallelisation. It also presents problems in terms of the structure of the algorithms and data.
This paper describes a new implementation of the free open-source GAP computational algebra system that places parallelism at the heart of the design, dealing with the key scalability and cross-platform portability problems. We provide three system layers that deal with the three most important classes of hardware: individual shared memory
multi-core nodes, mid-scale distributed clusters of (multi-core) nodes, and full-blown HPC systems, comprising large-scale tightly-connected networks of multi-core nodes. This requires us to develop new cross-layer programming abstractions in the form of new domain-specific skeletons that allow us to seamlessly target different hardware levels. Our results show that, using our approach, we can achieve good scalability and speedups for two realistic exemplars, on high-performance systems comprising up to 32,000 cores, as well as on ubiquitous multi-core systems and distributed clusters. The work reported here paves the way towards full scale exploitation of symbolic computation by high-performance computing systems, and we demonstrate the potential with two major case studies
Gröbner Bases of Ideals Invariant under a Commutative Group: the Non-Modular Case
International audienceWe propose efficient algorithms to compute the Gröbner basis of an ideal globally invariant under the action of a commutative matrix group , in the non-modular case (where doesn't divide ). The idea is to simultaneously diagonalize the matrices in , and apply a linear change of variables on corresponding to the base-change matrix of this diagonalization. We can now suppose that the matrices acting on are diagonal. This action induces a grading on the ring , compatible with the degree, indexed by a group related to , that we call -degree. The next step is the observation that this grading is maintained during a Gröbner basis computation or even a change of ordering, which allows us to split the Macaulay matrices into submatrices of roughly the same size. In the same way, we are able to split the canonical basis of (the staircase) if is a zero-dimensional ideal. Therefore, we derive \emph{abelian} versions of the classical algorithms , or FGLM. Moreover, this new variant of allows complete parallelization of the linear algebra steps, which has been successfully implemented. On instances coming from applications (NTRU crypto-system or the Cyclic-n problem), a speed-up of more than 400 can be obtained. For example, a Gröbner basis of the Cyclic-11 problem can be solved in less than 8 hours with this variant of . Moreover, using this method, we can identify new classes of polynomial systems that can be solved in polynomial time
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