Sign in. DOI: Andrew J. Sommese , Charles W. 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.
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Morgan Morgan, 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, 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.
Our approach throughout this book is to assume that we are trying to explain each topic to an engineer or scientist.
Numerical Solution Of Systems Of Polynomials Arising In Engineering And Science, The
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. These results generally rely upon growth conditions of the nonlinearity.
However, in general, one cannot forecast how many solutions a boundary value problem may possess or even determine the existence of a solution.
In recent years numerical continuation methods have been developed which permit the numerical approximation of all complex solutions of systems of polynomial equations. In this paper, numerical continuation methods are adapted to numerically calculate the solutions of finite difference discretizations of nonlinear two-point boundary value problems. The approach taken here is to perform a homotopy deformation to successively refine discretizations.
In this way additional new solutions on finer meshes are obtained from solutions on coarser meshes. The complicating issue which the complex polynomial system setting introduces is that the number of solutions grows with the number of mesh points of the discretization.
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Numerical algebraic geometry for model selection and its application to the life sciences
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Bates A. Sommese C. Article First Online: 25 August This is a preview of subscription content, log in to check access. Allgower, E.
In: Topics in numerical analysis II. Irish Acad. College, Dublin, , pp. London: Academic Press Google Scholar.