I have a quick question regarding pseudo-arclength continuation. As some background, I am a chemical engineer, not a mathematician, applied or otherwise, so my knowledge of numerical methods is limited to the "standard" stuff.
I've been reading up as extensively as I can on pseudo-arclength continuation, but unfortunately, it's all from second-hand sources; I don't have access to Keller's original paper. Here's what I understand at this point:
We want to solve a problem . We assume that the solution is known at and . To avoid the singularity of the Jacobian, and therefore the breakdown of Newton's method, at turning points, and both become parameterised by arclength ( ), and we end up with an augmented system of equations to solve:
While this seems simple enough, how does one obtain the derivatives w.r.t ? Not a single text seems to mention this. Ideas that spring to mind are forward differences using, say, cubic splines; however, that seems horrendously inefficient to me. There must be a better way!