Recursiveness in Matrix Rational Interpolation Problems 1 2

We consider the problem of computing solutions to a variety of matrix rational interpolation problems. These include the partial realization problem for matrix power series and Newton-Pad e, Hermite-Pad e, Simultaneous Pad e, M-Pad e and multipoint Pad e approximation problems along with their matrix generalizations. A general recurrence relation is given for solving these problems. Unlike other previous recursive methods, our recurrence works along arbitrary computational paths. When restricted to speciic paths, the recurrence relation generalizes previous work of Antoulas, Cabay and Labahn, Beckermann, Van Barel and Bultheel and Gutknecht along with others. Our results rely on the concept of extended M-Pad e approximation introduced in this paper. This is a natural generalization of the two-point Pad e approximation problem extended to multiple interpolation points (including innnity) and matrix Laurent and Newton series. By using module theoretic techniques we determine complete parameterizations of all solutions to this problem. Our recurrence relation then eeciently computes these parame-terizations. This recursion requires no conditions on the input data. We also discuss the concept of duality which was shown to be of particular interest for a stable computation of those approximants. Finally we show the invariance of our approximation problem under linear transformations of the extended complex plane.