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Abstract. A set of new formulae for the inverse of a block Hankel (or block Toeplitz) matrix is given. The formulae are expressed in terms of certain matrix Pad6 forms, which approximate a matrix power series associated with the block Hankel matrix.By using Frobenius-type identities between certain matrix Pad6 forms, the inversion formulae are shown to generalize the formulae of Gohberg-Heinig and, in the scalar case, the formulae of Gohberg-Semencul and Gohberg-Krupnik.The new formulae have the significant advantage of requiring only that the block Hankel matrix itself be nonsingular. The other formulae require, in addition, that certain submatrices be nonsingular.Since effective algorithms for computing the required matrix Pad6 forms are available, the formulae are practical. Indeed, some of the algorithms allow for the efficient calculation of the inverse not only of the given block Hankel matrix, but also of any nonsingular block principal minor.

For k + 1 power series a 0 (z); : : :; a k (z), we present a new iterative, look-ahead algorithm for numerically computing Pad e-Hermite systems and simultaneous Pad e systems along a diagonal of the associated Pad e tables. The algorithm computes the systems at all those points along the diagonal at which the associated striped Sylvester and mosaic Sylvester matrices are wellconditioned. The operation and the stability of the algorithm is controlled by a single parameter which serves as a threshold in deciding if the Sylvester matrices at a point are su ciently wellconditioned. We show that the algorithm is weakly stable, and provide bounds for the error in the computed solutions as a function of. Experimental results are given which show that the bounds re ect the actual behavior of the error. The algorithm requires O(knk 2 +s 3 knk) operations, to compute Pad e-Hermite and simultaneous Pad e systems of type n = n 0 ; : :: ; n k ], where knk = n 0 + +n k and s is the largest step-size taken along the diagonal. An additional application of the algorithm is the stable inversion of striped and mosaic Sylvester matrices.

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