A Stable Algorithm for Multi-dimensional Padé Systems and the Inversion of Generalized Sylvester Matrices

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  • Technical report TR94-07. For k+1 power series a_0(z), ..., a_k(z), we present a new iterative, look-ahead algorithm for numerically computing Pade'-Hermite systems and simultaneous Pade' systems along a diagonal of the associated Pade' tables. The algorithm computes the systems at all those points along the diagonal at which the associated striped Sylvester and mosaic Sylvester matrices are well-conditioned. It is shown that a good estimate for the condition numbers of these Sylvester matrices at a point is easily determined from the Pade'-Hermite system and simultaneous Pad'e system computed at that point. The operation and the stability of the algorithm is controlled by a single parameter tau which serves as a threshold in deciding if the Sylvester matrices at a point are sufficiently well-conditioned. We show that the algorithm is weakly stable, and provide bounds for the error in the computed solutions as a function of tau. Experimental results are given which show that the bounds reflect the actual behavior of the error. The algorithm requires O(||n||^2 + s^2 ||n||) operations, to compute Pade'-Hermite and simultaneous Pade' systems of type n=[n_0, ..., n_k], where ||n|| = 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. | TRID-ID TR94-07

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    Attribution 3.0 International