= Parameter continuation
{title2=$F(u(\lambda),\lambda)=0$}
<Parameter continuation> changes $\lambda$ by a small amount and solves again for $u$, using the previous solution as the initial guess for <Newton method>. If $F_u$ is an <invertible matrix>, the <implicit function theorem> gives a local <solution branch> $u(\lambda)$, with
$$
\frac{du}{d\lambda}=-F_u^{-1}F_\lambda.
$$
Local convergence of <Newton method> also needs regularity and a sufficiently small step: the cited result assumes a <Lipschitz continuous> <derivative> and an invertible $F_u$ at the seed.<Condition Estimates for Pseudo-Arclength Continuation by K. I. Dickson and others>[§2.1, Corollary 2.1, printed pp. 264–265: Lipschitz continuously differentiable map, nonsingular state Jacobian, and a sufficiently small parameter increment]{ref}
For the cubic <polynomial>,
$$
F_x=3x^2-1,\qquad F_\lambda=1,\qquad \frac{dx}{d\lambda}=\frac1{1-3x^2}.
$$
The <derivative> becomes unbounded at either <continuation fold>, and the scalar <Newton method> divides by $F_x=0$ there. Just beyond the upper <continuation fold>, no nearby real solution exists at the increased $\lambda$, although a distant negative-$x$ solution still exists. Converging to that solution would jump away from the local <solution branch>. Decreasing the parameter step cannot make $x(\lambda)$ a valid smooth local description at the bend.
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