Symbolic-numeric methods for improving structural analysis of
differential-algebraic equation systems
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Systems of differential-algebraic equations (DAEs) are generated routinely by
simulation and modeling environments such as Modelica and MapleSim. Before a
simulation starts and a numerical solution method is applied, some kind of
structural analysis is performed to determine the structure and the index of a
DAE. Structural analysis methods serve as a necessary preprocessing stage, and
among them, Pantelides's algorithm is widely used.
Recently Pryce's $\Sigma$-method is becoming increasingly popular, owing to
its straightforward approach and capability of analyzing high-order systems.
Both methods are equivalent in the sense that when one succeeds, producing a
nonsingular system Jacobian, the other also succeeds, and the two give the same
Although provably successful on fairly many problems of interest, the
structural analysis methods can fail on some simple, solvable DAEs and give
incorrect structural information including the index. In this report, we focus
on the $\Sigma$-method. We investigate its failures, and develop two
symbolic-numeric conversion methods for converting a DAE, on which the
$\Sigma$-method fails, to an equivalent problem on which this method succeeds.
Aimed at making structural analysis methods more reliable, our conversion
methods exploit structural information of a DAE, and require a symbolic tool
for their implementation.