  This paper is concerned with the connection coefficients between the local
fundamental solutions of a $2\times 2$ linear ordinary differential system with
two neighboring regular singular points at $z=0$ and $z=1$. We derive an
asymptotic formula for the connection coefficients which can be used for
numerical calculations and, in particular, for determining the eigenvalues of
some spectral problems arising in mathematical physics. As an application, new
algorithms for computing the eigenvalues of the ellipsoidal wave equation and
the spheroidal wave equation are presented.
