  Analytic interpolation problems with rationality and derivative constraints
are ubiquitous in systems and control. This paper provides a new method for
such problems, both in the scalar and matrix case, based on a non-standard
Riccati-type equation. The rank of the solution matrix is the same as the
degree of the interpolant, thus providing a natural approach to model
reduction. A homotopy continuation method is presented and applied to some
problems in modeling and robust control. We also address a question on the
positive degree of a covariance sequence originally posed by Kalman.
