Sequentialization of a graph as a sparse-matrix technique for predominantly tridiagonal matrices

Abstract

The concept of the sequentialization of a graph is proposed. It corresponds to the Thomas algorithm in the simplest case for which the ordinary method of the reduction of a graph by successive elimination of nodes corresponds to Gaussian elimination. The method can effectively be applied to an interconnected chain graph which corresponds to the tridiagonal matrix with off-diagonal elements. We derive three fundamental rules for the sequentialization of chain graphs. By examples we show that the combined application of these rules can sequentialize an important class of interconnected chain graphs. It is an extension of the Thomas algorithm to a class of sparse matrices which is mainly tridiagonal but has some off-diagonal elements. The amount of calculation in the proposed method is proportional to the number of nodes. The method is useful not only for numerical but also for analytical calculation. The method finds a useful application in calculating the material balance of the interconnected cascade system with reflux and backmixing.