Applications of the Cayley Approach in the Numerical Solution of Matrix Differential Systems on Quadratic Groups

In recent years several numerical methods have been developed to integrate matrix differential systems of ODEs whose solutions remain on a certain Lie group throughout the evolution. In this paper some results, derived for the orthogonal group in by Diele et al. (1998), will be extended to the class of quadratic groups including the symplectic and Lorentz group. We will show how this approach also applies to ODEs on the Stiefel manifold and the orthogonal factorization of the Lorentz group will be derived. Furthermore, we will consider the numerical solution of important problems such as the Penrose regression problem, the calculation of Lyapunov exponents of Hamiltonian systems, the solution of Hamiltonian isospectral problems. Numerical tests will show the performance of our numerical methods.