Problems and progress in vector Padé approximation

Some open problems in vector Padé approximation are stated, and some recent de Montessus-type theorems governing convergence of rows of the vector Padé table are contrasted. Weshow how these results indicate when it is more appropriate to use generalised inverse vector Padé approximants or their hybridised form. We show how a straightforward analogue of the Berlekamp-Massey algorithm may be used to calculate generalised inverse vector Padé approximants. This algorithm is applied to the derivation of a new low-order hybrid vector approximant. Related results include the case of a row convergence theorem for a complex-valued power series using a Clifford inverse instead of the Moore-Penrose inverse.