Mathematical aspects of numerical continuation and handling of singularities

Consider a smooth function $F: \ensuremath{\mathbf{R}}^{n+1} \to \ensuremath{\mathbf{R}}^n$. We want to compute a solution curve of the equation $F(x)=0$. Numerical continuation is a technique to compute a consecutive sequence of points which approximate the desired branch. Most continuation algorithms implement a predictor-corrector method. The idea behind this method is to generate a sequence of points $x_i$, $i=1,2,\dots$ along the curve, satisfying a chosen tolerance criterion: $\vert\vert F(x_i)\vert\vert\leq\epsilon$ for some $\epsilon > 0$ and an additional accuracy condition $\vert\vert\delta x_i\vert\vert\leq\epsilon'$ where $\epsilon' > 0$ and $\delta x_i$ is the last Newton correction.

To show how the points are generated, suppose we have found a point $x_i$ on the curve. Also suppose we have a normalized tangent vector $v_i$ at $x_i$, i.e. $F_x(x_i)v_i = 0,\ \langle v_i,v_i \rangle =1$.

The computation of the next point $x_{i+1}$ consists of 2 steps:



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