Globally Convergent, Iterative Path-Following for Algebraic Equations

Abstract

Homotopy methods are of great importance for the solution of systems of equations. It is a major problem to ensure well-defined iterations along the homotopy path. Many investigations have considered the complexity of path-following methods depending on the unknown distance of some given path to the variety of ill-posed problems. It is shown here that there exists a construction method for safe paths for a single algebraic equation. A safe path may be effectively determined with bounded effort. Special perturbation estimates for the zeros together with convergence conditions for Newton’s method in simultaneous mode allow our method to proceed on the safe path. This yields the first globally convergent, never-failing, uniformly iterative path-following algorithm. The maximum number of homotopy steps in our algorithm reaches a theoretical bound forecast by Shub and Smale i.e., the number of steps is at most quadratic in the condition number. A constructive proof of the fundamental theorem of algebra meeting demands by Gaus, Kronecker and Weierstras is a consequence of our algorithm.