Numerical continuation through folds with step control

We consider an efficient algorithm for tracing implicitly defined curves. The algorithm does not parametrize the solution in terms of arc length or in terms of the naturally occurring parameter λ, but rather does a parameter transformation that chooses the component relatively contributing the most change as the parameter. For comparison purposes, we present the predictor-corrector path following technique. The exact computation of fold points of the simple turning point type will also be presented. Numerical examples that demonstrate the efficiency of the algorithm will be done.