Applications of Newton Homotopies

Homotopy continuation is the fundamental method, or family of methods, of numerical algebraic geometry. Using homotopies, we compute solution sets for systems of polynomial equations. We focus on a particular class of homotopy, namely, the Newton homotopy, which is a homotopy for which only the constant terms are changed. Newton homotopies arise naturally when performing monodromy loops, moving end effectors of robots, or simply when trying to compute a solution to a square system of equations. In this thesis, we present an a posteriori certification scheme which uses the result of heuristic tracking methods as input, producing a certificate that solution the path was tracked correctly. We also describe certified tracking procedures wherein the predictor is guaranteed to produce a point in the quadratic convergence basin of Newton’s method for the next step. Finally, we present an application of Newton homotopies to the computation of saturation properties from fundamental equations of state, determining the coexistence curve for all vapor-liquid equilibrium conditions between the triple point and critical point. Examples and computational results are presented for all three applications of Newton homotopies.