This report is about finding clusters of complex solutions of triangular systems of polynomial equations. We introduce the local solution clustering problem for a system of polynomial equations, that is grouping all its complex solutions lying in an initial complex domain in clusters smaller than a given real number \epsilon^n, for any given \epsilon} > 0, and counting the sum of multiplicities of the solutions in each cluster. For triangular systems, we propose a criterion based on the Pellet theorem to count the sum of the multiplicities of the solutions in a cluster. We also propose an algorithm for solving the local solution clustering problem for triangular systems, based on a recent near-optimal algorithm  for clustering the complex roots of univariate polynomials . Our algorithm is numeric and certified . We implemented it and compared it with two homotopy solvers for randomly generated triangular systemsand regular chains for state of the art systems . Our solver always give correct answers, is often faster than the homotopy solver that gives often correct answers, and sometimes faster than the one that gives sometimes correct results.
This paper gives the first algorithm for finding a set of natural \epsilon-clusters of complex zeros of a triangular system of polynomials within a given polybox in \mathbb{C^n, for any given \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.
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report is about finding clusters of complex solutions of triangular systems of polynomial equations. We introduce the local solution clustering problem for a system of polynomial equations, that is grouping all its complex solutions lying in an initial complex domain in clusters smaller than a given real number \epsilon
paper gives the first algorithm for finding a set of natural \epsilon-clusters of complex zeros of a triangular system of polynomials within a given polybox in \mathbb{C
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0, and counting the sum of multiplicities of the solutions in each cluster. For triangular systems, we propose a criterion based on the Pellet theorem to count the sum of the multiplicities of the solutions in a cluster. We also propose an algorithm for solving the local solution clustering problem for triangular systems,
0. Our algorithm is
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of Becker et al (2016)
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univariate polynomials
a univariate polynomial where the coefficients are represented by number oracles
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and certified
, certified and based on subdivision
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homotopy solvers for randomly generated triangular systemsand regular chains for state of the art systems
well-known homotopy solvers on various triangular systems
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give
gives
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gives often
often gives