Note on unit tangent vector computation for homotopy curve tracking on a hypercube

Abstract

Probability-one homotopy methods are a class of methods for solving nonlinear systems of equations that are globally convergent from an arbitrary starting point. The essence of all such algorithms is the construction of an appropriate homotopy map ρ a (λ, x) and subsequent tracking of some smooth curve γ in the zero set of the homotopy map. Tracking a homotopy curve involves finding the unit tangent vector at different points along the zero curve, which amounts to calculating the kernel of the n × ( n + 1) Jacobian matrix Dρ a (λ, x). While computing the tangent vector is just one part of the curve tracking algorithm, it can require a significant percentage of the total tracking time. This note presents computational results showing the performance of several different parallel orthogonal factorization/triangular system solving algorithms for the tangent vector computation on a hypercube.