1,191 research outputs found
A Numerical Algorithm for Zero Counting. III: Randomization and Condition
In a recent paper (Cucker, Krick, Malajovich and Wschebor, A Numerical
Algorithm for Zero Counting. I: Complexity and accuracy, J. Compl.,24:582-605,
2008) we analyzed a numerical algorithm for computing the number of real zeros
of a polynomial system. The analysis relied on a condition number kappa(f) for
the input system f. In this paper, we look at kappa(f) as a random variable
derived from imposing a probability measure on the space of polynomial systems
and give bounds for both the tail P{kappa(f) > a} and the expected value E(log
kappa(f))
Convexity properties of the condition number II
In our previous paper [SIMAX 31 n.3 1491-1506(2010)], we studied the
condition metric in the space of maximal rank matrices. Here, we show that this
condition metric induces a Lipschitz-Riemann structure on that space. After
investigating geodesics in such a nonsmooth structure, we show that the inverse
of the smallest singular value of a matrix is a log-convex function along
geodesics (Theorem 1).
We also show that a similar result holds for the solution variety of linear
systems (Theorem 31).
Some of our intermediate results, such as Theorem 12, on the second covariant
derivative or Hessian of a function with symmetries on a manifold, and Theorem
29 on piecewise self-convex functions, are of independent interest.
Those results were motivated by our investigations on the com- plexity of
path-following algorithms for solving polynomial systems.Comment: Revised versio
The complexity and geometry of numerically solving polynomial systems
These pages contain a short overview on the state of the art of efficient
numerical analysis methods that solve systems of multivariate polynomial
equations. We focus on the work of Steve Smale who initiated this research
framework, and on the collaboration between Stephen Smale and Michael Shub,
which set the foundations of this approach to polynomial system--solving,
culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo,
Peter Buergisser and Felipe Cucker
High Probability Analysis of the Condition Number of Sparse Polynomial Systems
Let F:=(f_1,...,f_n) be a random polynomial system with fixed n-tuple of
supports. Our main result is an upper bound on the probability that the
condition number of f in a region U is larger than 1/epsilon. The bound depends
on an integral of a differential form on a toric manifold and admits a simple
explicit upper bound when the Newton polytopes (and underlying covariances) are
all identical.
We also consider polynomials with real coefficients and give bounds for the
expected number of real roots and (restricted) condition number. Using a Kahler
geometric framework throughout, we also express the expected number of roots of
f inside a region U as the integral over U of a certain {\bf mixed volume}
form, thus recovering the classical mixed volume when U = (C^*)^n.Comment: 29 pages, no figures. Extensive revision and streamlining of
math.NA/0012104. In particular, new theorem with explicit high probability
estimate of the condition number of a random sparse polynomial system
(Theorem 1) has been adde
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