Finding and certifying numerical roots of systems of equations

Abstract

Numerical algebraic geometry studies methods to approach problems in algebraic geometry numerically. Especially, finding roots of systems of equations using theory in algebraic geometry involves symbolic algorithm which requires expensive computations, numerical techniques often provides faster methods to tackle these problems. This thesis establishes numerical techniques to approximate roots of systems of equations and ways to certify its correctness. As techniques for approximating roots of systems of equations, homotopy continuation method will be introduced. Especially, combining homotopy method with monodromy group action, we solve parametrized polynomial systems. Since numerical approaches rely on heuristic method, we study how to certify numerical roots of systems of equations. Based on Newton’s method, we study Krawczyk method using interval arithmetic and Smale’s alpha theory. These two methods will be mainly used for certifying regular roots of systems. Furthermore, as an approach for multiple roots, we establish the local separation bound of a multiple root. For multiple roots whose deflation process terminates after only one iteration, we give their local separation bound and study how to certify an approximation of such multiple roots.