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

For k + 1 power series a0(z), . . . , ak(z), we present a new iterative, look-ahead algorithm for numerically computing Padé-Hermite systems and simultaneous Padé systems along a diagonal of the associated Padé 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 τ 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 τ . Experimental results are given which show that the bounds reflect the actual behavior of the error. The algorithm requires O(‖n‖2+s3‖n‖) operations, to compute Padé-Hermite and simultaneous Padé systems of type n = [n0, . . . , nk], where ‖n‖ = n0 + · · ·+nk 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.