Consider a smooth function
. We want to compute a solution
curve of the equation
. 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
,
along the curve, satisfying a chosen
tolerance criterion:
for some
and an additional accuracy condition
where
and
is the last Newton correction.
To show how the points are generated, suppose we have found a point on the
curve. Also suppose we have a normalized
tangent vector
at
, i.e.
.
The computation of the next point consists of 2 steps: