46 research outputs found
Generalized companion matrix for approximate GCD
We study a variant of the univariate approximate GCD problem, where the
coefficients of one polynomial f(x)are known exactly, whereas the coefficients
of the second polynomial g(x)may be perturbed. Our approach relies on the
properties of the matrix which describes the operator of multiplication by gin
the quotient ring C[x]=(f). In particular, the structure of the null space of
the multiplication matrix contains all the essential information about GCD(f;
g). Moreover, the multiplication matrix exhibits a displacement structure that
allows us to design a fast algorithm for approximate GCD computation with
quadratic complexity w.r.t. polynomial degrees.Comment: Submitted to MEGA 201
Over-constrained Weierstrass iteration and the nearest consistent system
We propose a generalization of the Weierstrass iteration for over-constrained
systems of equations and we prove that the proposed method is the Gauss-Newton
iteration to find the nearest system which has at least common roots and
which is obtained via a perturbation of prescribed structure. In the univariate
case we show the connection of our method to the optimization problem
formulated by Karmarkar and Lakshman for the nearest GCD. In the multivariate
case we generalize the expressions of Karmarkar and Lakshman, and give
explicitly several iteration functions to compute the optimum.
The arithmetic complexity of the iterations is detailed
On the Computation of the Topology of a Non-Reduced Implicit Space Curve
An algorithm is presented for the computation of the topology of a
non-reduced space curve defined as the intersection of two implicit algebraic
surfaces. It computes a Piecewise Linear Structure (PLS) isotopic to the
original space curve. The algorithm is designed to provide the exact result for
all inputs. It's a symbolic-numeric algorithm based on subresultant
computation. Simple algebraic criteria are given to certify the output of the
algorithm. The algorithm uses only one projection of the non-reduced space
curve augmented with adjacency information around some "particular points" of
the space curve. The algorithm is implemented with the Mathemagix Computer
Algebra System (CAS) using the SYNAPS library as a backend
On the isotopic meshing of an algebraic implicit surface
International audienceWe present a new and complete algorithm for computing the topology of an algebraic surface given by a squarefree polynomial in Q[X, Y, Z]. Our algorithm involves only subresultant computations and entirely relies on rational manipulation, which makes it direct to implement. We extend the work in [15], on the topology of non-reduced algebraic space curves, and apply it to the polar curve or apparent contour of the surface S. We exploit simple algebraic criterion to certify the pseudo-genericity and genericity position of the surface. This gives us rational parametrizations of the components of the polar curve, which are used to lift the topology of the projection of the polar curve. We deduce the connection of the two-dimensional components above the cell defined by the projection of the polar curve. A complexity analysis of the algorithm is provided leading to a bound in OB (d15 τ ) for the complexity of the computation of the topology of an implicit algebraic surface defined by integer coefficients polynomial of degree d and coefficients size τ . Examples illustrate the implementation in Mathemagix of this first complete code for certified topology of algebraic surfaces
Over-constrained Weierstrass iteration and the nearest consistent system
We propose a generalization of the Weierstrass iteration for over-constrained systems of equations and we prove that the proposed method allows us to find the nearest system which has at least common roots and which is obtained via a perturbation of prescribed structure. In the univariate case we show the connection of ourmethod to the optimization problem formulated by Karmarkar and Lakshmanfor the nearest GCD. In the multivariate case we generalize the expressions of Karmarkar and Lakshman, and give a simple iterative method to compute the optimum. The arithmetic complexity of the iteration is detailed