Related papers: Subtropical Real Root Finding
Quantifier-free nonlinear arithmetic (QF_NRA) appears in many applications of satisfiability modulo theories solving (SMT). Accordingly, efficient reasoning for corresponding constraints in SMT theory solvers is highly relevant. We propose…
We report an ongoing work on clustering algorithms for complex roots of a univariate polynomial $p$ of degree $d$ with real or complex coefficients. As in their previous best subdivision algorithms our root-finders are robust even for…
The topology of Liouville sets of the real forms of the complex generic Neumann system depends indirectly on the roots of the special polynomial $U_{\cal S}(\lambda)$. For certain polynomials, the existence and positions of the real roots,…
In most introductory numerical analysis textbooks, the treatment of a single nonlinear equation often consists of a collection of all-purpose methods that frequently do not work or are inefficient. These textbooks neglect to teach the…
This paper deals with the algorithmic aspects of solving feasibility problems of semidefinite programming (SDP), aka linear matrix inequalities (LMI). Since in some SDP instances all feasible solutions have irrational entries, numerical…
Given a polynomial $x \in {\mathbb R}^n \mapsto p(x)$ in $n=2$ variables, a symbolic-numerical algorithm is first described for detecting whether the connected component of the plane sublevel set ${\mathcal P} = \{x : p(x) \geq 0\}$…
Given a polynomial system f, a fundamental question is to determine if f has real roots. Many algorithms involving the use of infinitesimal deformations have been proposed to answer this question. In this article, we transform an approach…
In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the roots of the equation system are represented as linear combinations of roots of several univariate…
Isolating the real roots of univariate polynomials is a fundamental problem in symbolic computation and it is arguably one of the most important problems in computational mathematics. The problem has a long history decorated with numerous…
We describe new algorithms to compute Whitney stratifications of real algebraic varieties. Using either conormal or polar techniques, these algorithms stratify a complexification of a given real variety. We then show that the resulting…
The roots of any polynomial of degree m with integer coefficients, can be computed by manipulation of sequences made from 2m distinct symbols and counting the different symbols in the sequences. This method requires only 'primitive'…
In order to verify programs or hybrid systems, one often needs to prove that certain formulas are unsatisfiable. In this paper, we consider conjunctions of polynomial inequalities over the reals. Classical algorithms for deciding these not…
We depart from our approximation of 2000 of all root radii of a polynomial, which has readily extended Sch{\"o}nhage's efficient algorithm of 1982 for a single root radius. We revisit this extension, advance it, based on our simple but…
A fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. The studied hypergeometric polynomials satisfy a higher order differential equation and generalize Laguerre…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
Many problems in applied mathematics require root finding algorithms. Unfortunately, root finding methods have limitations. Firstly, regarding the convergence, there is a trade-off between the size of it's domain and it's rate. Secondly the…
We consider the problem of approximating all real roots of a square-free polynomial $f$. Given isolating intervals, our algorithm refines each of them to a width of $2^{-L}$ or less, that is, each of the roots is approximated to $L$ bits…
A challenging problem in computational mathematics is to compute roots of a high-degree univariate random polynomial. We combine an efficient multiprecision implementation for solving high-degree random polynomials with two certification…
In this paper methods for simultaneous finding all roots of generalized polynomials are developed. These methods are related to the case when the roots are multiple. They possess cubic rate of convergence and they are as labour-consuming as…
We describe, study, and experiment with an algorithm for finding all solutions of systems of polynomial equations using homotopy continuation and monodromy. This algorithm follows a framework developed in previous work and can operate in…