The Kronecker canonical form of matrix pencils plays an important role in various fields such as systems control and differential-algebraic equations. The Kronecker canonical form admits two characterizations: the highest degree of subdeterminants, which can be computed by combinatorial relaxation algorithms, and the ranks of larger constant matrices called expanded matrices. This article surveys the combinatorial approach to the analysis of the Kronecker canonical form.

Continuous stage Runge-Kutta (CSRK) methods, which were introduced around 2010, are a framework of iterative numerical methods for solving ordinary differential equations. It turned out that some CSRK methods preserve some underlying geometric structures of differential equations, such as symplecticity or energy-preservation of Hamiltonian systems. This paper reviews CSRK methods and their recent developments with emphasis on their structure-preservation properties.