Condition Estimates for Pseudo-Arclength Continuation

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. Pseudo-arclength continuation solves parameter dependent nonlinear equations $G(u,\lambda) = 0$ by introducing a new parameter $s$, which approximates arclength, and viewing the vector $x = (u,\lambda)$ as a function of $s$. In this way simple fold singularities can be computed directly by solving a larger system $F(x,s) = 0$ by simple continuation in the new parameter $s$. It is known that the Jacobian $F_x$ of $F$ with respect to $x=(u,\lambda)$ 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 $F(x,s)=0$ is essentially the same as that of the original problem $G(u,\lambda)=0$. 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.