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

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 Pad\'{e}--Hermite systems and simultaneous Pad\'e 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 Pad\'e--Hermite and simultaneous Pad\'e systems and, thereby, also for the numerical inversion of striped and mosaic Sylvester matrices.

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