Solving sparse linear systems of equations using the modified digraph approach

Abstract

We propose two different algorithms which depend on the modified digraph approach for solving a sparse system of linear equations. The main feature of the algorithms is that the solution of a sparse system of linear equations can be expressed exactly if all the non-zero entries, including the right-hand side, are integers and if none of the products exceeds the size of the largest integer that can be represented in the arithmetic of the computer used. The implementation of the algorithms is tested on five problems. The results are compared with those obtained using an algorithm proposed earlier. It is shown that the efficiency with which a sparse system of linear equations can be analysed by a digital computer using the proposed modified digraph approach as a tool depends mainly on the efficiency with which semifactors and k-semifactors are generated. Finally, in our implementation of the proposed algorithms, the input sparse matrix is stored using a row-ordered list of a modified uncompressed storage scheme.

Keywords:

• Modified digraph
• Sparse linear systems
• Semifactor
• k-Semifactor

Acknowledgements

The author would like to thank the referees for their valuable and constructive comments which led to an improvement in the presentation of the paper.