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A Stable Algorithm for Multidimensional Padé Systems and the Inversion of Generalized Sylvester Matrices

 Author(s) / Creator(s)

Technical report TR9407. For k+1 power series a_0(z), ..., a_k(z), we present a new iterative, lookahead 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 wellconditioned. 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 wellconditioned. 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 stepsize taken along the diagonal. An additional application of the algorithm is the stable inversion of striped and mosaic Sylvester matrices.  TRIDID TR9407

 Date created
 1994

 Type of Item
 Report

 License
 Attribution 3.0 International