10:00am - 12:00pmSymbolic-numeric methods for non-linear equations: Algorithms and applications
Chair(s): Angelos Mantzaflaris (Inria, France), Bernard Mourrain (Inria, France), Elias Tsigaridas (Inria, France)
Modeling real-world systems or processes in areas such as control theory, geometric modeling, biochemistry, coding theory, cryptology, and so on, almost certainly involves non-linear equations. Higher degree equations are the first step away from linear models. Available tools for recovering their solutions range from numerical methods such as Newton-Raphson, homotopy continuation algorithms, subdivision-based solvers, to symbolic tools such as Groebner bases, border bases, characteristic sets and multivariate resultants. There is continuous progress in combining symbolic methods and numerical solving, in order to devise new algorithms with varying blends of exactness, stability and robustness as well as computational complexity, that are tailored for different applications. Among the challenges which occur in the process is reliable root isolation, certification and approximation, treatment of singular solutions, the exploitation of structure coming from specific applications as well as efficient interpolation. The mini-symposium will host presentations related to state-of-the-art solution strategies for these problems, theoretical and algorithmic advances as well as emerging application areas.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
On hybrid univariate polynomial root-finders
Victor Pan
Lehman College CUNY
We combine various known methods developed for nearly optimal root-finding for univariate polynomials with our new techniques and obtain new algorithms with improved efficiency for both complex and real root-finding.
A robust path tracking algorithm for polynomial homotopy continuation
Marc Van Barel1, Simon Telen1, Jan Verschelde2
1KU Leuven, 2University of Illinois at Chicago
Homotopy continuation is an important strategy for solving systems of polynomial equations and for tackling other problems in computational algebraic geometry. State of the art implementations suffer from `path jumping', which often causes the loss of some solutions. We propose a new algorithm that uses Padé approximants for detecting difficult regions along the path. This results in an adaptive stepsize path tracker which proves to be more robust than existing algorithms.
On the relationship of well conditioned polynomials and elliptic Fekete points
Jinsan Cheng, Junyi Wen
Chinese Academy of Mathematics and Systems Science
In this talk, we present a method for isolating real roots of a bivariate polynomial system in a box. Our method is a subdivision method and based on the real root isolation of univariate polynomials and the geometry properties of the given system. By using the upper and lower bound polynomials of the system, we get some candidate boxes. We give the uniqueness and existence conditions to check if the system has a unique simple real root in the box. The method is complete for the system containing only simple real zeros. The experimental results show the superiority of our method.
A sequence of polynomials with optimal condition number
Maria De Ujue Etayo Rodriguez, Carlos Beltrán, Jordi Marzo, Joaquim Ortega-CerdÃ
University of Cantabria
During this talk we will solve a problem posed by Michael Shub and Stephen Smale in 1993, in their famous article "Complexity of Bezout’s theorem. III."
The problem ask to find an explicit sequence of univariate polynomials of degree N with normalized contidion number less or equal than N, using the definition of normalized condition number that can be found, for example, in the book "Complexity and real computation" by Blum, Cucker, Shub and Smale. We find such a sequence of polynomials. Actually, the condition number of our polynomials is bounded by the square root of N, which we prove is a lower bound for the normalized condition number, meaning that our sequence has, up to some constant, optimal condition number.