We give a unified (“basis free”) framework for the Descartes method for real root isolation of square-free real polynomials. This framework encompasses the usual Descartes ’ rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = P n i=0 aiXi with integer coefficients |ai | < 2 L, this yields a bound of O(n(L + log n)) on the size of recursion trees. We show that this bound is tight for L = Ω(log n), and we use it to derive the best known bit complexity bound for the integer case.

We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (-Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of eOB(d 4 τ 2). This matches the best known bounds for binary subdivision solvers. Moreover, we generalize this to cover the non square-free polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities, and we improve the known bounds at least by a factor of d. Finally, we present our C++ implementation in synaps and some preliminary experiments on various data sets.

real radical ideals

this paper -- we enumerate all 35 di#erent morphologies of QSIC, and characterize each of these morphologies using a signature sequence that can exactly be computed using rational arithmetic. The third problem, not handled here, leads to a lengthy case by case study which depends a lot on the application behind. Consider the intersection curve of two quadrics given by BX = 0, where X = (x, y, z, w) and A, B are 4 4 real symmetric matrices. The characteristic polynomial of (1) and f(#) = 0 is called the characteristic equation of B

The Descartes method for polynomial real root isolation can be performed with respect to monomial bases and with respect to Bernstein bases. The first variant uses Taylor shift by 1 as its main subalgorithm, the second uses de Casteljau’s algorithm. When applied to integer polynomials, the two variants have co-dominant, almost tight computing time bounds. Implementations of either variant can obtain speed-ups over previous state-of-the-art implementations by more than an order of magnitude if they use features of the processor architecture. We present an implementation of the Bernstein-bases variant of the Descartes method that automatically generates architecture-aware high-level code and leaves further optimizations to the compiler. We compare the performance of our implementation, algorithmically tuned implementations of the monomial and Bernstein variants, and architecture-unaware implementations of both variants on four different processor architectures and for three classes of input polynomials.

We present an algorithm RealRootIsolate for isolating the real roots of a system of multivariate polynomials given by a zerodimensional squarefree regular chain. The output of the algorithm is guaranteed in the sense that all real roots are obtained and are described by boxes of arbitrary precision. Real roots are encoded with a hybrid representation, combining a symbolic object, namely a regular chain, and a numerical approximation given by intervals. Our isolation algorithm is a generalization, for regular chains, of the algorithm proposed by Collins and Akritas. We have implemented RealRootIsolate as a command of the module SemiAlgebraicSetTools of the RegularChains library in Maple. Benchmarks are reported. 1

We introduce an exact subdivision algorithm CEVAL for isolating complex roots of a square-free polynomial. The subdivision predicates are based on evaluating the original polynomial or its derivatives, and hence is easy to implement. It can be seen as a generalization of a previous real root isolation algorithm called EVAL. Under suitable conditions, the algorithm is applicable for general analytic functions. We provide a complexity analysis of our algorithm on the benchmark problem of isolating all complex roots of a square-free polynomial with Gaussian integer coefficients. The analysis is based on a novel technique called δ-clusters. This analysis shows, somewhat surprisingly, that the simple EVAL algorithm matches (up to logarithmic factors) the bit complexity bounds of current practical exact algorithms such as those based on Descartes, Continued Fraction or Sturm methods. Furthermore, the more general CEVAL also achieves the same complexity.

Abstract. In this paper we propose a unified methodology for computing the set VK(I) of complex (K = C) or real (K = R) roots of an ideal I ⊆ R[x], assuming VK(I) is finite. We show how moment matrices, defined in terms of a given set of generators of the ideal I, can be used to (numerically) find not only the real variety VR(I), as shown in the authors ’ previous work, but also the complex variety VC(I), thus leading to a unified treatment of the algebraic and real algebraic problem. In contrast to the real algebraic version of the algorithm, the complex analogue only uses basic numerical linear algebra because it does not require positive semidefiniteness of the moment matrix and so avoids semidefinite programming techniques. The links between these algorithms and other numerical algebraic methods are outlined and their stopping criteria are related. Key words. Polynomial ideal, zero-dimensional ideal, complex roots, real roots, numerical linear algebra.

This paper presents some important issues and potential research tracks for Geometric Constraint Solving: the use of the simplicial Bernstein base to reduce the wrapping effect in interval methods, the computation of the dimension of the solution set with methods used to measure the dimension of fractals, the pitfalls of graph based decomposition methods, the alternative provided by linear algebra, the witness configuration method, the use of randomized provers to detect dependences between constraints, the study of incidence constraints, the search for intrinsic (coordinate-free) formulations and the need for formal specifications.

Abstract not found