Computation of Numerical Padé-Hermite and Simultaneous Padé Systems II: A Weakly Stable Algorithm

For $k+1$ power series $a_0(z), \ldots, a_k(z)$, 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 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 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 ${\cal O}(\|n\|^2 + s^3 \|n\|)$ operations to compute Pad\'{e}--Hermite and simultaneous Pad\'e systems of type $n=[n_0, \ldots, n_k]$, where $\|n\|=n_0+\cdots+n_k$ 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.