  Differential-algebraic equations (DAEs) have been used in modeling various
dynamical systems in science and engineering. Several preprocessing methods for
DAEs, such as consistent initialization and index reduction, use structural
information on DAEs. Unfortunately, these methods may fail when the system
Jacobian, which is a functional matrix, derived from the DAE is singular.
  To transform a DAE with a singular system Jacobian into a nonsingular system,
several regularization methods have been proposed. Most of all existing
regularization methods rely on symbolic computation to eliminate the system
Jacobian for finding a certificate of singularity, resulting in much
computational time. Iwata--Oki--Takamatsu (2019) proposed a method (IOT-method)
to find a certificate without symbolic computations. The IOT method
approximates the system Jacobian by a simpler symbolic matrix, called a layered
mixed matrix, which admits a fast combinatorial algorithm for singularity
testing. However, it often overlooks the singularity of the system Jacobian
since the approximation largely discards algebraic relationships among entries
in the original system Jacobian.
  In this study, we propose a new regularization method extending the idea of
the IOT method. Instead of layered mixed matrices, our method approximates the
system Jacobian by more expressive symbolic matrices, called rank-1 coefficient
mixed (1CM) matrices. This makes our method more widely applicable. We give a
fast combinatorial algorithm for finding a singularity certificate of
1CM-matrices, which is free from symbolic elimination. Our method is also
advantageous in that it globally preserves the solution set to the DAE. Through
numerical experiments, we confirmed that our method runs fast for large-scale
DAEs from real instances.
