@article{dickson_kelley_ipsen_kevrekidis_2007, title={Condition estimates for pseudo-arclength continuation}, volume={45}, ISSN={["1095-7170"]}, DOI={10.1137/060654384}, abstractNote={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.}, number={1}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Dickson, K. I. and Kelley, C. T. and Ipsen, I. C. F. and Kevrekidis, I. G.}, year={2007}, pages={263–276} }