Applications of symmetric spaces and Lie triple systems in Numerical Analysis

A remarkable number of different numerical algorithms can be understood and analyzed using the concepts of symmetric spaces and Lie triple systems, which are well known in differential geometry from the study of spaces of constant curvature and their tangents. This theory unifies a range of different topics, such as polar-type matrix decompositions, splitting methods for computation of the matrix exponential, composition of selfadjoint numerical integrators and dynamical systems with symmetries and reversing symmetries. In this paper we will give an introduction to the mathematical theory behind these constructions, and review recent results. Furthermore, we present a new result, related to time-reversal symmetries, selfadjoint numerical schemes and Yoshida-type composition techniques. The new technique allows to increase the order of preservation of symmetries by two units, with an appropriate choice of stepsize. Since all the time-steps are positive, the technique is particularly suited to stiff problems, where a negative time-step can cause instabilities.