  Accurate orthogonal tight-binding Hamiltonians are constructed for
ferromagnetic SrRuO$_3$ and the layered perovskite superconductor,
Sr$_2$RuO$_4$ by fitting to all-electron full-potential local density band
structures obtained by the linearized augmented planewave method. These
Hamiltonians allow the band structure to be computed on very fine meshes in the
Brillouin zone at low cost, and additionally have analytic band velocities,
while retaining the accuracy of the full-potential electronic structure
calculations. This greatly facilitates calculation of transport and
superconducting parameters related to the fermiology. These features are
exploited to calculate the Hall coefficient and vortex lattice geometry for
Sr$_2$RuO$_4$ with fine integration meshes. We find the lower limit for the
interband order parameter anisotropy to be compatible with the observed square
geometry. We also find that the sign reversal of the Hall coefficient can be
explained in a conventional way if the bands are shifted by a few mRy so as to
match the experimental de Haas-van Alphen areas exactly, {\it and} the
temperature dependence of the relaxation time is strongly dependent on the band
character.
