Basic Concepts of Continuation Methods

Most numerical path following methods are predictor-corrector methods. The predictor at step i provides a starting value for the Newton iteration which attempts to correct this value to x si) = x.  

We have already seen the predictor of the classical continuation method which just takes 

the predictor is just a constant function in each step. 

A more sophisticated predictor is used by the tangential continuation method where the predictor is defined as 

This is just a step of the explicit Euler method for ODEs applied to the Davidenko differential equation, cf. Sec. 4.1.1. 

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