  The present paper discusses the problem of least-squares over the real
symplectic group of matrices Sp(2n,R)$. The least-squares problem may be
extended from flat spaces to curved spaces by the notion of geodesic distance.
The resulting non-linear minimization problem on manifold may be tackled by
means of a gradient-descent algorithm tailored to the geometry of the space at
hand. In turn, gradient steepest descent on manifold may be implemented through
a geodesic-based stepping method. As the space Sp(2n,R) is a non-compact Lie
group, it is convenient to endow it with a pseudo-Riemannian geometry. Indeed,
a pseudo-Riemannian metric allows the computation of geodesic arcs and geodesic
distances in closed form on Sp(2n,R).
