The determination of sparse eigensystems and parallel linear system solvers

The algebraic eigenvalue problem occurring in a variety of problems in the Natural, Engineering and Social Sciences areas can with some advantage be solved by matrix methods. However, these problems become more difficult to handle when the matrices involved are large and sparse because the storage and manipulations of these types of matrices have to be achieved in such ways that firstly, no storage is wasted by retaining the zero elements of the matrix and secondly, saving valuable computer time by not operating on the zero elements when unnecessary. For this purpose, we have previously developed a software package on the storage and manipulation of sparse matrices, which consists of basic matrix operations (i.e. addition, multiplication, etc.) and the solution of linear systems by iterative methods. However, in that work we encountered a great deal of difficulty in handling the operations which generate non-zero elements during processes such as the Gaussian elimination process. [Continues.]