Extensions of Faddeev's algorithms to polynomial matrices

Starting from algorithms introduced in [Ky M. Vu, An extension of the Faddeev's algorithms, in: Proceedings of the IEEE Multi-conference on Systems and Control on September 3-5th, 2008, San Antonio, TX] which are applicable to one-variable regular polynomial matrices, we introduce two dual extensions of the Faddeev's algorithm to one-variable rectangular or singular matrices. Corresponding algorithms for symbolic computing the Drazin and the Moore-Penrose inverse are introduced. These algorithms are alternative with respect to previous representations of the Moore-Penrose and the Drazin inverse of one-variable polynomial matrices based on the Leverrier-Faddeev's algorithm. Complexity analysis is performed. Algorithms are implemented in the symbolic computational package MATHEMATICA and illustrative test examples are presented.