Conversion Methods, Block Triangularization, and Structural Analysis of Differential-Algebraic Equation Systems
For researchers working on numerical solution of DAEs, this provides a more efficient automated method to fix structural analysis failures.
The authors extend their conversion methods for structural analysis of DAEs by incorporating block triangularization, enabling targeted conversion of singular diagonal blocks and improving efficiency. The approach automates the process in a computer algebra system.
In a previous article, the authors developed two conversion methods to improve the $Σ$-method for structural analysis (SA) of differential-algebraic equations (DAEs). These methods reformulate a DAE on which the $Σ$-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.