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A Numerical Algorithm for Zero Counting. II: Distance to Illposedness and Smoothed Analysis
, 2009
"... We show a Condition Number Theorem for the condition number of zero counting for real polynomial systems. That is, we show that this condition number equals the inverse of the normalized distance to the set of illposed systems (i.e., those having multiple real zeros). As a consequence, a smoothed ..."
Abstract

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We show a Condition Number Theorem for the condition number of zero counting for real polynomial systems. That is, we show that this condition number equals the inverse of the normalized distance to the set of illposed systems (i.e., those having multiple real zeros). As a consequence, a smoothed analysis of this condition number follows.
A Numerical Algorithm for Zero Counting. II: Randomization and Condition
, 812
"... Abstract. In a recent paper [9] we analyzed a numerical algorithm for computing the number of real zeros of a polynomial system. The analysis relied on a condition number κ(f) for the input system f. In this paper we continue this analysis by looking at κ(f) as a random variable derived from imposin ..."
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Abstract. In a recent paper [9] we analyzed a numerical algorithm for computing the number of real zeros of a polynomial system. The analysis relied on a condition number κ(f) for the input system f. In this paper we continue this analysis by looking at κ(f) as a random variable derived from imposing a probability measure on the space of polynomial systems. We give bounds for both the tail P{κ(f)> a} and the expected value E(log κ(f)). 1