Consider a smooth function F: Rn+1 ® Rn. 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 xi, i=1,2,... along the curve, satisfying a chosen tolerance criterion: ||F(xi)|| £ e for some e > 0 and an additional accuracy condition ||dxi|| £ e¢ where e¢ > 0 and dxi is the last Newton correction.
To show how the points are generated, suppose we have found a point xi on the curve. Also suppose we have a normalized tangent vector vi at xi, i.e. Fx(xi)vi = 0, ávi,vi ñ=1.
The computation of the next point xi+1 consists of 2 steps: