Even if it is possible to define a non singular algorithm that computes the averaged solutions of Eqs.(1) just following the steps used in the proof of the existence theorem, we try to exploit the qualitative information, that is the properties of the singular solution appearing in our analytical theory, to define a simpler algorithm. We have used a Runge-Kutta-Gauss method which is symplectic (see Sanz-Serna 1988); this property is useful to maintain the Hamiltonian character of the solutions.

Like all the methods based upon the Gaussian quadrature, the choice of the intermediate nodes does not require to compute the right-hand sides of the equations, that is the derivatives of the averaged Hamiltonian, in the singular points (note that, if we are not on a node crossing line, we can compute the derivatives of as the averaged derivatives of ).

This is the main idea of this algorithm: it computes intermediate
values of the derivatives between two extreme points and, by
successive iterations, tries to reach the singularity in an exact way
with a second extreme. This is performed by a control on the nodal
distance based upon *regula falsi*. When we arrive to a node
crossing line we continue our extrapolation of the solution as an
analytical function, but we add the term that causes the jumps in the
derivatives (see Figure 3); the analytical expression of
this jump is available from the theory (Gronchi and Milani 1998 and
1999).

2000-05-16