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Polynomial homotopies on multicore workstations. Accepted for publication
 in the proceedings of PASCO 2010
"... Homotopy continuation methods to solve polynomial systems scale very well on parallel machines. In this paper we examine its parallel implementation on multiprocessor multicore workstations using threads. With more cores we can speed up pleasingly parallel path tracking jobs. In addition, we can com ..."
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Homotopy continuation methods to solve polynomial systems scale very well on parallel machines. In this paper we examine its parallel implementation on multiprocessor multicore workstations using threads. With more cores we can speed up pleasingly parallel path tracking jobs. In addition, we can compute solutions more accurately in the same amount of time with threads, and thus achieve quality up. Focusing on polynomial evaluation and linear system solving (the key ingredients of Newton’s method) we can double the accuracy of the results with the quad doubles of QD2.3.9 in less than double the time, if we use all available eight cores on our workstation. 1
Tropical Algebraic Geometry in Maple  a preprocessing algorithm for finding common factors to multivariate polynomials with approximate coefficients
, 2009
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3.2. Algebraic Geometric Modeling 2 3.3. Algebraic Geometric Computing 2
"... c t i v it y e p o r t 2009 Table of contents ..."
AF:Small: Solving Linear Differential Equations in terms of Special Functions, Project Description
, 2010
"... Linear differential equations with polynomial or rational function coefficients are very common in science. Many scientists use computer algebra systems to solve such equations. Computer algebra systems (Maple, Mathematica, etc.) contain numerous programs to match an equation with an equation in ..."
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Linear differential equations with polynomial or rational function coefficients are very common in science. Many scientists use computer algebra systems to solve such equations. Computer algebra systems (Maple, Mathematica, etc.) contain numerous programs to match an equation with an equation in
Envelope Computation in the Plane by Approximate
"... Given a rational family of planar rational curves in a certain region of interest, we are interested in computing an implicit representation of the envelope. The points of the envelope correspond to the zero set of a function (which represents the envelope condition) in the parameter space combining ..."
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Given a rational family of planar rational curves in a certain region of interest, we are interested in computing an implicit representation of the envelope. The points of the envelope correspond to the zero set of a function (which represents the envelope condition) in the parameter space combining the curve parameter and the motion parameter. We analyze the connection of this function to the implicit equation of the envelope. This connection enables us to use approximate implicitization for computing the (exact or approximate) implicit representation of the envelope. Based on these results, we formulate an algorithm for computing a piecewise algebraic approximation of low degree and illustrate its performance by several examples. 1
Computing Monodromy via Continuation Methods on Random Riemann Surfaces
"... We consider a Riemann surface X defined by a polynomial f(x,y) of degree d, whose coefficients are chosen randomly. Hence, we can suppose that X is smooth, that the discriminant δ(x) of f has d(d−1) simple roots, ∆, and that δ(0) = 0 i.e. the corresponding fiber has d distinct points {y1,...,yd}. W ..."
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We consider a Riemann surface X defined by a polynomial f(x,y) of degree d, whose coefficients are chosen randomly. Hence, we can suppose that X is smooth, that the discriminant δ(x) of f has d(d−1) simple roots, ∆, and that δ(0) = 0 i.e. the corresponding fiber has d distinct points {y1,...,yd}. When we lift a loop 0 ∈ γ ⊂ C− ∆ by a continuation method, we get d paths in X connecting {y1,...,yd}, hence defining a permutation of that set. This is called monodromy. Here we present experimentations in Maple to get statistics on the distribution of transpositions corresponding to loops around each point of ∆. Multiplying families of “neighbor ” transpositions, we construct permutations and the subgroups of the symmetricgrouptheygenerate.Thisallowsustoestablishandstudyexperimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups. Assuming that these two conjectures are true, we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization, under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions.
THEME Algorithms, Certification, and CryptographyTable of contents
"... 3.2. Algebraic representations for geometric modeling 2 3.3. Algebraic algorithms for geometric computing 3 3.4. Symbolic numeric analysis 3 ..."
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3.2. Algebraic representations for geometric modeling 2 3.3. Algebraic algorithms for geometric computing 3 3.4. Symbolic numeric analysis 3