Graph theory, irreducibility, and structural analysis of
differential-algebraic equation systems
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abstract

The $\Sigma$-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.