Multistep high order interpolants of Runge-Kutta methods

We consider a p-order Runge-Kutta method K(i)(n) = f(x(n) + c(i)h, y(n) + hSIGMA(j=1)(nu)a(ij)K(j)(n)), i = 1,..., nu, y(n+1) = y(n) + hSIGMA(i = 1)(nu)b(i)K(i)(n)), for solving an initial-value problem for ordinary differential equations. The aim of this paper is to construct p-order interpolants by using the values furnished by the method on N successive intervals of integration. By using Lagrange interpolation one can obtain a p-order interpolant over p intervals, but we are interested in finding the minimum number of intervals needed to obtain this. We provide the conditions to be satisfied and we obtain an estimation of the number N. Some examples are given.