4,490 research outputs found
Complexity Estimates for Two Uncoupling Algorithms
Uncoupling algorithms transform a linear differential system of first order
into one or several scalar differential equations. We examine two approaches to
uncoupling: the cyclic-vector method (CVM) and the
Danilevski-Barkatou-Z\"urcher algorithm (DBZ). We give tight size bounds on the
scalar equations produced by CVM, and design a fast variant of CVM whose
complexity is quasi-optimal with respect to the output size. We exhibit a
strong structural link between CVM and DBZ enabling to show that, in the
generic case, DBZ has polynomial complexity and that it produces a single
equation, strongly related to the output of CVM. We prove that algorithm CVM is
faster than DBZ by almost two orders of magnitude, and provide experimental
results that validate the theoretical complexity analyses.Comment: To appear in Proceedings of ISSAC'13 (21/01/2013
Multiplicity Preserving Triangular Set Decomposition of Two Polynomials
In this paper, a multiplicity preserving triangular set decomposition
algorithm is proposed for a system of two polynomials. The algorithm decomposes
the variety defined by the polynomial system into unmixed components
represented by triangular sets, which may have negative multiplicities. In the
bivariate case, we give a complete algorithm to decompose the system into
multiplicity preserving triangular sets with positive multiplicities. We also
analyze the complexity of the algorithm in the bivariate case. We implement our
algorithm and show the effectiveness of the method with extensive experiments.Comment: 18 page
Irredundant Triangular Decomposition
Triangular decomposition is a classic, widely used and well-developed way to
represent algebraic varieties with many applications. In particular, there
exist sharp degree bounds for a single triangular set in terms of intrinsic
data of the variety it represents, and powerful randomized algorithms for
computing triangular decompositions using Hensel lifting in the
zero-dimensional case and for irreducible varieties. However, in the general
case, most of the algorithms computing triangular decompositions produce
embedded components, which makes it impossible to directly apply the intrinsic
degree bounds. This, in turn, is an obstacle for efficiently applying Hensel
lifting due to the higher degrees of the output polynomials and the lower
probability of success. In this paper, we give an algorithm to compute an
irredundant triangular decomposition of an arbitrary algebraic set defined
by a set of polynomials in C[x_1, x_2, ..., x_n]. Using this irredundant
triangular decomposition, we were able to give intrinsic degree bounds for the
polynomials appearing in the triangular sets and apply Hensel lifting
techniques. Our decomposition algorithm is randomized, and we analyze the
probability of success
Special Algorithm for Stability Analysis of Multistable Biological Regulatory Systems
We consider the problem of counting (stable) equilibriums of an important
family of algebraic differential equations modeling multistable biological
regulatory systems. The problem can be solved, in principle, using real
quantifier elimination algorithms, in particular real root classification
algorithms. However, it is well known that they can handle only very small
cases due to the enormous computing time requirements. In this paper, we
present a special algorithm which is much more efficient than the general
methods. Its efficiency comes from the exploitation of certain interesting
structures of the family of differential equations.Comment: 24 pages, 5 algorithms, 10 figure
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