COMPUTATION OF NUMERICAL PADE-HERMITE AND SIMULTANEOUS PADE SYSTEMS .2. A WEAKLY STABLE ALGORITHM

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 t he diagonal at which the associated striped Sylvester and mosaic Sylve ster matrices are well conditioned. 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 suff iciently well conditioned. We show that the algorithm is weakly stable and provide bounds for the error in the computed solutions as a funct ion 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(3)\\n\\) Operations to compute Pade-Hermite and simultaneous Fade 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 additiona l application of the algorithm is the stable inversion of striped and mosaic Sylvester matrices.