arXiv subjects

Sascha Timme

Publications and source records attributed to Sascha Timme.

9 recordsLinked to original sources

arXiv2020

Certifying zeros of polynomial systems using interval arithmetic

We establish interval arithmetic as a practical tool for certification in numerical algebraic geometry. Our software HomotopyContinuation.jl now has a built-in function certify, which proves the correctness of an isolated nonsingular solution to a square system of polynomial equations. The implementation rests on Krawczyk's method. We demonstrate that it dramatically outperforms earlier approaches to certification. We see this contribution as powerful new tool in numerical algebraic geometry, that can make certification the default and not just an option.

math.AG↗

arXiv2020

Catastrophe in Elastic Tensegrity Frameworks

We discuss elastic tensegrity frameworks made from rigid bars and elastic cables, depending on many parameters. For any fixed parameter values, the stable equilibrium position of the framework is determined by minimizing an energy function subject to algebraic constraints. As parameters smoothly change, it can happen that a stable equilibrium disappears. This loss of equilibrium is called `catastrophe' since the framework will experience large-scale shape changes despite small changes of parameters. Using nonlinear algebra we characterize a semialgebraic subset of the parameter space, the catastrophe set, which detects the merging of local extrema from this parametrized family of constrained optimization problems, and hence detects possible catastrophe. Tools from numerical nonlinear algebra allow reliable and efficient computation of all stable equilibrium positions as well as the catastrophe set itself.

math.MG↗

arXiv2020

Polymake.jl: A new interface to polymake

We present the Julia interface Polymake.jl to polymake, a software for research in polyhedral geometry. We describe the technical design and how the integration into Julia makes it possible to combine polymake with state-of-the-art numerical software.

math.CO↗

arXiv2020

Backward Error Measures for Roots of Polynomials

We analyze different measures for the backward error of a set of numerical approximations for the roots of a polynomial. We focus mainly on the element-wise mixed backward error introduced by Mastronardi and Van Dooren, and the tropical backward error introduced by Tisseur and Van Barel. We show that these measures are equivalent under suitable assumptions. We also show relations between these measures and the classical element-wise and norm-wise backward error measures.

math.NA↗

arXiv2019

96120: The degree of the linear orbit of a cubic surface

The projective linear group $\pgl(\comp,4)$ acts on cubic surfaces, considered as points of $\mathbb{P}_{\mathbb{C}}^{19}$. We compute the degree of the $15$-dimensional projective variety given by the Zariski closure of the orbit of a general cubic surface. The result, 96120, is obtained using methods from numerical algebraic geometry.

math.AG↗

arXiv2019

Estimating linear covariance models with numerical nonlinear algebra

Numerical nonlinear algebra is applied to maximum likelihood estimation for Gaussian models defined by linear constraints on the covariance matrix. We examine the generic case as well as special models (e.g. Toeplitz, sparse, trees) that are of interest in statistics. We study the maximum likelihood degree and its dual analogue, and we introduce a new software package LinearCovarianceModels.jl for solving the score equations. All local maxima can thus be computed reliably. In addition we identify several scenarios for which the estimator is a rational function.

stat.CO↗

arXiv2019

3264 Conics in a Second

Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given instance. This article illustrates how these two fields complement each other. Our focus lies on the 3264 conics that are tangent to five given conics in the plane. We present a web interface for computing them. It uses the software HomotopyContinuation.jl, which makes this process fast and reliable. We discuss an instance where all 3264 solutions are real.

math.AG↗

arXiv2019

Mixed Precision Path Tracking for Polynomial Homotopy Continuation

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 guess if it is not an approximate zero. The algorithm also uses an adaptive step size control which builds on a local understanding of the region of convergence of Newton's method and the distance to the closest singularity. To handle numerically challenging situations the algorithm uses mixed precision arithmetic. The efficiency and robustness are demonstrated in several numerical examples.

math.NA↗

arXiv2017

HomotopyContinuation.jl: A package for homotopy continuation in Julia

We present the Julia package HomotopyContinuation.jl, which provides an algorithmic framework for solving polynomial systems by numerical homotopy continuation. We introduce the basic capabilities of the package and demonstrate the software on an illustrative example. We motivate our choice of Julia and how its features allow us to improve upon existing software packages with respect to usability, modularity and performance. Furthermore, we compare the performance of HomotopyContinuation.jl to the existing packages Bertini and PHCpack.

cs.MS↗