427 research outputs found

    Clustering Complex Zeros of Triangular Systems of Polynomials

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    This paper gives the first algorithm for finding a set of natural ϵ\epsilon-clusters of complex zeros of a triangular system of polynomials within a given polybox in Cn\mathbb{C}^n, for any given ϵ>0\epsilon>0. Our algorithm is based on a recent near-optimal algorithm of Becker et al (2016) for clustering the complex roots of a univariate polynomial where the coefficients are represented by number oracles. Our algorithm is numeric, certified and based on subdivision. We implemented it and compared it with two well-known homotopy solvers on various triangular systems. Our solver always gives correct answers, is often faster than the homotopy solver that often gives correct answers, and sometimes faster than the one that gives sometimes correct results.Comment: Research report V6: description of the main algorithm update

    A Generic Position Based Method for Real Root Isolation of Zero-Dimensional Polynomial Systems

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    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 O~B(N10)\tilde{O}_B(N^{10}) for the bivariate case, where N=max(d,τ)N=\max(d,\tau), dd resp., τ\tau 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

    An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks

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    We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of points. First, the amount of purely symbolic operations is significantly reduced, that is, only resultant computation and square-free factorization is still needed. Second, our algorithm neither assumes generic position of the input system nor demands for any change of the coordinate system. The latter is due to a novel inclusion predicate to certify that a certain region is isolating for a solution. Our implementation exploits graphics hardware to expedite the resultant computation. Furthermore, we integrate a number of filtering techniques to improve the overall performance. Efficiency of the proposed method is proven by a comparison of our implementation with two state-of-the-art implementations, that is, LPG and Maple's isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201

    On the asymptotic and practical complexity of solving bivariate systems over the reals

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    This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of \sOB(N^{14}) for the purely projection-based method, and \sOB(N^{12}) for two subresultant-based methods: this notation ignores polylogarithmic factors, where NN bounds the degree and the bitsize of the polynomials. The previous record bound was \sOB(N^{14}). Our main tool is signed subresultant sequences. We exploit recent advances on the complexity of univariate root isolation, and extend them to sign evaluation of bivariate polynomials over two algebraic numbers, and real root counting for polynomials over an extension field. Our algorithms apply to the problem of simultaneous inequalities; they also compute the topology of real plane algebraic curves in \sOB(N^{12}), whereas the previous bound was \sOB(N^{14}). All algorithms have been implemented in MAPLE, in conjunction with numeric filtering. We compare them against FGB/RS, system solvers from SYNAPS, and MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is among the most robust, and its runtimes are comparable, or within a small constant factor, with respect to the C/C++ libraries. Key words: real solving, polynomial systems, complexity, MAPLE softwareComment: 17 pages, 4 algorithms, 1 table, and 1 figure with 2 sub-figure

    On isolation of singular zeros of multivariate analytic systems

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    We give a separation bound for an isolated multiple root xx of a square multivariate analytic system ff satisfying that an operator deduced by adding Df(x)Df(x) and a projection of D2f(x)D^2f(x) in a direction of the kernel of Df(x)Df(x) is invertible. We prove that the deflation process applied on ff and this kind of roots terminates after only one iteration. When xx is only given approximately, we give a numerical criterion for isolating a cluster of zeros of ff near xx. We also propose a lower bound of the number of roots in the cluster.Comment: 17 page

    Complete Subdivision Algorithms, II: Isotopic Meshing of Singular Algebraic Curves

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    Given a real valued function f(X,Y), a box region B_0 in R^2 and a positive epsilon, we want to compute an epsilon-isotopic polygonal approximation to the restriction of the curve S=f^{-1}(0)={p in R^2: f(p)=0} to B_0. We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga and Vegter gave a numerical subdivision algorithm that is exact when the curve S is bounded and non-singular. They used a computational model that relied only on function evaluation and interval arithmetic. We generalize their algorithm to any bounded (but possibly non-simply connected) region that does not contain singularities of S. With this generalization as a subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete purely numerical method to compute isotopic approximations of algebraic curves with isolated singularities
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