The focus of this paper is on Birkhoff’s matrix polynomial interpolation. Given a list \(Z_n=[z_{0},...,z_{n}]\) with \((n+1)\) distinct nodes, of \(\mathbb {K}=\mathbb {R}\) or \(\mathbb {C}\) , we will study the existence and uniqueness of a \(s \times m\) matrix polynomial P such that P and a number of its derivatives take, in these nodes, given values. Recently, Messaoudi and Sadok [10] presented a new algorithm for computing the Hermite matrix interpolation polynomial called the Generalized Recursive Matrix Polynomial Interpolation Algorithm (GRMPIA). In this paper, we present a new formulation of the Birkhoff matrix polynomial interpolation problem and derive a new algorithm called the Recursive Hermite-Birkhoff Matrix Polynomial Interpolation Algorithm (RHBMPIA) to solve the Birkhoff interpolation problem and extend the GRMPIA. A new existing result will be established. The numerical stability of this algorithm will also be studied and some examples will be given.
