Graph theory, irreducibility, and structural analysis of differential-algebraic equation systems
Provides structural insights for DAE system analysis, but is incremental for specialists.
The paper compares fine and coarse block-triangular forms derived from DAE systems, defining a Fine-Block Graph to analyze block ordering and structure, and shows that the set of normalized offset vectors is finite iff there is one coarse block.
The $Σ$-method for structural analysis of a differential-algebraic equation (DAE) system produces offset vectors from which the sparsity pattern of a system Jacobian is derived. This pattern implies a block-triangular form (BTF) of the DAE that can be exploited to speed up numerical solution. The paper compares this fine BTF with the usually coarser BTF derived from the sparsity pattern of the \sigmx. It defines a Fine-Block Graph with weighted edges, which gives insight into the relation between coarse and fine blocks, and the permitted ordering of blocks to achieve BTF. It also illuminates the structure of the set of normalised offset vectors of the DAE, e.g.\ this set is finite if and only if there is just one coarse block.