= Continuation method
{wiki}
A <continuation method> follows a <solution branch> of a parameter-dependent equation from a known solution, using nearby solutions to guide the next solve. This is useful when one wants to understand how solutions change, rather than obtain just one isolated answer. A <continuation predictor-corrector step> follows a <tangent vector> and then uses a <Newton method> correction on a transverse section. The same geometry supports both <pseudo-arclength continuation> through a <continuation fold> and <transversal surface intersection tracing>.
Consider the original scalar <polynomial> example
$$
F(x,\lambda)=x^3-x+\lambda=0,\qquad \lambda=x-x^3.
$$
Its <solution branch> bends back twice in $\lambda$. Increasing $\lambda$ alone eventually stops following the same nearby solutions; following the <curve> in the $(x,\lambda)$ plane can continue around both bends. The <continuation folds> are
$$
(x_-,\lambda_-)=\left(-\frac1{\sqrt3},-\frac2{3\sqrt3}\right),\qquad
(x_+,\lambda_+)=\left(\frac1{\sqrt3},\frac2{3\sqrt3}\right).
$$
For $|\lambda|<2/(3\sqrt3)$ there are three distinct real <zeros of a function> at that fixed $\lambda$, all lying on this one connected <curve>. A <continuation method> keeps their geometric relationship visible.
\Image[/continuation-cubic-solution-curve-folds.png]
{title=A connected cubic solution curve with two folds and three solutions at the same parameter value}
{description=The arrows follow increasing $x$ along $\lambda=x-x^3$, even where $\lambda$ changes direction. The dashed horizontal line $\lambda=0.2$ meets the <solution branch> three times. The marked <continuation folds> are extrema of $\lambda$, not singularities of the full <curve>.}
Every <continuation method> needs a seed, supplied by an independent solve, an explicitly known solution, or another search. Following one <solution branch> does not guarantee finding every <connected component> of the solution set. A large step can miss detail or make the correction converge to a distant solution. At branch intersections or a loss of <rank> of the full <Jacobian matrix>, the regular-<curve> argument needs separate analysis. The seed and component limitations also matter in <transversal surface intersection tracing>.
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