97,950 research outputs found
Polynomial-Division-Based Algorithms for Computing Linear Recurrence Relations
Sparse polynomial interpolation, sparse linear system solving or modular
rational reconstruction are fundamental problems in Computer Algebra. They come
down to computing linear recurrence relations of a sequence with the
Berlekamp-Massey algorithm. Likewise, sparse multivariate polynomial
interpolation and multidimensional cyclic code decoding require guessing linear
recurrence relations of a multivariate sequence.Several algorithms solve this
problem. The so-called Berlekamp-Massey-Sakata algorithm (1988) uses polynomial
additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on
linear algebra operations on a multi-Hankel matrix, a multivariate
generalization of a Hankel matrix. The Artinian Gorenstein border basis
algorithm (2017) uses a Gram-Schmidt process.We propose a new algorithm for
computing the Gr{\"o}bner basis of the ideal of relations of a sequence based
solely on multivariate polynomial arithmetic. This algorithm allows us to both
revisit the Berlekamp-Massey-Sakata algorithm through the use of polynomial
divisions and to completely revise the Scalar-FGLM algorithm without linear
algebra operations.A key observation in the design of this algorithm is to work
on the mirror of the truncated generating series allowing us to use polynomial
arithmetic modulo a monomial ideal. It appears to have some similarities with
Pad{\'e} approximants of this mirror polynomial.As an addition from the paper
published at the ISSAC conferance, we give an adaptive variant of this
algorithm taking into account the shape of the final Gr{\"o}bner basis
gradually as it is discovered. The main advantage of this algorithm is that its
complexity in terms of operations and sequence queries only depends on the
output Gr{\"o}bner basis.All these algorithms have been implemented in Maple
and we report on our comparisons
Formal Representation of the SS-DB Benchmark and Experimental Evaluation in EXTASCID
Evaluating the performance of scientific data processing systems is a
difficult task considering the plethora of application-specific solutions
available in this landscape and the lack of a generally-accepted benchmark. The
dual structure of scientific data coupled with the complex nature of processing
complicate the evaluation procedure further. SS-DB is the first attempt to
define a general benchmark for complex scientific processing over raw and
derived data. It fails to draw sufficient attention though because of the
ambiguous plain language specification and the extraordinary SciDB results. In
this paper, we remedy the shortcomings of the original SS-DB specification by
providing a formal representation in terms of ArrayQL algebra operators and
ArrayQL/SciQL constructs. These are the first formal representations of the
SS-DB benchmark. Starting from the formal representation, we give a reference
implementation and present benchmark results in EXTASCID, a novel system for
scientific data processing. EXTASCID is complete in providing native support
both for array and relational data and extensible in executing any user code
inside the system by the means of a configurable metaoperator. These features
result in an order of magnitude improvement over SciDB at data loading,
extracting derived data, and operations over derived data.Comment: 32 pages, 3 figure
Effective partitioning method for computing weighted Moore-Penrose inverse
We introduce a method and an algorithm for computing the weighted
Moore-Penrose inverse of multiple-variable polynomial matrix and the related
algorithm which is appropriated for sparse polynomial matrices. These methods
and algorithms are generalizations of algorithms developed in [M.B. Tasic, P.S.
Stanimirovic, M.D. Petkovic, Symbolic computation of weighted Moore-Penrose
inverse using partitioning method, Appl. Math. Comput. 189 (2007) 615-640] to
multiple-variable rational and polynomial matrices and improvements of these
algorithms on sparse matrices. Also, these methods are generalizations of the
partitioning method for computing the Moore-Penrose inverse of rational and
polynomial matrices introduced in [P.S. Stanimirovic, M.B. Tasic, Partitioning
method for rational and polynomial matrices, Appl. Math. Comput. 155 (2004)
137-163; M.D. Petkovic, P.S. Stanimirovic, Symbolic computation of the
Moore-Penrose inverse using partitioning method, Internat. J. Comput. Math. 82
(2005) 355-367] to the case of weighted Moore-Penrose inverse. Algorithms are
implemented in the symbolic computational package MATHEMATICA
Computing Small Certificates of Inconsistency of Quadratic Fewnomial Systems
B{\'e}zout 's theorem states that dense generic systems of n multivariate
quadratic equations in n variables have 2 n solutions over algebraically closed
fields. When only a small subset M of monomials appear in the equations
(fewnomial systems), the number of solutions may decrease dramatically. We
focus in this work on subsets of quadratic monomials M such that generic
systems with support M do not admit any solution at all. For these systems,
Hilbert's Nullstellensatz ensures the existence of algebraic certificates of
inconsistency. However, up to our knowledge all known bounds on the sizes of
such certificates -including those which take into account the Newton polytopes
of the polynomials- are exponential in n. Our main results show that if the
inequality 2|M| -- 2n \sqrt 1 + 8{\nu} -- 1 holds for a quadratic
fewnomial system -- where {\nu} is the matching number of a graph associated
with M, and |M| is the cardinality of M -- then there exists generically a
certificate of inconsistency of linear size (measured as the number of
coefficients in the ground field K). Moreover this certificate can be computed
within a polynomial number of arithmetic operations. Next, we evaluate how
often this inequality holds, and we give evidence that the probability that the
inequality is satisfied depends strongly on the number of squares. More
precisely, we show that if M is picked uniformly at random among the subsets of
n + k + 1 quadratic monomials containing at least (n 1/2+)
squares, then the probability that the inequality holds tends to 1 as n grows.
Interestingly, this phenomenon is related with the matching number of random
graphs in the Erd{\"o}s-Renyi model. Finally, we provide experimental results
showing that certificates in inconsistency can be computed for systems with
more than 10000 variables and equations.Comment: ISSAC 2016, Jul 2016, Waterloo, Canada. Proceedings of ISSAC 201
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