Index Reduction for Differential-algebraic Equations with Mixed Matrices

Differential-algebraic equations (DAEs) are widely used for the modeling of dynamical systems. The difficulty in numerically solving a DAE is measured by its differentiation index. For highly accurate simulation of dynamical systems, it is important to convert high-index DAEs into low-index DAEs. Most of the existing simulation software packages for dynamical systems are equipped with an index-reduction algorithm given by Mattsson and Söderlind. Unfortunately, this algorithm fails if there are numerical cancellations. These numerical cancellations are often caused by accurate constants in structural equations. Distinguishing those accurate constants from generic parameters that represent physical quantities, Murota and Iri introduced the notion of a mixed matrix as a mathematical tool for faithful model description in a structural approach to systems analysis. For DAEs described with the use of mixed matrices, efficient algorithms to compute the index have been developed by exploiting matroid theory. This article presents an index-reduction algorithm for linear DAEs whose coefficient matrices are mixed matrices, i.e., linear DAEs containing physical quantities as parameters. Our algorithm detects numerical cancellations between accurate constants and transforms a DAE into an equivalent DAE to which Mattsson–Söderlind’s index-reduction algorithm is applicable. Our algorithm is based on the combinatorial relaxation approach, which is a framework to solve a linear algebraic problem by iteratively relaxing it into an efficiently solvable combinatorial optimization problem. The algorithm does not rely on symbolic manipulations but on fast combinatorial algorithms on graphs and matroids. Our algorithm is proved to work for any linear DAEs whose coefficient matrices are mixed matrices. Furthermore, we provide an improved algorithm under an assumption based on dimensional analysis of dynamical systems. Through numerical experiments, it is confirmed that our algorithms run sufficiently fast for large-scale DAEs and output DAEs such that physical meanings of coefficients are easy to interpret. Our algorithms can also be applied to nonlinear DAEs by regarding nonlinear terms as parameters.