Conversion Methods, Block Triangularization, and Structural Analysis of
Differential-Algebraic Equation Systems
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abstract
In a previous article, the authors developed two conversion methods to
improve the $\Sigma$-method for structural analysis (SA) of
differential-algebraic equations (DAEs). These methods reformulate a DAE on
which the $\Sigma$-method fails into an equivalent problem on which this SA is
more likely to succeed with a generically nonsingular Jacobian. The basic
version of these methods processes the DAE as a whole. This article presents
the block version that exploits block triangularization of a DAE. Using a block
triangular form of a Jacobian sparsity pattern, we identify which diagonal
blocks of the Jacobian are identically singular and then perform a conversion
on each such block. This approach improves the efficiency of finding a suitable
conversion for fixing SA's failures. All of our conversion methods can be
implemented in a computer algebra system so that every conversion can be
automated.