Computing Real Roots of Real Polynomials ... and now For Real!

Very recent work introduces an asymptotically fast subdivision algorithm, denoted ANewDsc, for isolating the real roots of a univariate real polynomial. The method combines Descartes' Rule of Signs to test intervals for the existence of roots, Newton iteration to speed up convergence against clusters of roots, and approximate computation to decrease the required precision. It achieves record bounds on the worst-case complexity for the considered problem, matching the complexity of Pan's method for computing all complex roots and improving upon the complexity of other subdivision methods by several magnitudes. In the article at hand, we report on an implementation of ANewDsc on top of the RS root isolator. RS is a highly efficient realization of the classical Descartes method and currently serves as the default real root solver in Maple. We describe crucial design changes within ANewDsc and RS that led to a high-performance implementation without harming the theoretical complexity of the underlying algorithm. With an excerpt of our extensive collection of benchmarks, available online at http://anewdsc.mpi-inf.mpg.de/, we illustrate that the theoretical gain in performance of ANewDsc over other subdivision methods also transfers into practice. These experiments also show that our new implementation outperforms both RS and mature competitors by magnitudes for notoriously hard instances with clustered roots. For all other instances, we avoid almost any overhead by integrating additional optimizations and heuristics.Comment: Accepted for presentation at the 41st International Symposium on Symbolic and Algebraic Computation (ISSAC), July 19--22, 2016, Waterloo, Ontario, Canad

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

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

SqFreeEVAL: An (almost) optimal real-root isolation algorithm

Let f be a univariate polynomial with real coefficients, f in R[X]. Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the real roots of f in a given interval. In this paper, we consider a simple subdivision algorithm whose primitives are purely numerical (e.g., function evaluation). The complexity of this algorithm is adaptive because the algorithm makes decisions based on local data. The complexity analysis of adaptive algorithms (and this algorithm in particular) is a new challenge for computer science. In this paper, we compute the size of the subdivision tree for the SqFreeEVAL algorithm. The SqFreeEVAL algorithm is an evaluation-based numerical algorithm which is well-known in several communities. The algorithm itself is simple, but prior attempts to compute its complexity have proven to be quite technical and have yielded sub-optimal results. Our main result is a simple O(d(L+ln d)) bound on the size of the subdivision tree for the SqFreeEVAL algorithm on the benchmark problem of isolating all real roots of an integer polynomial f of degree d and whose coefficients can be written with at most L bits. Our proof uses two amortization-based techniques: First, we use the algebraic amortization technique of the standard Mahler-Davenport root bounds to interpret the integral in terms of d and L. Second, we use a continuous amortization technique based on an integral to bound the size of the subdivision tree. This paper is the first to use the novel analysis technique of continuous amortization to derive state of the art complexity bounds

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

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