The Numerical Solution of Systems of Polynomials Arising in Engineering and Science

Abstract

PrefaceThis book started with the goal of explaining, to engineers and scientists, the advances made in the numerical computation of the isolated solutions of systems of nonlinear multivariate complex polynomials since the book of A. Morgan (Morgan, 1987). The writing of this book was delayed because of a number of surprising developments, which made possible numerically describing not just the isolated solutions, but also positive-dimensional solution sets of polynomial systems. The most recent advances allow one to work with individual solution components, which opens up new ways of solving a large system of polynomials by intersecting the solution sets of subsets of the equations. This collection of ideas, methods, and problems makes up the new area of Numerical Algebraic Geometry.The heavy dependence of the new developments since (Morgan, 1987) on algebraic geometric ideas poses a serious challenge for an exposition aimed at engineers, scientists, and numerical analysts -most of whom have had little or no exposure to algebraic geometry. Furthermore most of the introductory books on algebraic geometry are oriented towards computational algebra, and give short shrift at best to the geometric results which underly the numerical analysis of polynomial systems. Even worse, from the standpoint of an engineer or scientist, such books typically aim to resolve algebraic questions and so do not directly address the numerical/geometric questions coming from applications.Our approach throughout this book is to assume that we are trying to explain each topic to an engineer or scientist. We want to be accurate: we do not cut corners on giving precise definitions and statements. We give illustrative examples exhibiting all the phenomena involved, but we only give proofs to the extent that they further understanding.The set of common zeros of a system of polynomials is not a manifold, but it is close to being one in the sense that exceptional points are rare. This vague statement can be made mathematically precise, and indeed, the theoretical underpinnings of our methods imply that we avoid such trouble spots "with probability one." The usual algebraic approaches to the subject do not show how familiar geometric notions from calculus relate to these solution sets. The geometric approach is harder, since to link concepts like prime ideals to algebraic sets with certain very nice There remains a tension that we see no way to completely resolve. Dealing with polynomials and algebraic subsets of Euclidean space is basic, but this is not general enough to cover the applications common in engineering and science. For example, the use of products of projective spaces and multihomogeneous polynomials which live on them is extraordinarily useful, but these polynomials are not "functions" on the products of projective spaces. Working in an appropriate generality to cover everything needed would cast a pall over the whole book. Moreover, the early parts of the book need only advanced calculus and a few concepts from algebraic geometr...

Mechanism mobility and a local dimension test

“…Numerical algebraic geometry [22,23], which is based on continuation methods, is able to solve larger systems than symbolic methods. The strength of this approach is that it is fully automatic and does not require foreknowledge of an assembly configuration.…”

Galois groups of Schubert problems via homotopy computation

“…Numerical homotopy continuation [26] gives a method for finding all solutions to a system of polynomials with finitely many solutions. Current parallel implementations [18] can solve systems with over 40 million solutions [19].…”

“…Current parallel implementations [18] can solve systems with over 40 million solutions [19]. The emerging field of numerical algebraic geometry [25,26] uses numerical homotopy continuation as a foundation for algorithms to study algebraic varieties. While numerical algebraic geometry was developed for applications of mathematics, we apply it in pure mathematics, computing Galois groups of enumerative-geometric problems from the Schubert calculus, called Schubert problems.…”

The Approach of Moments for Polynomial Equations

“…Such methods rely on Bertini's theorem allowing to deform an easier instance with known solutions of the class of problems to be solved into the original system, without encountering singularities along the path (cf. [40] for details). Keeping track of the roots during this deformation allows to compute the desired roots.…”