97 research outputs found
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
H\"older foliations, revisited
We investigate transverse H\"older regularity of some canonical leaf
conjugacies in partially hyperbolic dynamical systems and transverse H\"older
regularity of some invariant foliations. Our results validate claims made
elsewhere in the literature.Comment: 52 pages, to appear in Journal of Modern Dynamic
Adaptative Step Size Selection for Homotopy Methods to Solve Polynomial Equations
Given a C^1 path of systems of homogeneous polynomial equations f_t, t in
[a,b] and an approximation x_a to a zero zeta_a of the initial system f_a, we
show how to adaptively choose the step size for a Newton based homotopy method
so that we approximate the lifted path (f_t,zeta_t) in the space of (problems,
solutions) pairs.
The total number of Newton iterations is bounded in terms of the length of
the lifted path in the condition metric
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