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.

翻译：微分代数方程（DAEs）已广泛应用于科学和工程中各类动力学系统的建模。DAEs的若干预处理方法（如一致初始化与指标约简）依赖于其结构信息。然而，当从DAEs导出的系统雅可比矩阵（一种函数矩阵）奇异时，这些方法可能失效。为将奇异系统雅可比矩阵的DAE转化为非奇异系统，已提出多种正则化方法。现有正则化方法大多依赖符号计算消除系统雅可比矩阵以寻找奇异性凭证，导致计算时间较长。Iwata-Oki-Takamatsu (2019) 提出一种无需符号计算的IOT方法（IOT-method），该方法将系统雅可比矩阵近似为更简单的符号矩阵（称为分层混合矩阵），并采用快速组合算法进行奇异性检验。然而，由于这种近似大量舍弃了原始系统雅可比矩阵中元素间的代数关系，常导致无法识别其奇异性。本研究提出一种扩展IOT方法思想的新正则化方法。我们采用表达能力更强的符号矩阵——秩1系数混合（1CM）矩阵替代分层混合矩阵进行近似，使方法具有更广泛的适用性。我们提出了一种无符号消除的1CM矩阵奇异性凭证快速组合算法。该方法同时具备全局保持DAE解集的优势。通过数值实验，验证了该方法对实际大规模DAEs的计算效率。