Many problems from the real world and from Scientific Computing are modeled by matrix equations or by matrix functions. For instance, the algebraic Riccati equation is related with the analysis of stability of dynamical systems. Quadratic equations like $AX^2+BX+C=0$ model damped vibration problems as well as stochastic models encountered in queuing theory.
The goal of the research in this area is to develop tools for designing fast and effective algorithms to solve this kind of
equations. These equations have a rich linear algebra structure can often be recast as generalized eigenvalue problems. The techniques needed combine the ones used for general nonlinear equations (multivariate Newton methods, fixed-point iterations) and eigenvalue problems (Schur decompositions, orthogonal reductions, rational approximations).
Matrix structures, such as entrywise nonnegativity, symmetry and symplecticity, play a crucial role: they are needed for defining the solutions of these equations in the first place, and to ensure feasibility, accuracy and computational efficiency of the numerical algorithms.
Applications include control and system theory, queuing theory and structured Markov chain modelling in applied probability, and time series estimation in statistics. It is useful to interface directly with researchers working in these application fields: looking at the problems from different points of view gives useful insight, and the algorithms can be better tailored to the needs of the practitioners.