Accuracy of a Multistep Method

For starting a multistep method k starting values xo. Xk-i are required. We will assume for the moment that these are values of one solution trajectory of the ODE, i.e. 

The quantity of interest is the global error of the method at a given time point tn 

If for exact starting values Cn = 0 h), then the method is said to be convergent More precisely, a method is convergent of order p, if Cn = 0 hP). 

We first introduce the so-called local residual of the method  

Sometimes one considers the local residual per unit step l x,tn, h)/h. 

---