Condition Estimates for Pseudo-Arclength Continuation

Abstract

We bound the condition number of the Jacobian in pseudo arclength continuation problems, and we quantify the effect of this condition number on the linear system solution in a Newton GMRES solve. In pseudo arclength continuation one repeatedly solves systems of nonlinear equations $F(u(s),λ(s))=0$ for a real-valued function $u$ and a real parameter $λ$, given different values of the arclength $s$. It is known that the Jacobian $F_x$ of $F$ with respect to $x=(u,λ)$ is nonsingular, if the path contains only regular points and simple fold singularities. We introduce a new characterization of simple folds in terms of the singular value decomposition, and we use it to derive a new bound for the norm of $F_x^{-1}$. We also show that the convergence rate of GMRES in a Newton step for $F(u(s),λ(s))=0$ is essentially the same as that of the original problem $G(u,λ)=0$. In particular we prove that the bounds on the degrees of the minimal polynomials of the Jacobians $F_x$ and $G_u$ differ by at most 2. We illustrate the effectiveness of our bounds with an example from radiative transfer theory.