4 research outputs found
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
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
Complexity of Sparse Polynomial Solving 2: Renormalization
Renormalized homotopy continuation on toric varieties is introduced as a tool
for solving sparse systems of polynomial equations, or sparse systems of
exponential sums. The cost of continuation depends on a renormalized condition
length, defined as a line integral of the condition number along all the lifted
renormalized paths.
The theory developed in this paper leads to a continuation algorithm tracking
all the solutions between two generic systems with the same structure. The
algorithm is randomized, in the sense that it follows a random path between the
two systems. The probability of success is one. In order to produce an expected
cost bound, several invariants depending solely of the supports of the
equations are introduced. For instance, the mixed area is a quermassintegral
that generalizes surface area in the same way that mixed volume generalizes
ordinary volume. The facet gap measures for each direction in the 0-fan, how
close is the supporting hyperplane to the nearest vertex. Once the supports are
fixed, the expected cost depends on the input coefficients solely through two
invariants: the renormalized toric condition number and the imbalance of the
absolute values of the coefficients. This leads to a non-uniform complexity
bound for polynomial solving in terms of those two invariants. Up to
logarithms, the expected cost is quadratic in the first invariant and linear in
the last one.Comment: 90 pages. Major revision from the previous versio