Related papers: Certifying zeros of polynomial systems using inter…
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…
We propose an algorithm to construct a certified approximation of a surface by generalizing the Krawczyk test. The Krawczyk test is based on interval arithmetic, and confirms the existence and uniqueness of a solution to a square system of…
We consider numerical certification of approximate solutions to a system of polynomial equations with more equations than unknowns by first certifying solutions to a square subsystem. We give several approaches that certifiably select which…
The package \texttt{NumericalCertification} implements methods for certifying numerical approximations of solutions for a given system of polynomial equations. For certifying regular solutions, the package implements Smale's $\alpha$-theory…
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 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…
We develop a new symbolic-numeric algorithm for the certification of singular isolated points, using their associated local ring structure and certified numerical computations. An improvement of an existing method to compute inverse systems…
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…
We present a proof procedure for univariate real polynomial problems in Isabelle/HOL. The core mathematics of our procedure is based on univariate cylindrical algebraic decomposition. We follow the approach of untrusted certificates,…
Smale's alpha-theory uses estimates related to the convergence of Newton's method to give criteria implying that Newton iterations will converge quadratically to solutions to a square polynomial system. The program alphaCertified implements…
Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…
A new algorithm for real root isolation of polynomial equations based on hybrid computation is presented in this paper. Firstly, the approximate (complex) zeros of the given polynomial equations are obtained via homotopy continuation…
In this paper we present a simple technique to derive certificates of non-realizability for an abstract polytopal sphere. Our approach uses a variant of the classical algebraic certificates introduced by Bokowski and Sturmfels in…
We develop algorithms for certifying an approximation to a nonsingular solution of a square system of equations built from univariate analytic functions. These algorithms are based on the existence of oracles for evaluating basic data about…
Polynomial systems occur in many fields of science and engineering. Polynomial homotopy continuation methods apply symbolic-numeric algorithms to solve polynomial systems. We describe the design and implementation of our web interface and…
We develop a certified numerical algorithm for computing Galois/monodromy groups of parametrized polynomial systems. Our approach employs certified homotopy path tracking to guarantee the correctness of the monodromy action produced by the…
Let $\mathbb{Q}$ (resp. $\mathbb{R}$) be the field of rational (resp. real) numbers and $X = (X_1, \ldots, X_n)$ be variables. Deciding the non-negativity of polynomials in $\mathbb{Q}[X]$ over $\mathbb{R}^n$ or over semi-algebraic domains…
Numerical algebraic geometry revolves around the study of solutions to polynomial systems via numerical methods. The polyhedral homotopy of Huber and Sturmfels for computing isolated solutions and the concept of witness sets as numerical…
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…
We reexamine Smale's alpha theory as a way to certify a numerical solution to an analytic system. For a given point and a system, Smale's alpha theory determines whether Newton's method applied to this point shows the quadratic convergence…