Block computation and representation of a sparse nullspace basis of a rectangular matrix

In this paper, we propose a new method to efficiently compute a representation of an orthogonal basis of the nullspace of a sparse matrix operator BT with B∈Rn×m, n>m. We assume that B has full rank, i.e., rank(B)=m. It is well-known that the last n-m columns of the orthogonal matrix Q in a QR factorization B=QR form such a desired null basis. The orthogonal matrix Q can be represented either explicitly as a matrix, or implicitly as a matrix H of Householder vectors. Typically, the matrix H represents the orthogonal factor much more compactly than Q. We will employ this observation to design an efficient block algorithm that computes a sparse representation of the nullspace basis in almost optimal complexity. This new algorithm may, e.g., be used to construct a null space basis of the discrete divergence operator in the finite element context, and we will provide numerical results for this particular application.