Conversion Methods, Block Triangularization, and Structural Analysis of Differential-Algebraic Equation Systems

TL;DR

This paper introduces a block triangularization approach to enhance the structural analysis of differential-algebraic equations by efficiently identifying and converting problematic blocks to improve solvability.

Contribution

It presents a novel block version of conversion methods that exploits block triangularization to improve the efficiency of fixing structural analysis failures in DAEs.

Findings

The block triangularization method effectively identifies singular Jacobian blocks.

Conversion methods can be automated within computer algebra systems.

The approach improves the success rate of structural analysis for complex DAEs.

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.