Matrix Padé Fractions and Their Computation

Abstract

For matrix power series with coefficients over a field, the notion of a matrix power series remainder sequence and its corresponding cofactor sequence are introduced and developed. An algorithm for constructing these sequences is presented. It is shown that the cofactor sequence yields directly a sequence of Pade fractions for a matrix power series represented as a quotient $B(z)^{ - 1} A(z)$. When $B(z)^{ - 1} A(z)$ is normal, the complexity of the algorithm for computing a Pade fraction of type $(m,n)$ is $O(p^3 (m + n)^2 )$, where p is the order of the matrices $A(z)$ and $B(z)$. For a power series that are abnormal for a given $(m,n)$, Padé fractions may not exist. However, it is shown that a generalized notion of Padé fraction, the Padé form, which is introduced in this paper, does always exist and can be computed by the algorithm. In the abnormal case, the algorithm can reach a complexity of $O(p^3 (m + n)^3 )$, depending on the nature of the abnormalities. In the special case of a scalar power series, however, the algorithm complexity is $O(m + n)^2 $, even in the abnormal case.