6,300 research outputs found

    Computing Real Roots of Real Polynomials

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    Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for approximating all complex roots, the best algorithm known is based on an almost optimal method for approximate polynomial factorization, introduced by Pan in 2002. Pan's factorization algorithm goes back to the splitting circle method from Schoenhage in 1982. The main drawbacks of Pan's method are that it is quite involved and that all roots have to be computed at the same time. For the important special case, where only the real roots have to be computed, much simpler methods are used in practice; however, they considerably lag behind Pan's method with respect to complexity. In this paper, we resolve this discrepancy by introducing a hybrid of the Descartes method and Newton iteration, denoted ANEWDSC, which is simpler than Pan's method, but achieves a run-time comparable to it. Our algorithm computes isolating intervals for the real roots of any real square-free polynomial, given by an oracle that provides arbitrary good approximations of the polynomial's coefficients. ANEWDSC can also be used to only isolate the roots in a given interval and to refine the isolating intervals to an arbitrary small size; it achieves near optimal complexity for the latter task.Comment: to appear in the Journal of Symbolic Computatio

    A Near-Optimal Algorithm for Computing Real Roots of Sparse Polynomials

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    Let pZ[x]p\in\mathbb{Z}[x] be an arbitrary polynomial of degree nn with kk non-zero integer coefficients of absolute value less than 2τ2^\tau. In this paper, we answer the open question whether the real roots of pp can be computed with a number of arithmetic operations over the rational numbers that is polynomial in the input size of the sparse representation of pp. More precisely, we give a deterministic, complete, and certified algorithm that determines isolating intervals for all real roots of pp with O(k3log(nτ)logn)O(k^3\cdot\log(n\tau)\cdot \log n) many exact arithmetic operations over the rational numbers. When using approximate but certified arithmetic, the bit complexity of our algorithm is bounded by O~(k4nτ)\tilde{O}(k^4\cdot n\tau), where O~()\tilde{O}(\cdot) means that we ignore logarithmic. Hence, for sufficiently sparse polynomials (i.e. k=O(logc(nτ))k=O(\log^c (n\tau)) for a positive constant cc), the bit complexity is O~(nτ)\tilde{O}(n\tau). We also prove that the latter bound is optimal up to logarithmic factors

    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

    Character Sums and Deterministic Polynomial Root Finding in Finite Fields

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    We obtain a new bound of certain double multiplicative character sums. We use this bound together with some other previously obtained results to obtain new algorithms for finding roots of polynomials modulo a prime pp
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