Computation of Numerical Padé-Hermite and Simultaneous Padé Systems I: Near Inversion of Generalized Sylvester Matrices

Abstract. We present new formulae for the “near” inverses of striped Sylvester and mosaic Sylvester matrices. The formulae assume computation over floating-point rather than exact arithmetic domains. The near inverses are expressed in terms of numerical Padé-Hermite systems and simultaneous Padé systems. These systems are approximants for the power series determined from the coefficients of the Sylvester matrices. The inverse formulae provide good estimates for the condition numbers of these matrices, and serve as primary tools in a companion paper for the development of a fast, weakly stable algorithm for the computation of Padé-Hermite and simultaneous Padé systems and, thereby, also for the numerical inversion of striped and mosaic Sylvester matrices.