Adams method continuous representation

The goal is to construct and to evaluate a continuous representation without additional evaluations of the right hand side function /. The construction of a continuous representation is straight forward for integration methods based on a polynomial representation of the solution or its derivative, like Adams or BDF multistep methods or Runge-Kutta methods based on collocation. 

However, for the use in switching algorithms, see Sec. 6, the interpolation error is propagated as it influences the determination of the switching point and the point of restart. Thus in this case, the interpolation error must be controlled. In [SG84] it is shown for Adams methods that the continuous representation is error controlled, for BDF methods the corresponding result can be found in [EichQl]. 

Prom Eq. (4.1.5) a continuous representation for Adams-Moulton methods can easily be derived ... 

By construction the order of the continuous representation is the order of the Adams-Moulton method, i.e. g = fc -f 1. 

It should be noted that the way an Adams method is implemented influences the continuous representation of the solution. Special care has to be taken in P EC) implementations, see Sec. 4.1.1. In that case X (l) a n+i- For this type of implementation (4.5.1) deflnes no continuous representation of the solution. Using (4.5.1) in that case may lead to problems when passing from one interval to the next in connection with switching algorithms, which will be described in Sec. 6. 

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