arXiv subjects

Simon Telen

Publications and source records attributed to Simon Telen.

46 records · Page 3Linked to original sources

arXiv2022

Yet another eigenvalue algorithm for solving polynomial systems

In latest years, several advancements have been made in symbolic-numerical eigenvalue techniques for solving polynomial systems. In this article, we add to this list. We design an algorithm which solves systems with isolated solutions reliably and efficiently. In overdetermined cases, it reduces the task to an eigenvalue problem in a simpler and considerably faster way than in previous methods, and it can outperform the homotopy continuation approach. We provide many examples and an implementation in the proof-of-concept Julia package EigenvalueSolver.jl.

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arXiv2020

Likelihood Equations and Scattering Amplitudes

We relate scattering amplitudes in particle physics to maximum likelihood estimation for discrete models in algebraic statistics. The scattering potential plays the role of the log-likelihood function, and its critical points are solutions to rational function equations. We study the ML degree of low-rank tensor models in statistics, and we revisit physical theories proposed by Arkani-Hamed, Cachazo and their collaborators. Recent advances in numerical algebraic geometry are employed to compute and certify critical points. We also discuss positive models and how to compute their string amplitudes.

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arXiv2020

Polyhedral Homotopies in Cox Coordinates

We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety $X_Σ$. The algorithm lends its name from a construction, described by Cox, of $X_Σ$ as a GIT quotient $X_Σ= (\mathbb{C}^k \setminus Z) // G$ of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space $\mathbb{C}^k$ of $X_Σ$ and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of $X_Σ$. It furthermore generalizes the commonly used path tracking algorithms in (multi)projective spaces in that it tracks a set of homogeneous coordinates contained in the $G$-orbit corresponding to each solution. The Cox homotopy combines the advantages of polyhedral homotopies and (multi)homogeneous homotopies, tracking only mixed volume many solutions and providing an elegant way to deal with solutions on or near the special divisors of $X_Σ$. In addition, the strategy may help to understand the deficiency of the root count for certain families of systems with respect to the BKK bound.

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arXiv2020

A Robust Numerical Path Tracking Algorithm for Polynomial Homotopy Continuation

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 arithmetic. It is based on an adaptive stepsize predictor that uses Padé techniques to detect local difficulties for function approximation and danger for path jumping. We show the potential of the new path tracking algorithm through several numerical examples and compare with existing implementations.

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arXiv2020

Robust Numerical Tracking of One Path of a Polynomial Homotopy on Parallel Shared Memory Computers

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 top of that, it computes singular values of the Hessians of the polynomials in the homotopy to estimate the distance to the nearest different path. Together, these estimates are used to compute an appropriate adaptive stepsize. For n-dimensional problems, the cost overhead of our robust path tracker is O(n), compared to the commonly used predictor-corrector methods. This cost overhead can be reduced by a multithreaded program on a parallel shared memory computer.

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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.

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arXiv2020

Numerical Root Finding via Cox Rings

We present a new eigenvalue method for solving a system of Laurent polynomial equations defining a zero-dimensional reduced subscheme of a toric compactification $X$ of $(\mathbb{C} \setminus \{0\})^n$. We homogenize the input equations to obtain a homogeneous ideal in the Cox ring of $X$ and generalize the eigenvalue, eigenvector theorem for rootfinding in affine space to compute homogeneous coordinates of the solutions. Several numerical experiments show the effectiveness of the resulting method. In particular, the method outperforms existing solvers in the case of (nearly) degenerate systems with solutions on or near the torus invariant prime divisors.

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arXiv2018

Truncated Normal Forms for Solving Polynomial Systems: Generalized and Efficient Algorithms

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. Normal form algorithms provide an algebraic approach to solve this problem. The framework presented in Telen et al. (2018) uses truncated normal forms (TNFs) to compute the algebra structure of R/I and the solutions of I. This framework allows for the use of much more general bases than the standard monomials for R/I. This is exploited in this paper to introduce the use of two special (nonmonomial) types of basis functions with nice properties. This allows, for instance, to adapt the basis functions to the expected location of the roots of I. We also propose algorithms for efficient computation of TNFs and a generalization of the construction of TNFs in the case of non-generic zero-dimensional systems. The potential of the TNF method and usefulness of the new results are exposed by many experiments.

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arXiv2018

A Stabilized Normal Form Algorithm for Generic Systems of Polynomial Equations

We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring $\mathbb{C}[x]/I$ associated to a zero-dimensional ideal $I$ generated by $n$ $\mathbb{C}$-polynomials in $n$ variables. We assume that the polynomials are generic in the sense that the number of solutions in $\mathbb{C}^n$ equals the Bézout number. The main contribution of this paper is an automated choice of basis for $\mathbb{C}[x]/I$, which is crucial for the feasibility of normal form methods in finite precision arithmetic. This choice is based on numerical linear algebra techniques and governed by the numerical properties of the given generators of $I$.

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arXiv2017

Solving Polynomial Systems via a Stabilized Representation of Quotient Algebras

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to compute the structure of the quotient ring R/I from the null space of a Macaulay-type matrix. The affine dense, affine sparse, homogeneous and multi-homogeneous cases are treated. In the presented framework, the concept of a border basis is generalized by relaxing the conditions on the set of basis elements. This allows for algorithms to adapt the choice of basis in order to enhance the numerical stability. We present such an algorithm and show numerical results.

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