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13
Fast arithmetic for triangular sets: from theory to practice
 ISSAC'07
, 2007
"... We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, ..."
Abstract

Cited by 30 (25 self)
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We study arithmetic operations for triangular families of polynomials, concentrating on multiplication in dimension zero. By a suitable extension of fast univariate Euclidean division, we obtain theoretical and practical improvements over a direct recursive approach; for a family of special cases, we reach quasilinear complexity. The main outcome we have in mind is the acceleration of higherlevel algorithms, by interfacing our lowlevel implementation with languages such as AXIOM or Maple. We show the potential for huge speedups, by comparing two AXIOM implementations of van Hoeij and Monagan's modular GCD algorithm.
Almost tight recursion tree bounds for the Descartes method
 In Proc. Int. Symp. on Symbolic and Algebraic Computation
, 2006
"... We give a unified (“basis free”) framework for the Descartes method for real root isolation of squarefree 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 ..."
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Cited by 30 (3 self)
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We give a unified (“basis free”) framework for the Descartes method for real root isolation of squarefree 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.
HIGHPERFORMANCE IMPLEMENTATIONS OF THE DESCARTES METHOD
, 2006
"... 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 vari ..."
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Cited by 9 (0 self)
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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 codominant, almost tight computing time bounds. Implementations of either variant can obtain speedups over previous stateoftheart implementations by more than an order of magnitude if they use features of the processor architecture. We present an implementation of the Bernsteinbases variant of the Descartes method that automatically generates architectureaware highlevel 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 architectureunaware implementations of both variants on four different processor architectures and for three classes of input polynomials.
An efficient and exact subdivision algorithm for isolating complex roots of a polynomial and its complexity analysis
, 2009
"... We introduce an exact subdivision algorithm CEVAL for isolating complex roots of a squarefree 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 isolati ..."
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Cited by 3 (3 self)
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We introduce an exact subdivision algorithm CEVAL for isolating complex roots of a squarefree 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 squarefree polynomial with Gaussian integer coefficients. The analysis is based on a novel technique called δclusters. 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.
Cache Complexity and Multicore Implementation for Univariate Real Root Isolation
"... Abstract. We present parallel algorithms with optimal cache complexity for the kernel routine of many real root isolation algorithms, namely the Taylor shift by 1. We then report on multicore implementation for isolating the real roots of univariate polynomials with integer coefficients based on a c ..."
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Cited by 2 (2 self)
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Abstract. We present parallel algorithms with optimal cache complexity for the kernel routine of many real root isolation algorithms, namely the Taylor shift by 1. We then report on multicore implementation for isolating the real roots of univariate polynomials with integer coefficients based on a classical algorithm due to Vincent, Collins and Akritas. For processing some wellknown benchmark examples with sufficiently large size, our software tool reaches linear speedup on an 8core machine. In addition, we show that our software is able to fully utilize the many cores and the memory space of a 32core machine to tackle large problems that are out of reach for a desktop implementation. 1.
Fast Algorithms, Modular Methods, Parallel Approaches and Software Engineering for Solving Polynomial Systems Symbolically
, 2007
"... Symbolic methods are powerful tools in scientific computing. The implementation of symbolic solvers is, however, a highly difficult task due to the extremely high time and space complexity of the problem. In this thesis, we study and apply fast algorithms, modular methods, parallel approaches and so ..."
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Symbolic methods are powerful tools in scientific computing. The implementation of symbolic solvers is, however, a highly difficult task due to the extremely high time and space complexity of the problem. In this thesis, we study and apply fast algorithms, modular methods, parallel approaches and software engineering techniques to improve the efficiency of symbolic solvers for computing triangular decomposition, one of the most promising methods for solving nonlinear systems of equations symbolically. We first adapt nearly optimal algorithms for polynomial arithmetic over fields to direct products of fields for polynomial multiplication, inversion and GCD computations. Then, by introducing the notion of equiprojectable decomposition, a sharp modular method for triangular decompositions based on Hensel lifting techniques is obtained. Its implementation also brings to the Maple computer algebra system a unique capacity for automatic case discussion and recombination. A highlevel categorical parallel framework is developed, written in the Aldor language, to support highperformance computer algebra on symmetric multi
Overview 8
"... Taylor shift of random polynomial on 8core machine dnc, Base=8 blocking, Base=50 Isolatingtherealrootsofaunivariatepolynomialisadriving subject in computer algebra. Many researchers have studied this problem under various angles from algebraic algorithms to implementation techniques [1, 2, 7, 3, ..."
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Taylor shift of random polynomial on 8core machine dnc, Base=8 blocking, Base=50 Isolatingtherealrootsofaunivariatepolynomialisadriving subject in computer algebra. Many researchers have studied this problem under various angles from algebraic algorithms to implementation techniques [1, 2, 7, 3, 5]. Today, multicores have become the most popular parallel hardware architectures. Besides, understanding the implications of hierarchical memory on performance software engineering has becomeessential. Theseobservationsmotivatetheworkpresented in this poster. First, we analyze the cache complexity The work and span of this algorithm are respectively Θ(n2) andΘ(nlog23),Inaddition,thisalgorithmcanberuninplace, inspaceΘ(n). Usingtheidealcachemodel[4],foracacheof Z wordswithcachelinesizeL,wehaveshownthatthisalgorithm incurs Θ(n2 /ZL) cache misses. Using the HongKung lower bounds, we deduce that this latter result is optimal. Speedup using 8 cores 6 5