Numerical integrations show a high probability of collisions among the members of the prograde group during the lifespan of the solar system (e.g. on average 1.5 collisions between Himalia and Elara).
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The Numerical Recipes books cover a range of topics that include both classical numerical analysis (interpolation, integration, linear algebra, differential equations, and so on), signal processing (Fourier methods, filtering), statistical treatment of data, and a few topics in machine learning (hidden Markov models, support vector machines).
Numerical integration methods such as Runge Kutta, or Bulirsch-Stoer could be used to solve this partictular system of ODEs.
In a famous 1967 paper he used what is now known as Verlet integration (a method for the numerical integration of equations of motion) and the Verlet list (a data structure that keeps track of each molecule's immediate neighbors in order to speed computer calculations of molecule to molecule interactions).
In 1989, Jacques Laskar of the Bureau des Longitudes in Paris published the results of his numerical integration of the Solar System over 200 million years.