Matrix Padd Fractions

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 algor!thm for constructing these sequences is presented. It is shown that the cofactor sequence yields directly a sequence of Padd fractions for a matrix power series represented as a quotient B(z)-lA(z). When B(z)-lA(z) is normal, the complexity of the algorithm for computing a Padd fraction of type (re,n) is O(pa(m+n)z), where p is the order of the matrices A(z) and B(z). For power series which are abnormal, for a given (re,n), Padd fractions may not exist. However, it is shown that a generalized notion of Padd fraction, the Padd form, 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(pa(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. 1. I n t r o d u c t i o n . Let = a , z ' , (:.1) i--@ where a~, i ----0, ... , is a p x p matrix with coefficients from a field K, be a formal power series. Loosely speaking, a matrix Padd approximant of A(z) is an expression of the form U(z).V(z) -1, or Y~z)-l.U(z), where U(z) and V(z) are matrix polynomials of degree at most m and n, respectively, whose expansion agrees with A(z) up to and including the term am+". The definition of a Padd approximant can be made more formal in a variety of ways. For example, Rissanen [9] restricts V(z) to be a scalar polynomial and allows U(z) to be a p x q matrix. Typically, however, U(z) and V(z) are p x p polynomial matrices, and V(z) is further restricted by the condition that the constant term, V(0), is invertible (c.f., Bose and Basu[1], Bultheel[3], and Starkand[12]). In this paper, we cMl such approximants matrix Padd fractions, which is consistent with the scalar (p----l) case (c.f., Gragg[7]).