Uniformly well-conditioned pseudo-arclength continuation

Kelly I. Dickson
243 Harrelson Hall, Campus Box 8205
North Carolina State University, Raleigh NC 27695
kidickso@unity.ncsu.edu
C.T. Kelley, I. C. F. Ipsen, I. G. Kevrekidis

Numerical continuation is the process of solving nonlinear equations of the form

$\displaystyle G(x,\lambda)=0$

for various real number parameter values, $ \lambda$. The obvious approach, called natural parameterization, is to perturb $ \lambda$ with each continuation step and find the corresponding solution $ x$ via a nonlinear solver (Newton's method). While this approach is reasonable for paths containing only regular points (points $ (x,\lambda)$ where the Jacobian matrix of $ G$ is nonsingular), the approach breaks down at simple fold points where the Jacobian matrix of $ G$ becomes singular and Newton's method fails.

In order to remedy this, one may implement pseudoarclength continuation (PAC) which introduces a new parameter based on the arclength $ s$ of the solution path. In order to implement PAC, one converts the old problem $ G(x,\lambda)=0$ to a new problem

$\displaystyle F(x(s),\lambda(s))=0.$

Using PAC on the new problem requires the Jacobian matrix of $ F$, $ F'$, which ought to be nonsingular at both regular points and simple folds if we have indeed bypassed the problem that natural parameterization presents.

While the nonsingularity of $ F'$ at regular points and simple folds is a known fact, we present a theorem that gives conditions under which $ F'$ is uniformly nonsingular for a path containing simple folds. We do this by bounding the smallest singular value of $ F'$ from below. The theorem justifies the use of PAC in a practical way for solution curves containing nothing ``worse'' than a simple fold.