1,310 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
On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection
Given a zero-dimensional polynomial system consisting of n integer
polynomials in n variables, we propose a certified and complete method to
compute all complex solutions of the system as well as a corresponding
separating linear form l with coefficients of small bit size. For computing l,
we need to project the solutions into one dimension along O(n) distinct
directions but no further algebraic manipulations. The solutions are then
directly reconstructed from the considered projections. The first step is
deterministic, whereas the second step uses randomization, thus being
Las-Vegas.
The theoretical analysis of our approach shows that the overall cost for the
two problems considered above is dominated by the cost of carrying out the
projections. We also give bounds on the bit complexity of our algorithms that
are exclusively stated in terms of the number of variables, the total degree
and the bitsize of the input polynomials
Novel Approach to Real Polynomial Root-finding and Matrix Eigen-solving
Univariate polynomial root-finding is both classical and important for modern
computing. Frequently one seeks just the real roots of a polynomial with real
coefficients. They can be approximated at a low computational cost if the
polynomial has no nonreal roots, but typically nonreal roots are much more
numerous than the real ones. We dramatically accelerate the known algorithms in
this case by exploiting the correlation between the computations with matrices
and polynomials, extending the techniques of the matrix sign iteration, and
exploiting the structure of the companion matrix of the input polynomial. We
extend some of the proposed techniques to the approximation of the real
eigenvalues of a real nonsymmetric matrix.Comment: 17 pages, added algorithm
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