7 research outputs found
On Continued Fraction Expansion of Real Roots of Polynomial Systems, Complexity and Condition Numbers
International audienceWe elaborate on a correspondence between the coeffcients of a multivariate polynomial represented in the Bernstein basis and in a tensor-monomial basis, which leads to homography representations of polynomial functions, that use only integer arithmetic (in contrast to Bernstein basis) and are feasible over unbounded regions. Then, we study an algorithm to split this representation and we obtain a subdivision scheme for the domain of multivariate polynomial functions. This implies a new algorithm for real root isolation, MCF, that generalizes the Continued Fraction (CF) algorithm of univariate polynomials. A partial extension of Vincent's Theorem for multivariate polynomials is presented, which allows us to prove the termination of the algorithm. Bounding functions, projection and preconditioning are employed to speed up the scheme. The resulting isolation boxes have optimized rational coordinates, corresponding to the first terms of the continued fraction expansion of the real roots. Finally, we present new complexity bounds for a simplified version of the algorithm in the bit complexity model, and also bounds in the real RAM model for a family of subdivision algorithms in terms of the real condition number of the system. Examples computed with our C++ implementation illustrate the practical aspects of our method
A Generic Position Based Method for Real Root Isolation of Zero-Dimensional Polynomial Systems
We improve the local generic position method for isolating the real roots of
a zero-dimensional bivariate polynomial system with two polynomials and extend
the method to general zero-dimensional polynomial systems. The method mainly
involves resultant computation and real root isolation of univariate polynomial
equations. The roots of the system have a linear univariate representation. The
complexity of the method is for the bivariate case, where
, resp., is an upper bound on the degree, resp., the
maximal coefficient bitsize of the input polynomials. The algorithm is
certified with probability 1 in the multivariate case. The implementation shows
that the method is efficient, especially for bivariate polynomial systems.Comment: 24 pages, 5 figure
On the Boolean complexity of real root refinement
International audienceWe assume that a real square-free polynomial has a degree , a maximum coefficient bitsize and a real root lying in an isolating interval and having no nonreal roots nearby (we quantify this assumption). Then, we combine the {\em Double Exponential Sieve} algorithm (also called the {\em Bisection of the Exponents}), the bisection, and Newton iteration to decrease the width of this inclusion interval by a factor of . The algorithm has Boolean complexity . Our algorithms support the same complexity bound for the refinement of roots, for any
The Complexity of Subdivision for Diameter-Distance Tests
We present a general framework for analyzing the complexity of
subdivision-based algorithms whose tests are based on the sizes of regions and
their distance to certain sets (often varieties) intrinsic to the problem under
study. We call such tests diameter-distance tests. We illustrate that
diameter-distance tests are common in the literature by proving that many
interval arithmetic-based tests are, in fact, diameter-distance tests. For this
class of algorithms, we provide both non-adaptive bounds for the complexity,
based on separation bounds, as well as adaptive bounds, by applying the
framework of continuous amortization.
Using this structure, we provide the first complexity analysis for the
algorithm by Plantinga and Vegeter for approximating real implicit curves and
surfaces. We present both adaptive and non-adaptive a priori worst-case bounds
on the complexity of this algorithm both in terms of the number of subregions
constructed and in terms of the bit complexity for the construction. Finally,
we construct families of hypersurfaces to prove that our bounds are tight