An Implementation of Tarjan's Algorithm for the Block Triangularization of a Matrix

Abstract

An implementation of Tarj an's algorithm for symmetrically permuting a given matrix to block tmangular form is described. The discussion includes a flowchart of the algorithm, a complexity analysis, and a comparison with the earlier widely used algorithm of Sargent and Westerberg. T~ming results are presented from several experiments using the code developed by the authors.

Survey of appropriate matching algorithms for large scale systems of differential algebraic equations

“…For further information, the reader is referred to [5], [6], [8], [9], [2], [10]. As mentioned above and shown in detail by Elmqvist [10] matching algorithms are provide the information how a system of equations can be transformed symbolically into a system of assignments.…”

“…• Find a traversal of the directed graph which means to sort the equations and identify algebraic loops For the second step Tarjan's Algorithm [26] is very efficient and offers time linear complexity with respect to the number of equations [6]. To understand the first step one has to look at the Adjacency Matrix of a system of equations.…”

An XML Representation of DAE Systems Obtained from Modelica Models

“…The goal is now to compute the state derivativesẋ and the algebraic variables v. To this end, the equations and the variables of the problem can be re-ordered so that the incidence matrix (equations on the rows, unknowns on the columns) is brought in Block-Lower-Triangular (BLT) form. This task is accomplished by using the well-known Tarjan algorithm [7], applied to the equations-variables bipartite graph, which is equivalent to the incidence matrix of the system. The strongly connected components of the graph correspond to the blocks on the diagonal, and a partial ordering among equations can be deduced from the graph after the algorithm has terminated.…”

Modular representation of complex hybrid systems: application to the simulation of batch processes

“…It lets you identify the causal graph of the system and lets you order the system into a sequence of sub-problems [10]. This is achieved by symmetric row and column permutations so that the incidence matrix becomes block triangular, with minimal blocks [6] [17].…”