Fast and Efficient Parallel Solution of Sparse Linear Systems

Abstract

This paper presents a parallel algorithm for the solution of a linear system $A{\bf x} = {\bf b}$ with a sparse $n \times n$ symmetric positive definite matrix A, associated with the graph $G(A)$ that has n vertices and has an edge for each nonzero entry of A. If $G(A)$ has an $s(n)$-separator family and a known $s(n)$-separator tree, then the algorithm requires only $O(\log ^3 n)$ time and $(|E| + {{M(s(n)))} / {\log n}}$ processors for the evaluation of the solution vector ${\bf x} = A^{ - 1} {\bf b}$, where $|E|$ is the number of edges in $G(A)$ and $M(n)$ is the number of processors sufficient for multiplying two $n \times n$ rational matrices in time $O(\log n)$. Furthermore, for this computational cost the algorithm computes a recursive factorization of A such that the solution of any other linear system $A{\bf x} = {\bf b}'$ with the same matrix A requires only $O(\log ^2 n)$ time and $({{|E|} / {\log n}}) + s(n)^2 $ processors.