  We address the problem of computing the Maslov index for large linear
symplectic systems on the real line. The Maslov index measures the signed
intersections (with a given reference plane) of a path of Lagrangian planes.
The natural chart parameterization for the Grassmannian of Lagrangian planes is
the space of real symmetric matrices. Linear system evolution induces a Riccati
evolution in the chart. For large order systems this is a practical approach as
the computational complexity is quadratic in the order. The Riccati solutions,
however, also exhibit singularites (which are traversed by changing charts).
Our new results involve characterizing these Riccati singularities and two
trace formulae for the Maslov index as follows. First, we show that the number
of singular eigenvalues of the symmetric chart representation equals the
dimension of intersection with the reference plane. Second, the Cayley map is a
diffeomorphism from the space of real symmetric matrices to the manifold of
unitary symmetric matrices. We show the logarithm of the Cayley map equals the
arctan map (modulo 2i) and its trace measures the angle of the Langrangian
plane to the reference plane. Third, the Riccati flow under the Cayley map
induces a flow in the manifold of unitary symmetric matrices. Using the natural
unitary action on this manifold, we pullback the flow to the unitary Lie
algebra and monitor its trace. This avoids singularities, and is a natural
robust procedure. We demonstrate the effectiveness of these approaches by
applying them to a large eigenvalue problem. We also discuss the extension of
the Maslov index to the infinite dimensional case.
