# Pseudo-arclength continuation implementation

## Main Question or Discussion Point

Hi there,

Apologies if this is the wrong section for this question; I'm not a regular here. I assume a mod will move it if it's not the right place.

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" engineering stuff, which, unfortunately, does not include this.

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 (not that I'm sure that would help me). Here's what I understand at this point:

We want to solve a problem $F(x,\lambda)=0$. We assume that the solution is known at $x^0$ and $\lambda^0$. To avoid the singularity of the Jacobian, and therefore the breakdown of Newton's method, at turning points, $x$ and $\lambda$ both become parameterised by arclength ($s$), and we end up with an augmented system of equations to solve:

$F(x,\lambda)=0$
$\left(u-u^{0}\right)\mathrm{d}u^{0}/\mathrm{d}s+\left(\lambda-\lambda^{0}\right)\mathrm{d}\lambda^{0}/\mathrm{d}s-\Delta S=0$

While this seems simple enough, how does one obtain the derivatives w.r.t $s$? 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!

Thanks,
TM

I suppose the question boils down to: is there a straightforward way of expressing $x$ and $\lambda$ as functions of $s$?