Analysis of Function Component Complexity for Hypercube Homotopy Algorithms

Probability-one homotopy algorithms are a class of methods for solving nonlinear systems of equations that globally convergent from an arbitrary starting point with probability one. The essence of these homotopy algorithms is the construction of a homotopy map p-sub a and the subsequent tracking of a smooth curve y in the zero set p-sub a to the -1 (0) of p-sub a. Tracking the zero curve y requires repeated evaluation of the map p-sub a, its n x (v + 1) Jacobian matrix Dp-sub a and numerical linear algebra for calculating the kernel of Dp-sub a. This paper analyzes parallel homotopy algorithms on a hypercube, considering the numerical algebra, several communications topologies and problem decomposition strategies, functions component complexity, problem size, and the effect of different component complexity distributions. These parameters interact in complicated ways, but some general principles can be inferred based on empirical results.