Related papers: Robust certified numerical homotopy tracking
We design a homotopy continuation algorithm, that is based on numerically tracking Viro's patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients…
This article develops a new predictor-corrector algorithm for numerical path tracking in the context of polynomial homotopy continuation. In the corrector step it uses a newly developed Newton corrector algorithm which rejects an initial…
We propose a new algorithm for numerical path tracking in polynomial homotopy continuation. The algorithm is `robust' in the sense that it is designed to prevent path jumping and in many cases, it can be used in (only) double precision…
We use our recent implementation of a certified homotopy tracking algorithm to search for start systems that minimize the average complexity of finding all roots of a regular system of polynomial equations. While finding optimal start…
Given a homotopy connecting two polynomial systems we provide a rigorous algorithm for tracking a regular homotopy path connecting an approximate zero of the start system to an approximate zero of the target system. Our method uses recent…
A path tracking algorithm that adaptively adjusts precision is presented. By adjusting the level of precision in accordance with the numerical conditioning of the path, the algorithm achieves high reliability with less computational cost…
We consider the problem of tracking one solution path defined by a polynomial homotopy on a parallel shared memory computer. Our robust path tracker applies Newton's method on power series to locate the closest singular parameter value. On…
The homotopy continuation method has been widely used in solving parametric systems of nonlinear equations. But it can be very expensive and inefficient due to singularities during the tracking even though both start and end points are…
Polynomial systems occur in many areas of science and engineering. Unlike general nonlinear systems, the algebraic structure enables to compute all solutions of a polynomial system. We describe our massive parallel predictor-corrector…
A theory of numerical path-following in toric varieties was suggested in two previous papers. The motivation is solving systems of polynomials with real or complex coefficients. When those polynomials are not assumed 'dense', solving them…
We revisit the problem of certifying the correctness of approximate solution paths computed by numerical homotopy continuation methods. We propose a conceptually simple approach based on a parametric variant of the Krawczyk method from…
The goal of this paper is to study the path-following method for univariate polynomials. We propose to study the complexity and condition properties when the Newton method is applied as a correction operator. Then we study the geodesics and…
Homotopy methods are attractive due to their capability of solving difficult optimisation and optimal control problems. The underlying idea is to construct a homotopy, which may be considered as a continuous (zero) curve between the…
Motivated by Wilmshurst's conjecture, we investigate the zeros of harmonic polynomials. We utilize a certified counting approach which is a combination of two methods from numerical algebraic geometry: numerical polynomial homotopy…
Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input…
Using validated numerical methods, interval arithmetic and Taylor models, we propose a certified predictor-corrector loop for tracking zeros of polynomial systems with a parameter. We provide a Rust implementation which shows tremendous…
We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph $H$ to $n$-vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new…
Computing an optimal cycle in a given homology class, also referred to as the homology localization problem, is known to be an NP-hard problem in general. Furthermore, there is currently no known optimality criterion that localizes classes…
A special homotopy continuation method, as a combination of the polyhedral homotopy and the linear product homotopy, is proposed for computing all the isolated solutions to a special class of polynomial systems. The root number bound of…
We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy based solvers in terms of decorated graphs. Under the…