= Pseudo-arclength continuation
{title2=$F(z)=0,\quad t_k^{\mathsf T}(z-z_k)-h=0$}
<Pseudo-arclength continuation> allows both the state and the parameter to change. Write $z=(u,\lambda)$ and choose a <unit vector> $t_k$ in the <kernel> of the full <Jacobian matrix> $DF(z_k)$. A <continuation predictor-corrector step> seeks the next solution on a <hyperplane> perpendicular to $t_k$, a distance $h>0$ along the old <tangent vector>. This adds one equation to $F(z)=0$ and makes the correction a square system. The constraint approximates <arc length>; $h$ is the displacement projected onto the old <tangent vector>, rather than the exact curved distance.<Condition Estimates for Pseudo-Arclength Continuation by K. I. Dickson and others>[§2.2, equations (2.2)–(2.5), printed pp. 265–266: unit tangent and hyperplane normalization; Corollary 2.3 gives local Newton convergence for sufficiently small steps]{ref}
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