Nordsieck vector

Nordsieck proposed an efficient way of collecting the information called the Nordsieck vector. 

When we solve a differential system, each element of the Nordsieck vector is, in turn, a vector in other words, z is usually a matrix. In such cases, it is preferable to consider the Nordsieck vector as a vector of vectors. 

If the exact (no errors) elements of the Nordsieck vector z were known at t , we could write the following Taylor expansion series for each element... 

In this case, it is sufficient to replace each derivative with the corresponding component of the Nordsieck vector to have a nonlinear system in the single unknown b. For example, in the case of implicit first-order multivalue methods, the following system must be solved ... 

To estimate the local error of a p-order multivalue method, it is necessary to have an approximate value for the (p + l)-order derivative. It is trivial since we collected the derivatives m the Nordsieck vector up to the order p. 

It is worth remarking that the Nordsieck vector allows us to perform an interpolation in the neighborhood of the current point f consequently, the prediction in a point hmv from t is obtained using the relation... 

The Nordsieck vector in t allows us to perform the interpolation with the required precision only between t -i and t , which is for negative values of the ratio (h ew/h) and with hnew < h = t -... 

Analogously, we can estimate the derivatives of higher orders or the elements of the Nordsieck vector. Specifically, if we wish to evaluate Zp+2, it is particularly easy if we collect three adjacent values of the last element Zp in correspondence with t -2, t -i, and t . The result is... 

As previously demonstrated, it is possible to exploit the last element of the Nordsieck vector Zp to calculate both Zp+i and Zp+2- Moreover, since the pt i element of z is... 

Once a satisfactory vector b has been obtained, it is used to calculate the Nordsieck vector at the n + l)th point. 

Check that the Nordsieck vector has decreasing elements otherwise, reduce the order of the method. 

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