We analyze the Matrix Berlekamp/Massey algorithm, which generalizes the Berlekamp/Massey algorithm [Massey 1969] for computing linear generators of scalar sequences. The Matrix Berlekamp/Massey algorithm computes a minimal matrix generator of a linearly generated matrix sequence and has been first introduced by Rissanen [1972a], Dickinson et al. [1974], and Coppersmith [1994]. Our version of the algorithm makes no restrictions on the rank and dimensions of the matrix sequence. We also give new proofs of correctness and complexity for the algorithm, which is based on self-contained loop invariants and includes an explicit termination criterion for a given determinantal degree bound of the minimal matrix generator.

We describe a fraction free version of the Matrix Berlekamp/Massey algorithm. The algorithm computes a minimal matrix generator of linearly generated square matrix sequences over an integral domain. The algorithm performs all operations in the integral domain, so all divisions performed are exact. For scalar sequences, the matrix algorithm specializes to a more efficient algorithm than the algorithm currently in the literature. The proof of integrality of the matrix algorithm gives a new proof of integrality for the scalar specialization.

We describe the design and implementation of two components in the LinBox library. The first is an implementation of black box matrix multiplication as a lazy matrixtimes-matrix product. The implementation uses template meta-programming to set the intermediate vector type used during application of the matrix product. We also describe an interface mechanism that allows incorporation of external components with native memory management such as garbage collection into LinBox. An implementation of the interface based on SACLIB's field arithmetic procedures is presented.

We describe a fraction free matrix Berlekamp/Massey algorithm. The algorithm behaves like a scalar algorithm by performing block eliminations through multiplication by adjoint matrices. The adjoints are computed using fraction free diagonalization. We also describe an interesting classification of the unique minimal generators of a linearly generated scalar integer sequence. A counter example to the scalar theorem is given for linearly generated integer matrix sequences.