Unified Divide-and-Conquer Algorithm

Abstract

In this chapter, we describe a superfast divide-and-conquer algorithm for recursive triangular factorization of structured matrices. The algorithm applies over any field of constants. As a by-product, we obtain the rank of an input matrix and a basis for its null space. For a non-singular matrix, we also compute its inverse and determinant. The null space basis and the inverse are represented in compressed form, with their short displacement generators. The presentation is unified over various classes of structured matrices. Treatment of singularities is specified separately for numerical and symbolic implementations of the algorithm. The algorithm enables computations for several fundamental and celebrated problems of computer algebra and numerical rational computations.