A Hierarchical Representation of the Inverse for Sparse Matrices

We present a representation of the inverse of a matrix for solving large sparse systems of linear algebraic equations $Ax = b$, that arises from the bordering method. We analyse the properties of the method for sparse matrices permuted to a suitable form. The relevant feature of the method is that it creates nonzero elements only in spike columns above the main diagonal. For an $n \times n$ matrix A with $\tau _0 $ nonzero elements, the number of nonzero elements in the representation of the inverse satisfies the inequality $\tau < \tau _0 + h \cdot n$, where h is a certain constant. It is proved that, under suitable assumptions, $h\leqq \log _2 s + 1$, where s is the number of spikes. The computation of the hierarchical form of the inverse requires at most $h \cdot \tau $ additions and $h \cdot \tau $ multiplications. For the known representation of the inverse, the solution of the system $Ax = b$ requires no more than $\tau - n$ additions and $\tau $ multiplications.