Partitioning and Tearing Systems of Equations

Abstract

Introduction. Kron [1] has developed a technique for tearing large, sparse linear systems of algebraic equations into smaller systems, then putting the solutions of these smaller systems together to form the solution of the original problem. This can result in a great savings over solving the whole system in one piece. The trick is to find where to tear so that on the one hand the remaining pieces are small and easily solvable, but on the other hand not so much violence is done to the equations so that the task of putting their solutions back together becomes too difncult. Kron's pro? cedure for solving the pieces and putting their solutions together is based upon representing systems of equations as electric networks. He makes his choice of where to tear from his insight into these networks and the physical processes the equations represent. In the hands of someone such as Kron who has an understanding of the methods and good insight into the prob? lem, this technique has had outstanding success. The widespread use of Kron's technique has perhaps been limited severely for lack of practitioners skilled in this art of determining effective ways of tearing specific problems. Many problems are too large to submit to such insight. It is the purpose of this paper to contribute to the development of procedures which can be performed on a computer for analyzing the structures of the systems of equations themselves as an aid in choosing how to break them up for easier solution. The methods developed here require only that we consider what variables occur in each equation, but not how they appear. Thus, the tech? niques are not limited to linear systems.